Realism VS Accuracy: Event Sequence Forecasting from a Generative Modeling Perspective

# Event Sequence Forecasting from a Generative Modeling Perspective
Source: [https://arxiv.org/html/2610.12586](https://arxiv.org/html/2610.12586)
Egor SurkovPetr MokrovIgor UdovichenkoAffiliation:Vega Institute FoundationIvan KarpukhinAffiliation:SB AI LabEvgeny BurnaevAffiliation:Applied AI InstituteAffiliation:AXXXEgor ShvetsovAffiliation:Applied AI InstituteAlexander KorotinAffiliation:Applied AI InstituteAffiliation:AXXX

###### Abstract

Event sequences are chains of random events, each described by heterogeneous \(numerical and categorical\) features\. The problem of event sequence forecasting occurs in finance, healthcare, business process intelligence, and other fields\. In this work, we study multi\-horizon forecasting of event sequences as a conditional generative modeling problem\. We evaluate forecasts not only for prediction accuracy, but also fordistributional realism\. To this end, we present abenchmarkcomprising metrics for both quality aspects, strong statistical baselines, recent models adapted from adjacent fields, and curated real\-world datasets\. Careful evaluation reveals that no method excels at both accuracy and realism, highlighting a fundamental trade\-off between the two\. Finally, we introduceWasserstein BaryBooster, an algorithm that aggregates samples from a realism\-strong diffusion model into a state\-of\-the\-art predictor, demonstrating that distributional realism can be converted into prediction accuracy\. Code is available at[https://github\.com/xaosina/event\-sequence\-forecasting](https://github.com/xaosina/event-sequence-forecasting)\.

## 1Introduction

Many high\-stakes decisions rest on modeling chains of future events: the next purchases on a customer’s card\([Bazarova et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib28)\), the upcoming entries in a patient’s clinical record\([Johnson et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib26)\), the next actions of a player in an online game\([Zhao et al\., 2023](https://arxiv.org/html/2610.12586#bib.bib27)\), and similar processes across other domains\([Lopes and Ferreira, 2019](https://arxiv.org/html/2610.12586#bib.bib29);[Liguori et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib30)\)\. Each event is timestamped and described by a heterogeneous set of features — typically a mix of numerical attributes \(e\.g\., transaction amount\) and categorical attributes \(e\.g\., merchant category code\)\. We refer to such data as*event sequences*\(Figure[1](https://arxiv.org/html/2610.12586#S1.F1)\) and study the problem of forecasting them over long horizons given a recorded history\.

The closest body of prior work is*temporal point processes*\(TPP\), which model event timing\([Lüdke et al\., 2023](https://arxiv.org/html/2610.12586#bib.bib7);[Lüdke et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib9);[Lüdke et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib8)\)and, in marked variants, a single discrete event type\([Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19);[Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20)\)\. While effective in their own setting, TPP methods do not natively handle the heterogeneous numerical and categorical features that real event sequences carry \(see §[2\.1](https://arxiv.org/html/2610.12586#S2.SS1)\)\. To the best of our knowledge, prior work has not studied long\-horizon forecasting of general event sequences systematically\.

We frame event sequence forecasting as a*conditional generative modeling*problem: given a history, the model produces samples that should be plausible both relative to that history and to the broader data distribution\. This framing implies two distinct perspectives on forecast quality\.*Prediction accuracy*measures how close a forecast is to the actual continuation of the sequence\.*Distributional realism*measures how indistinguishable forecasts are from genuine future trajectories drawn from the same distribution\. Prior evaluations, in TPP and beyond, have focused almost exclusively on prediction accuracy, overlooking realism\.

Figure 1:A toy event sequence\. After a bet on black at roulette, the evening follows one of two realistic chains of transactions: the person either lost \(51%\) or won \(49%\)\. The prediction that minimizes MSE and maximizes per\-event accuracy takes the categories of the more likely chain and averages the amounts, pairing cheap categories with large sums \(e\.g\., an $18 bus ride\)\. It scores best on predictive metrics, yet matches neither chain\. On the other hand, each possible outcome is realistic, but its expected MSE is about twice as large\.To the best of our knowledge, no prior benchmark treats both dimensions as first\-class citizens\. We make the followingcontributions:

1. 1\.We implement a suite of statistical baselines \(§[3\.1](https://arxiv.org/html/2610.12586#S3.SS1)\) and adapt cutting\-edge generative diffusion models and SoTA predictive models \(§[3\.3](https://arxiv.org/html/2610.12586#S3.SS3)–§[3\.5](https://arxiv.org/html/2610.12586#S3.SS5)\) to the task of forecasting event sequences with heterogeneous features\.
2. 2\.In §[4](https://arxiv.org/html/2610.12586#S4)we present abenchmarkcomprising several large\-scale event sequence datasets and a set of metrics that capture both perspectives on forecast quality, including but not limited tocDFS\(Conditional Detection Fooling Score\), an adaptation of the classifier\-two\-sample test\([Lopez\-Paz and Oquab, 2017](https://arxiv.org/html/2610.12586#bib.bib2)\), andTMS\(Temporal Matching Score\), our novel matching\-based metric inspired by the optimal transport distance used extensively in TPP research\([Mei et al\., 2019](https://arxiv.org/html/2610.12586#bib.bib31);[Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19);[Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20)\)\.
3. 3\.We find \(§[6](https://arxiv.org/html/2610.12586#S6)\) that no method excels at both TMS and cDFS, exposing afundamental trade\-off between prediction accuracy and distributional realism\.
4. 4\.In §[5](https://arxiv.org/html/2610.12586#S5)we introduceWasserstein BaryBooster, an aggregation procedure that, when applied on top of samples from a diffusion model, produces SoTA prediction accuracy\.

## 2Problem Statement and Related Work

We formalize the problem of event sequence forecasting as follows\. Letttbe the timestamp of an event, and let𝐱\\mathbf\{x\}and𝐤\\mathbf\{k\}denote its numerical and categorical feature vectors, respectively\. An event sequence is a sequence of events:S=\(E1,…,En\)S=\\left\(E\_\{1\},\\dots,E\_\{n\}\\right\), where each event is a 3\-tupleEi=\(ti,𝐱i,𝐤i\)E\_\{i\}=\(t\_\{i\},\\mathbf\{x\}\_\{i\},\\mathbf\{k\}\_\{i\}\)\. We denote bySm:l=\(Em,…,El\)S\_\{m:l\}=\\left\(E\_\{m\},\\dots,E\_\{l\}\\right\)a slice of the event sequence\. Given a historyS1:HS\_\{1:H\}, the goal is to forecast the*sequence*of future eventsSH\+1:H\+LS\_\{H\+1:H\+L\}\.

We position our paper among three relevant bodies of work: generative models applicable to event\-sequence data \(§[2\.1](https://arxiv.org/html/2610.12586#S2.SS1)\), existing benchmarks in the space \(§[2\.2](https://arxiv.org/html/2610.12586#S2.SS2)\), and metrics previously used to evaluate such models \(§[2\.3](https://arxiv.org/html/2610.12586#S2.SS3)\)\.

### 2\.1Generative Models for Event Data

Table 1:Comparison of related generative models\. “Features” indicates the supported feature type: “none” \(time only\), “cat” \(a single event type, as in marked TPP, MTPP\), “num” \(numerical features only, as in spatio\-temporal TPP, STPP\), or “mixed” \(both numerical and categorical\)\. “Seq\.” indicates whether the method natively handles sequential \(as opposed to single\-row tabular\) data\. “Forecasting” indicates whether the method allows history\-conditional generation\. “Diffusion” indicates whether the method is a latent diffusion model, a data\-space diffusion model, or non\-diffusion \(ND\)\. Methods applicable to event sequence forecasting are highlighted\.MethodFeaturesSeq\.ForecastingDiffusionCDiff\([Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1)\)mixed✓✓✓dataTabSyn\([Zhang et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib4)\)mixed✓✘✘latentTabDiff\([Shi et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib6)\)mixed✓✘✘dataAdd and Thin\([Lüdke et al\., 2023](https://arxiv.org/html/2610.12586#bib.bib7)\)none✘✓✓dataPSDiff\([Lüdke et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib9)\)num✘✓✓dataEdiTPP\([Lüdke et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib8)\)none✘✓✓dataDEF\([Karpukhin and Savchenko, 2026](https://arxiv.org/html/2610.12586#bib.bib3)\)mixed✓✓✓NDUniTTab\([Luetto et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib10)\)mixed✓✓✘NDTabDiT\([Garuti et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib11)\)mixed✓✓✘latentADiff4TPP\([Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\)mixed✓✓✓latentEvent sequences consist of events described by both numerical and categorical features, along with their timestamps\. This structure places them at the intersection of three lines of generative\-modeling work, which we summarize in Table[1](https://arxiv.org/html/2610.12586#S2.T1)\. We characterize methods by \(i\) their model type —*latent*diffusion trained in a learned representation space,*data*diffusion designed directly in feature space, or non\-diffusion \(ND\); \(ii\) whether they natively handle sequential data; and \(iii\) the feature types they can generate\. Appendix[A](https://arxiv.org/html/2610.12586#A1)discusseseach of these three classes in detail— tabular\-data generative models, temporal point processes, and models for general heterogeneous event sequences — together with the specific methods that we adapt as baselines and evaluate throughout this benchmark\.

Although many of these methods*can be applied*to forecasting of event sequences with heterogeneous features, to the best of our knowledge,*no prior work has done so*\. We fill this gap by adapting representatives of all three classes to the general event\-sequence forecasting setting: a data\-space diffusion model, a latent diffusion model, and a non\-diffusion method\.

### 2\.2Benchmarks for Event Sequences

Published benchmarks are scarce\. The three closest to our work are EasyTPP\([Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19)\), HoTPP\([Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20)\), and EBES\([Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22)\)\. Wereview each of them in detailin Appendix[B](https://arxiv.org/html/2610.12586#A2); here we summarize how our benchmark relates to all three\.

Our benchmark occupies a distinct region of the space delineated by these three efforts\. We adopt the heterogeneous\-event data setting of EBES but target aforecastingtask rather than sequence\-level classification\. Within the forecasting setting, we complement EasyTPP and HoTPP in two ways\. First, we lift the MTPP restriction, so events may carry heterogeneous features\. Second, we evaluate both*prediction accuracy*and*distributional realism*and use their combination to probe the trade\-off between the two\.

### 2\.3Metrics for Generative Sequence Models

We divide metrics for generative event\-sequence models into two groups\.

#### Prediction accuracy metrics\.

They measure how close, on average, the forecastS^H\+1:H\+L\\hat\{S\}\_\{H\+1:H\+L\}is to the ground\-truth continuationSH\+1:H\+LS\_\{H\+1:H\+L\}\. The traditional TPP metrics measure how well the model predicts a single step ahead\. The classic examples are time MAE or RMSE, mark accuracy, and next\-event negative log\-likelihood\([Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19)\)\. They are straightforward to compute but are known to correlate only weakly with long\-horizon forecasting quality\([Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20)\)\.

Optimal Transport Distance \(OTD\) for event sequences aligns predicted and ground\-truth events via a minimum\-cost edit over insertions, deletions, and event moves, with per\-operation costs chosen by hand\. It is widely used in TPP and MTPP evaluation\([Karpukhin and Savchenko, 2026](https://arxiv.org/html/2610.12586#bib.bib3);[Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1)\)and handles length mismatches, but its method ranking is sensitive to the cost hyperparameters\([Yang et al\., 2022](https://arxiv.org/html/2610.12586#bib.bib16)\)\. Temporal mean Average Precision \(T\-mAP\) instead aligns events by a one\-to\-one matching over a time tolerance and averages the resulting per\-label average precision, replacing the per\-operation costs with a single tolerance hyperparameter\([Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20)\)\. Neither extends to events with numerical side features\. Our TMS \(§[4\.2](https://arxiv.org/html/2610.12586#S4.SS2)\) is motivated by the same alignment intuition and extends it: it replaces the hand\-chosen costs with data\-driven normalization and scores an arbitrary number of numerical and categorical features on a common scale\.

#### Distributional realism metrics\.

They measure how closely the distribution of generated continuations matches the distribution of true ones\. The classifier two\-sample test \(C2ST\)\([Lopez\-Paz and Oquab, 2017](https://arxiv.org/html/2610.12586#bib.bib2)\)measures distributional similarity of generated samples by training a discriminator to separate real from generated samples; the resulting accuracy serves as a proxy for distributional distance\. The idea applies to any data types and underlies our cDFS metric \(§[4\.2](https://arxiv.org/html/2610.12586#S4.SS2)\) with necessary modifications to support heterogeneous types and conditioning on history\.

TabSyn\([Zhang et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib4)\)popularized*Shape*\(Kolmogorov–Smirnov distance per feature\) and*Trend*\(closeness of pairwise feature dependencies\) scores, which have since been used by follow\-up tabular\-diffusion work\. These scores audit the marginal and low\-order joint structure of generated rows but ignore the temporal structure\. We include them asauxiliary distributional metrics for contextin Appendix[E\.1](https://arxiv.org/html/2610.12586#A5.SS1)\.

None of these metrics is fully suitable for assessing forecasting accuracy or distributional realism on heterogeneous event sequences\. We bridge this gap by introducing in §[4\.2](https://arxiv.org/html/2610.12586#S4.SS2)two novel metrics: TMS for prediction accuracy, and cDFS for distributional realism\.

## 3Models for Evaluation

We choose representative methods, one from each major class of generative approaches relevant to event sequences: a latent diffusion model \(our tailoredLDM, §[3\.3](https://arxiv.org/html/2610.12586#S3.SS3)\), a data diffusion model \(DDM, §[3\.4](https://arxiv.org/html/2610.12586#S3.SS4)\), and a SoTA non\-diffusion method \(DEF, §[3\.5](https://arxiv.org/html/2610.12586#S3.SS5)\)\. We adapted them for general event sequences \(seeimplementation detailsin Appendix[I](https://arxiv.org/html/2610.12586#A9)\)\. For completeness, we include GPT\-style methods \(§[3\.2](https://arxiv.org/html/2610.12586#S3.SS2)\) and statistical baselines \(§[3\.1](https://arxiv.org/html/2610.12586#S3.SS1)\) in the comparison\.

### 3\.1Baselines

We include three simple methods\.Repeatreturns the lastLLevents of the history with their inter\-arrival times\.HistSamplerdrawsLLevents from the history uniformly at random with replacement\. Baselines of this kind are rarely reported in the event\-sequence literature; HoTPP\([Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20)\)is a notable exception for classical MTPP, but we did not encounterRepeatin the literature, although it performs competitively well \(see §[6](https://arxiv.org/html/2610.12586#S6)\)\.Modesets each categorical dimension to its historical mode and each numerical dimension to the historical median; the time channel uses the median historical inter\-arrival time\.

### 3\.2GPT\-based

Because event sequences are fundamentally sequential, we borrow architectural ideas from large language model \(LLM\) practice rather than confining ourselves to classical TPP designs\. We instantiate a GPT\-2\-style decoder in two regimes\.ARis autoregressive: conditioned on the prefix history, it factorizes forecasting through successive next\-event predictions\.MTuses the same backbone but is*not*autoregressive: it maps the history toLLparallel predictions of future events in a single forward pass \(no causal chain over future steps\), trained with the sum of per\-event negative log\-likelihoods across thoseLLslots\.

### 3\.3Latent Diffusion \(LDM\)

Our latent diffusion method adapts TabSyn\([Zhang et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib4)\)to the event\-sequence setting\. We first train aβ\\beta\-VAE\([Higgins et al\., 2017](https://arxiv.org/html/2610.12586#bib.bib14)\)to encode and decode the features of a single event, treating the time stamp as an additional feature alongside the numerical and categorical ones\. We then train a Gaussian diffusion model\([Karras et al\., 2022](https://arxiv.org/html/2610.12586#bib.bib17)\)on sequences of latent event representations, treating a window ofLLevents as a single vector inℝL​d\\mathbb\{R\}^\{Ld\}, whereddis the per\-event latent dimensionality\. To sampleLLevents, we draw a noise vector inℝL​d\\mathbb\{R\}^\{Ld\}, run the reverse diffusion process, split the resulting vector intoLLchunks, and decode each with the VAE decoder\.

We implement forecasting through conditional generation\. A history embedding produced by an RNN over the past events is used as the primary conditioning signal for the diffusion model\. This embedding summarizes long\-range history but may lose information about the most recent events\. To preserve recency, we also feed the latent codes of the lastLLevents in the history as a “raw” context\. The overall scheme is shown in Figure[10\(a\)](https://arxiv.org/html/2610.12586#A9.F10.sf1)\.

Our LDM is structurally close to the ADiff4TPP method proposed in[Mukherjee et al\. \(2026\)](https://arxiv.org/html/2610.12586#bib.bib12)\. We compare themstructurally\(Appendix[I\.1](https://arxiv.org/html/2610.12586#A9.SS1)\) andempirically\(Appendix[E\.7](https://arxiv.org/html/2610.12586#A5.SS7)\)\.

### 3\.4Data\-space Diffusion Model \(DDM\)

We use CDiff\([Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1)\)as the basis for our representative data\-space diffusion method\. CDiff was originally proposed for MTPP forecasting and combines Gaussian diffusion over inter\-arrival times with uniform categorical diffusion over event marks\([Hoogeboom et al\., 2021](https://arxiv.org/html/2610.12586#bib.bib15)\), training the model to predict multiple future events simultaneously\. The method extends naturally to general event sequences with minimal modification: we retain Gaussian diffusion for inter\-arrival times and for numerical features, and use uniform categorical diffusion for each categorical feature\. The method is illustrated in Figure[10\(b\)](https://arxiv.org/html/2610.12586#A9.F10.sf2)\.

We apply classifier\-free guidance as in our LDM \(Appendix[I\.3](https://arxiv.org/html/2610.12586#A9.SS3)\); theremaining implementation choicesfollow Appendix[I](https://arxiv.org/html/2610.12586#A9)\.

### 3\.5DEF

DEF\([Karpukhin and Savchenko, 2026](https://arxiv.org/html/2610.12586#bib.bib3)\)is a state\-of\-the\-art method for long\-horizon forecasting of MTPP\. The method is built on a backbone model that generates event candidates\. The authors introduced a Horizon Matching Loss that finds the optimal matching between ground truth and event candidates\. The backbone is trained to generate more candidates than typically occur within the horizon, so some candidates remain unmatched\. Each candidate also receives a presence probability, trained jointly to predict whether it will be matched\. At inference time, these probabilities select the events from the candidate pool to form the prediction\. For the schematic illustration of the method, refer to the Figure[10\(c\)](https://arxiv.org/html/2610.12586#A9.F10.sf3)\. We adapt DEF to general event sequences by predicting the full heterogeneous feature vector at each candidate event instead of a time and a single mark, and, since our setting fixes the horizon toLLevents, we select the top\-LLcandidates by presence score rather than by a time cutoff \(Appendix[I](https://arxiv.org/html/2610.12586#A9)\)\.

## 4Benchmark Description

### 4\.1Datasets

Table 2:Dataset statistics\. Number of sequences in each split, average sequence length, number of categorical and numerical features describing each event, and overall time span\.Dataset\# seq\. \(train / test\)avg\. seq\. len\.\# num\. \(incl\. time\)\# cat\.time spanGender11K / 2\.8K404\.31±292\.48404\.31\\pm 292\.482215 monthsAge24K / 6K881\.00±124\.75881\.00\\pm 124\.75222 yearsAlphabattle100K / 35K366\.91±212\.58366\.91\\pm 212\.58242 yearsAfter a careful review of the available public corpora, we found three that meet our criteria:Gender222[https://www\.kaggle\.com/competitions/python\-and\-analyze\-data\-final\-project/data](https://www.kaggle.com/competitions/python-and-analyze-data-final-project/data),Age333[https://ods\.ai/competitions/sberbank\-sirius\-lesson](https://ods.ai/competitions/sberbank-sirius-lesson), andAlphabattle444[https://www\.kaggle\.com/datasets/mrmorj/alfabattle\-20](https://www.kaggle.com/datasets/mrmorj/alfabattle-20)\. All three are card transaction datasets with long enough sequences, enough users, and no data artifacts\. Each event includes at least a timestamp, a transaction amount, and an MCC code\. The key statistics are in Table[2](https://arxiv.org/html/2610.12586#S4.T2)\. Notably, no non\-financial public corpus met the size, length, and structure criteria simultaneously\. This is a limitation of the current public data landscape rather than of the benchmark design: although the forecasting problem itself spans many domains, the data behind it is produced in privacy\-sensitive industries such as banking and healthcare, and rich corpora are rarely released publicly\. Thecorpora we rejected and the reasons whyare listed in Appendix[K](https://arxiv.org/html/2610.12586#A11);further preprocessing and splitting detailsare provided in Appendix[H](https://arxiv.org/html/2610.12586#A8)\.

To probe transferability beyond transactions, we additionally evaluate the benchmark ontwo TPP corpora \(Retweet, Amazon\)in Appendix[L](https://arxiv.org/html/2610.12586#A12); both fail our selection criteria, having far fewer and much shorter sequences, and each event carries only a timestamp and a categorical mark\. Despite this, the accuracy–realism trade\-off and the leading models on each axis carry over\.

### 4\.2Metrics

No existing metric covers our setting \(§[2](https://arxiv.org/html/2610.12586#S2)\): accuracy metrics look one step ahead or, like OTD and T\-mAP, leave numerical features aside, and realism metrics ignore the history\. We therefore introduce a pair of metrics for an arbitrary number of heterogeneous features\.

The*Temporal Matching Score*\(TMS\) measures*accuracy*— how closely, event by event, the prediction matches the ground\-truth continuation\. The*Conditional Detection Fooling Score*\(cDFS\) measures*distributional realism*— how difficult it is for a learned discriminator to tell generated continuations apart from real ones given the same history\. As auxiliary per\-feature distributional diagnostics, we also describe \(Appendix[D\.3](https://arxiv.org/html/2610.12586#A4.SS3)\) and report \(Appendix[E\.1](https://arxiv.org/html/2610.12586#A5.SS1)\) theShape and Trend scoresfrom[Zhang et al\. \(2024\)](https://arxiv.org/html/2610.12586#bib.bib4); thecorrelation of TMS with matched per\-feature submetricsis in Appendix[E\.3](https://arxiv.org/html/2610.12586#A5.SS3)\.

#### Conditional Detection Fooling Score \(cDFS\)\.

cDFS is a conditional variant of the classifier two\-sample test\([Lopez\-Paz and Oquab, 2017](https://arxiv.org/html/2610.12586#bib.bib2)\)\. For each test example, we form two sequences sharing the same historyS1:HS\_\{1:H\}: the true continuationSH\+1:H\+LS\_\{H\+1:H\+L\}and a model\-generated continuationS^H\+1:H\+L\\hat\{S\}\_\{H\+1:H\+L\}\. We train a discriminator to predict, from the generated or true continuation, optionally prepended with the lastH′H^\{\\prime\}events of history, whether it is real or generated\. The metric is defined as

cDFS=def2⋅\(1−AUROC\),\\mathrm\{cDFS\}\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}2\\cdot\(1\-\\mathrm\{AUROC\}\),where AUROC is the ROC\-AUC of the discriminator on a held\-out portion of the real/generated pairs\. Clipping AUROC from below at0\.50\.5, cDFS is bounded in\[0,1\]\[0,1\]: it equals00when the discriminator separates real from generated perfectly and11when it does no better than chance, as expected when the conditional distribution of generated continuations matches that of real continuations\. Absolute values depend on the discriminator, so we interpret the relative ranking\. Varying the discriminator’s architecture and capacity confirms thatthe ranking is stable\(Spearmanρ≥0\.91\\rho\\geq 0\.91; Appendix[D\.2](https://arxiv.org/html/2610.12586#A4.SS2)\)\. Unlike classical C2ST, our discriminator sees the history, so it detects continuations that are plausible in isolation but inconsistent with the preceding events\. We describetechnical details of cDFS computationin Appendix[D\.1](https://arxiv.org/html/2610.12586#A4.SS1)\.

#### Temporal Matching Score \(TMS\)\.

We propose TMS to measure prediction accuracy for event sequences\. It is a dataset\-level score inspired by the Optimal Transport Distance \(OTD\)\([Karpukhin and Savchenko, 2026](https://arxiv.org/html/2610.12586#bib.bib3);[Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1)\)\. OTD aligns predicted and true events via a minimum\-cost edit over insertions, deletions, and moves, but its hand\-chosen edit costs\([Yang et al\., 2022](https://arxiv.org/html/2610.12586#bib.bib16)\)do not extend naturally to events with heterogeneous numerical and categorical features\. TMS uses per\-feature, baseline\-rescaled normalization and a one\-to\-one permutation of horizon positions instead of hand\-tuned costs and edit alignment\.

For each numerical featuremmand for time, the paired scoreR1mR\_\{1\}^\{m\}\(resp\.R1tR\_\{1\}^\{t\}\) is one minus the mean absolute prediction error divided by the mean absolute error of a horizon\-naive baseline built from the history\. It can be viewed as a skill\-score variant of the Mean Absolute Scaled Error\([Hyndman and Koehler, 2006](https://arxiv.org/html/2610.12586#bib.bib23)\):11is perfect,00ties the baseline, and lower values are worse\. For each categorical featurenn, the paired scoreF1n∈\[0,1\]F\_\{1\}^\{n\}\\in\[0,1\]is the per\-pair macro\-F1F\_\{1\}between predicted and true horizon labels\. The MASE\-style baseline rescaling puts heterogeneous features on a common scale\.

To separate “wrong content” from “right content in the wrong order,” each per\-feature score is upgraded to aσ\\sigma\-parameterized version under a per\-pair permutationσ\\sigmabetween predicted and true horizon positions, with the paired score recovered atσ=id\\sigma=\\mathrm\{id\}\. TMS averages these scores under a single sharedσ\\sigmaoptimized jointly over all features,

TMS⁡\(𝒯\)=defmaxσ⁡11\+M\+N​\(R1t​\(σ\)\+∑m=1MR1m​\(σ\)\+∑n=1NF1n​\(σ\)\)\.\\mathrm\{TMS\}\(\\mathcal\{T\}\)\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\max\_\{\\sigma\}\\,\\frac\{1\}\{1\+M\+N\}\\\!\\left\(R\_\{1\}^\{t\}\(\\sigma\)\+\\sum\_\{m=1\}^\{M\}R\_\{1\}^\{m\}\(\\sigma\)\+\\sum\_\{n=1\}^\{N\}F\_\{1\}^\{n\}\(\\sigma\)\\right\)\.\(1\)The maximum decomposes per pair into a Hungarian assignment\([Kuhn, 1955](https://arxiv.org/html/2610.12586#bib.bib25)\)on anL×LL\\times Lfeature\-averaged cost matrix\. Because time is one of the matched features, reordering events still incurs a penalty when it worsens temporal alignment\. Because the sameσ\\sigmaenters every term, TMS is a single matched score rather than an average of independently aligned per\-feature diagnostics\.Full definitionsof the per\-feature scores, the present\-class normalization forF1nF\_\{1\}^\{n\}, the proof that the per\-feature numerators decompose into per\-pair contributions, and matched per\-feature variants are deferred to Appendix[C](https://arxiv.org/html/2610.12586#A3)\.

## 5Wasserstein BaryBooster

What distinguishes a generative forecaster is that it predicts many possible future trajectories rather than one, and for a good model these trajectories follow the distribution of the real data\. Accuracy, however, is awarded to a single continuation, so this potential is left unused\. To harvest it, we introduce*Wasserstein BaryBooster*, a post\-hoc procedure that turnsNNstochastic samples from any base generator into a single continuation optimal under TMS\.

Given a historySS, we drawS^1,…,S^N\\hat\{S\}\_\{1\},\\ldots,\\hat\{S\}\_\{N\}from the base model and returnS^⋆=defarg⁡min⁡∑n=1NS^⁡dTMS​\(S^,S^n\),\\hat\{S\}^\{\\star\}\\;\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\;\\arg\\min\_\{\\hat\{S\}\}\\;\\sum\_\{n=1\}^\{N\}d\_\{\\mathrm\{TMS\}\}\\\!\\left\(\\hat\{S\},\\,\\hat\{S\}\_\{n\}\\right\),\(2\)wheredTMS​\(A,B\)=min⁡∑i∈ℋσ⁡C¯\(A,B\)​\(i,σ⁡\(i\)\)d\_\{\\mathrm\{TMS\}\}\(A,B\)=\\min\_\{\\sigma\}\\sum\_\{i\\in\\mathcal\{H\}\}\\bar\{C\}\_\{\(A,B\)\}\(i,\\sigma\(i\)\)is the TMS transport cost between two continuations under the feature\-averaged cost matrixC¯\\bar\{C\}\(Appendix[C\.6](https://arxiv.org/html/2610.12586#A3.SS6)\)\.

Equation \([2](https://arxiv.org/html/2610.12586#S5.E2)\) is the discrete TMS analog of a Wasserstein barycenter\([Agueh and Carlier, 2011](https://arxiv.org/html/2610.12586#bib.bib24)\), which motivates the name; the*booster*suffix reflects that the procedure boosts the predictive accuracy of any underlying generator without retraining\. Note that the procedure is not limited to TMS: any other metric can take its place in Eq\. \([2](https://arxiv.org/html/2610.12586#S5.E2)\)\. We useN=30N=30; theoptimization procedureis given in Appendix[F](https://arxiv.org/html/2610.12586#A6)\.

## 6Experiments

Table 3:TMS and cDFS at horizonL=32L=32\. Within the baseline group, we highlightthe best,the second best, andthe third bestmethods\. BaryBoosted methods are highlighted when they would rank first/second/third after being added to the pool\.Redcells mark cDFS<0\.07<0\.07\(easily detected\)\.MethodTMS ↑cDFS ↑AlphabattleGenderAgeAlphabattleGenderAgeHist\.0\.3820\.2210\.3660\.5440\.4760\.234Mode0\.2030\.0470\.2630\.0000\.0000\.000Rep\.0\.4400\.3610\.4540\.0090\.0500\.065AR0\.2860\.1880\.3540\.0000\.0020\.000MT0\.2860\.1530\.3600\.0000\.0000\.000LDM0\.4050\.3180\.4260\.8560\.9071\.000DDM0\.2510\.1070\.4410\.0040\.0020\.202DEF0\.4240\.4130\.5080\.0000\.0020\.000LDM\-WB \(ours\)0\.4950\.4360\.5220\.1770\.2560\.224DDM\-WB \(ours\)0\.2740\.1360\.4800\.0010\.0010\.102### 6\.1Evaluation Protocol

We forecast at fixed horizonsL∈\{4,16,32,64\}L\\in\\\{4,16,32,64\\\}\. Main results useL=32L\{=\}32\(Table[3](https://arxiv.org/html/2610.12586#S6.T3)\); seeL∈\{4,16,64\}L\\in\\\{4,16,64\\\}in Appendix[E\.4](https://arxiv.org/html/2610.12586#A5.SS4), where the key conclusions hold\. We report both cDFS and TMS \(§[4\.2](https://arxiv.org/html/2610.12586#S4.SS2)\) so comparisons reflect both distributional realism and prediction accuracy, not just a single score\. We also evaluate*cross\-horizon*models: those trained at shorterLLare semi\-autoregressively rolled out to longerLL\(e\.g\., LDM trained onL=16L=16and evaluated on a longer horizon is denotedLDM/16\), to compare this approach to direct one\-shot forecasting\. Additionally, we apply*Wasserstein BaryBooster*\(§[5](https://arxiv.org/html/2610.12586#S5)\) to pretrained diffusion models and mark them with the\-WBsuffix\. Details ondata sampling and splittingare in Appendix[H](https://arxiv.org/html/2610.12586#A8)\. We reportrun time and GPU memory usagein Appendix[J](https://arxiv.org/html/2610.12586#A10)\. We addressstatistical significance and metric stability across runsin Appendix[E\.5](https://arxiv.org/html/2610.12586#A5.SS5)\.

### 6\.2Results

#### Performance overview\.

Table[3](https://arxiv.org/html/2610.12586#S6.T3)reveals a sharp split between the two metrics\. LDM is the only generative model to achieve a non\-trivial cDFS, except for DDM on Age\. Every remaining learned model, including the state\-of\-the\-art DEF, collapses to cDFS values near zero, indicating that a simple discriminator separates their continuations from real ones with near\-perfect accuracy\. Appendix[E\.6](https://arxiv.org/html/2610.12586#A5.SS6)showsgenerated samplesfor a single client\. Among the remaining methods, HistSampler is the closest competitor to LDM, but it produces less realistic sequences because it ignores temporal dependencies\. Table[3](https://arxiv.org/html/2610.12586#S6.T3)reports boosted variants denotedLDM\-WBandDDM\-WB; applying the same procedure to other generators yields little or no comparable improvement, so we omit those boosted variants from the main tables and report them in Appendix[G](https://arxiv.org/html/2610.12586#A7)\.

The prediction accuracy ranking tells a different story\. DEF is the best or second\-best method across all datasets, confirming its position as the strongest non\-diffusion forecaster\. Repeat is the strongest baseline on Alphabattle and second\-best on the other two datasets, beating every learned method on Alphabattle and outperforming both LDM and DDM almost everywhere\. AR and MT remain below HistSampler on both cDFS and TMS\. Their numerical features collapse to the conditional median, losing nearly all variance — the pathology of Figure[1](https://arxiv.org/html/2610.12586#S1.F1)\(Appendix[E\.6](https://arxiv.org/html/2610.12586#A5.SS6)\)\. Cross\-horizon models are almost never better than direct one\-shot models trained at the target horizon\.

Figure 2:Trade\-off between TMS and cDFS for each dataset on horizonL=32L=32\. The solid line shows the Pareto front over baseline methods\.
#### Accuracy–realism trade\-off\.

Figure[2](https://arxiv.org/html/2610.12586#S6.F2)shows the trade\-off explicitly\. Methods with high prediction accuracy show low distributional realism and vice versa\. Interestingly, Repeat lies on the Pareto front of the baseline pool across all datasets, despite being a one\-line, non\-parametric procedure\. This highlights the room for specialized generative models that surpass Repeat’s accuracy without sacrificing realism\.

#### BaryBoosting diffusion models\.

Wasserstein BaryBooster \(§[5](https://arxiv.org/html/2610.12586#S5)\) realizes the predictive potential of the LDM: when applied, it produces the highest TMS across all datasets while still retaining a non\-trivial cDFS \(see Table[3](https://arxiv.org/html/2610.12586#S6.T3)\)\. Applying the same procedure to DDM yields only modest TMS gains, reflecting DDM’s weaker base generations\. Boosting helps most for diffusion\-based generators \(Appendix[G](https://arxiv.org/html/2610.12586#A7)\): the gain is largest for diffusion models, absent for DEF, and small for the remaining non\-diffusion methods\.

Figure[2](https://arxiv.org/html/2610.12586#S6.F2)clearly shows that BaryBoosting moves the Pareto front: by aggregating multiple LDM samples around their TMS barycenter, it converts part of the parent model’s realism into the highest TMS\. This converts a generative model into a competitive predictor while leaving the original LDM available whenever distributional realism matters more than prediction accuracy\. The effect is strongest atL=4L=4\(Figure[5](https://arxiv.org/html/2610.12586#A5.F5)\), where boosting shifts LDM from near\-bottom TMS to best\-in\-class predictor\.

The gain is not confined to the optimized cost: Table[4](https://arxiv.org/html/2610.12586#S6.T4)shows that BaryBoosting LDM also improves many other prediction metrics that BaryBooster does not optimize directly\. Appendix[E\.2](https://arxiv.org/html/2610.12586#A5.SS2)extends thisindependent evaluationto all methods; DDM also improves after boosting, with a few exceptions\.

Table 4:BaryBoosting improves prediction metrics beyond the optimized TMS \(L=32L=32; excerpt of Table[9](https://arxiv.org/html/2610.12586#A5.T9)\)\.MethodR1 \(amount\) ↑R1 \(time\) ↑F1 \(mcc\) ↑AlphabattleGenderAgeAlphabattleGenderAgeAlphabattleGenderAgeLDM0\.4790\.1800\.2160\.1480\.1440\.0740\.4000\.3640\.323LDM\-WB \(ours\)0\.5830\.3340\.4380\.3370\.3310\.3310\.4590\.4480\.371

## 7Discussion

#### Outlook

A realistic generative model captures the whole distribution of possible futures, mirroring the real world, where the same history can lead to many different futures\. We show how this property can be put to use: aggregating samples with the BaryBooster yields a state\-of\-the\-art point forecast with no additional training for the target metric\. This opens the way to foundation models for event sequences: a single generative model, trained once, from which many downstream tasks can be solved\.

#### Limitations

Our empirical study uses a relatively small number of datasets and focuses narrowly on card transactions\. Although we made a best\-effort attempt to diversify the benchmark, practical constraints on data quality and schema compatibility limited the final pool; details onconsidered but excluded sourcesare reported in Appendix[K](https://arxiv.org/html/2610.12586#A11)\. In addition, the Wasserstein BaryBooster introduces substantial computational overhead, since it requires generating multiple candidate trajectories and solving an iterative barycentric aggregation procedure at inference time\. While this improves predictive performance, the extra compute may be prohibitive in latency\-sensitive or resource\-constrained deployments\. The overhead is, however, an inference\-time choice:N=5N=5already retains≈75%\{\\approx\}75\\%of the TMS gain at one sixth of the sampling cost \(Appendix[F\.1](https://arxiv.org/html/2610.12586#A6.SS1)\), and theNNsamplings parallelize across GPUs\.

#### Broader impacts

The forecasts studied in this paper can support planning and risk analysis\. Since a generative model produces many plausible continuations, analysts can explore the different paths a client’s future may take, rather than a single average forecast\. Our metrics, benchmark, and methods are not specific to transactions and can bring these benefits to event sequences in other domains, such as healthcare and e\-commerce\. On the other hand, realistic synthetic transactions could be misused, e\.g\., to evade fraud detection; in this work, all models are trained on public, anonymized data\.

## Acknowledgments and Disclosure of Funding

The work was supported by the grant for research centers in the field of AI provided by the Ministry of Economic Development of the Russian Federation in accordance with the agreement 000000C313925P4F0002 and the agreement No\. 139\-10\-2025\-033\. The authors declare no competing interests\.

## References

- Agueh and Carlier \(2011\)M\. Agueh and G\. CarlierBarycenters in the Wasserstein space\.SIAM Journal on Mathematical Analysis43\(2\),pp\. 904–924\.Cited by:[§5](https://arxiv.org/html/2610.12586#S5.p3.1)\.
- Bazarovaet al\.\(2025\)A\. Bazarova, M\. Kovaleva, I\. Kuleshov, E\. Romanenkova, A\. Stepikin, A\. Yugay, D\. Mollaev, I\. Kireev, A\. Savchenko, and A\. ZaytsevLearning transactions representations for information management in banks: mastering local, global, and external knowledge\.International Journal of Information Management Data Insights5\(1\),pp\. 100323\.Cited by:[§1](https://arxiv.org/html/2610.12586#S1.p1.1)\.
- Garutiet al\.\(2025\)F\. Garuti, E\. Sangineto, S\. Luetto, L\. Forni, and R\. CucchiaraDiffusion transformers for tabular data time series generation\.InThe Thirteenth International Conference on Learning Representations,Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px3.p1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.14.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.15.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.16.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.17.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.18.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.19.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.35.2.1.1),[Appendix K](https://arxiv.org/html/2610.12586#A11.p1.1),[§I\.3](https://arxiv.org/html/2610.12586#A9.SS3.SSS0.Px1.p1.2),[Table 1](https://arxiv.org/html/2610.12586#S2.T1.5.10.1)\.
- Havasiet al\.\(2025\)M\. Havasi, B\. Karrer, I\. Gat, and R\. T\. ChenEdit flows: variable length discrete flow matching with sequence\-level edit operations\.Advances in Neural Information Processing Systems38\.Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px2.p1.1)\.
- Higginset al\.\(2017\)I\. Higgins, L\. Matthey, A\. Pal, C\. Burgess, X\. Glorot, M\. Botvinick, S\. Mohamed, and A\. Lerchnerβ\\beta\-VAE: learning basic visual concepts with a constrained variational framework\.InInternational Conference on Learning Representations,Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px1.p1.1),[§3\.3](https://arxiv.org/html/2610.12586#S3.SS3.p1.1)\.
- Ho and Salimans \(2022\)J\. Ho and T\. SalimansClassifier\-free diffusion guidance\.arXiv preprint arXiv:2207\.12598\.Note:NeurIPS 2021 Workshop on Deep Generative Models and Downstream ApplicationsCited by:[§I\.3](https://arxiv.org/html/2610.12586#A9.SS3.SSS0.Px1.p1.1),[§I\.3](https://arxiv.org/html/2610.12586#A9.SS3.SSS0.Px7.p1.1)\.
- Hoogeboomet al\.\(2021\)E\. Hoogeboom, D\. Nielsen, P\. Jaini, P\. Forré, and M\. WellingArgmax flows and multinomial diffusion: learning categorical distributions\.Advances in neural information processing systems34,pp\. 12454–12465\.Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px2.p1.1),[§3\.4](https://arxiv.org/html/2610.12586#S3.SS4.p1.1)\.
- Hyndman and Koehler \(2006\)R\. J\. Hyndman and A\. B\. KoehlerAnother look at measures of forecast accuracy\.International journal of forecasting22\(4\),pp\. 679–688\.Cited by:[§C\.2](https://arxiv.org/html/2610.12586#A3.SS2.p2.2),[§4\.2](https://arxiv.org/html/2610.12586#S4.SS2.SSS0.Px2.p2.1)\.
- Johnsonet al\.\(2024\)A\. Johnson, L\. Bulgarelli, T\. Pollard, B\. Gow, B\. Moody, S\. Horng, L\. A\. Celi, and R\. MarkMIMIC\-IV\.PhysioNet\.Note:Version 3\.1External Links:[Document](https://dx.doi.org/10.13026/kpb9-mt58),[Link](https://doi.org/10.13026/kpb9-mt58)Cited by:[Appendix K](https://arxiv.org/html/2610.12586#A11.SS0.SSS0.Px2.p1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.13.2.1.1),[§1](https://arxiv.org/html/2610.12586#S1.p1.1)\.
- Karpukhin and Savchenko \(2026\)I\. Karpukhin and A\. SavchenkoDetecting the future: all\-at\-once event sequence forecasting with horizon matching\.InProceedings of the AAAI Conference on Artificial Intelligence,Vol\.40,pp\. 22536–22544\.Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px2.p1.1),[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px3.p1.1),[§I\.3](https://arxiv.org/html/2610.12586#A9.SS3.SSS0.Px12.p1.1),[§2\.3](https://arxiv.org/html/2610.12586#S2.SS3.SSS0.Px1.p2.1),[Table 1](https://arxiv.org/html/2610.12586#S2.T1.5.8.1.1),[§3\.5](https://arxiv.org/html/2610.12586#S3.SS5.p1.1),[§4\.2](https://arxiv.org/html/2610.12586#S4.SS2.SSS0.Px2.p1.1)\.
- Karpukhinet al\.\(2026\)I\. Karpukhin, F\. Shipilov, and A\. SavchenkoHoTPP benchmark: are we good at the long horizon events forecasting?\.Neurocomputing672,pp\. 132771\.Cited by:[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.13.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.35.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.7.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.8.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.9.2.1.1),[Appendix K](https://arxiv.org/html/2610.12586#A11.p1.1),[Appendix B](https://arxiv.org/html/2610.12586#A2.SS0.SSS0.Px2.p1.1),[item 2](https://arxiv.org/html/2610.12586#S1.I1.i2.p1.1),[§1](https://arxiv.org/html/2610.12586#S1.p2.1),[§2\.2](https://arxiv.org/html/2610.12586#S2.SS2.p1.1),[§2\.3](https://arxiv.org/html/2610.12586#S2.SS3.SSS0.Px1.p1.1),[§2\.3](https://arxiv.org/html/2610.12586#S2.SS3.SSS0.Px1.p2.1),[§3\.1](https://arxiv.org/html/2610.12586#S3.SS1.p1.1)\.
- Karraset al\.\(2022\)T\. Karras, M\. Aittala, T\. Aila, and S\. LaineElucidating the design space of diffusion\-based generative models\.Advances in neural information processing systems35,pp\. 26565–26577\.Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px1.p1.1),[§I\.3](https://arxiv.org/html/2610.12586#A9.SS3.SSS0.Px14.p1.1),[§3\.3](https://arxiv.org/html/2610.12586#S3.SS3.p1.1)\.
- Kuhn \(1955\)H\. W\. KuhnThe Hungarian method for the assignment problem\.Naval research logistics quarterly2\(1\-2\),pp\. 83–97\.Cited by:[§4\.2](https://arxiv.org/html/2610.12586#S4.SS2.SSS0.Px2.p3.2)\.
- Liguoriet al\.\(2025\)A\. Liguori, L\. Caroprese, M\. Minici, B\. Veloso, F\. Spinnato, M\. Nanni, G\. Manco, and J\. GamaModeling events and interactions through temporal processes: a survey\.Neurocomputing653,pp\. 131191\.Cited by:[§1](https://arxiv.org/html/2610.12586#S1.p1.1)\.
- Lopes and Ferreira \(2019\)I\. F\. Lopes and D\. R\. FerreiraA survey of process mining competitions: the BPI challenges 2011–2018\.InBusiness Process Management Workshops,pp\. 263–274\.Cited by:[§1](https://arxiv.org/html/2610.12586#S1.p1.1)\.
- Lopez\-Paz and Oquab \(2017\)D\. Lopez\-Paz and M\. OquabRevisiting classifier two\-sample tests\.InInternational Conference on Learning Representations,Cited by:[item 2](https://arxiv.org/html/2610.12586#S1.I1.i2.p1.1),[§2\.3](https://arxiv.org/html/2610.12586#S2.SS3.SSS0.Px2.p1.1),[§4\.2](https://arxiv.org/html/2610.12586#S4.SS2.SSS0.Px1.p1.1)\.
- Lüdkeet al\.\(2023\)D\. Lüdke, M\. Biloš, O\. Shchur, M\. Lienen, and S\. GünnemannAdd and Thin: diffusion for temporal point processes\.Advances in Neural Information Processing Systems36,pp\. 56784–56801\.Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px2.p1.1),[§1](https://arxiv.org/html/2610.12586#S1.p2.1),[Table 1](https://arxiv.org/html/2610.12586#S2.T1.5.5.1)\.
- Lüdkeet al\.\(2026\)D\. Lüdke, M\. Lienen, M\. Kollovieh, and S\. GünnemannEdit\-based flow matching for temporal point processes\.InThe Fourteenth International Conference on Learning Representations,Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px2.p1.1),[§1](https://arxiv.org/html/2610.12586#S1.p2.1),[Table 1](https://arxiv.org/html/2610.12586#S2.T1.5.7.1)\.
- Lüdkeet al\.\(2025\)D\. Lüdke, E\. R\. Raventós, M\. Kollovieh, and S\. GünnemannUnlocking point processes through point set diffusion\.InThe Thirteenth International Conference on Learning Representations,Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px2.p1.1),[§1](https://arxiv.org/html/2610.12586#S1.p2.1),[Table 1](https://arxiv.org/html/2610.12586#S2.T1.5.6.1)\.
- Luettoet al\.\(2025\)S\. Luetto, F\. Garuti, E\. Sangineto, L\. Forni, and R\. CucchiaraOne transformer for all time series: representing and training with time\-dependent heterogeneous tabular data\.Machine Learning114\(6\),pp\. 145\.Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px3.p1.1),[Table 1](https://arxiv.org/html/2610.12586#S2.T1.5.9.1)\.
- Meiet al\.\(2019\)H\. Mei, G\. Qin, and J\. EisnerImputing missing events in continuous\-time event streams\.InInternational Conference on Machine Learning,pp\. 4475–4485\.Cited by:[Appendix F](https://arxiv.org/html/2610.12586#A6.p1.1),[item 2](https://arxiv.org/html/2610.12586#S1.I1.i2.p1.1)\.
- Mukherjeeet al\.\(2026\)A\. Mukherjee, R\. Deng, H\. Zhao, Y\. Mao, L\. Sigal, and F\. TungADiff4TPP: asynchronous diffusion models for temporal point processes\.Transactions on Machine Learning Research\.Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px2.p1.1),[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px3.p1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.10.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.11.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.7.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.8.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.9.2.1.1),[Appendix K](https://arxiv.org/html/2610.12586#A11.p1.1),[§E\.7](https://arxiv.org/html/2610.12586#A5.SS7.p1.1),[§I\.1](https://arxiv.org/html/2610.12586#A9.SS1),[§I\.3](https://arxiv.org/html/2610.12586#A9.SS3.SSS0.Px14.p1.1),[Table 13](https://arxiv.org/html/2610.12586#A9.T13.5.1.2),[Table 1](https://arxiv.org/html/2610.12586#S2.T1.5.11.1.1),[§3\.3](https://arxiv.org/html/2610.12586#S3.SS3.p3.1)\.
- Osinet al\.\(2025\)D\. Osin, I\. Udovichenko, E\. Shvetsov, V\. Moskvoretskii, and E\. BurnaevEBES: easy benchmarking for event sequences\.InProceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining V\. 2,pp\. 5730–5741\.Cited by:[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.21.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.22.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.23.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.24.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.25.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.26.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.27.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.28.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.29.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.35.2.1.1),[Appendix K](https://arxiv.org/html/2610.12586#A11.p1.1),[Appendix B](https://arxiv.org/html/2610.12586#A2.SS0.SSS0.Px3.p1.1),[§2\.2](https://arxiv.org/html/2610.12586#S2.SS2.p1.1)\.
- Sahooet al\.\(2024\)S\. S\. Sahoo, M\. Arriola, Y\. Schiff, A\. Gokaslan, E\. Marroquin, J\. T\. Chiu, A\. Rush, and V\. KuleshovSimple and effective masked diffusion language models\.Advances in Neural Information Processing Systems37,pp\. 130136–130184\.Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px1.p1.1)\.
- Shiet al\.\(2024\)J\. Shi, K\. Han, Z\. Wang, A\. Doucet, and M\. TitsiasSimplified and generalized masked diffusion for discrete data\.Advances in neural information processing systems37,pp\. 103131–103167\.Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px1.p1.1)\.
- Shiet al\.\(2025\)J\. Shi, M\. Xu, H\. Hua, H\. Zhang, S\. Ermon, and J\. LeskovecTabDiff: a mixed\-type diffusion model for tabular data generation\.InThe Thirteenth International Conference on Learning Representations,Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px1.p1.1),[Table 1](https://arxiv.org/html/2610.12586#S2.T1.5.4.1)\.
- Surkovet al\.\(2025\)E\. Surkov, D\. Osin, E\. Burnaev, and E\. ShvetsovHow to model human actions distribution with event sequence data\.arXiv preprint arXiv:2510\.05856\.Cited by:[Appendix K](https://arxiv.org/html/2610.12586#A11.SS0.SSS0.Px2.p1.1),[Appendix K](https://arxiv.org/html/2610.12586#A11.SS0.SSS0.Px3.p1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.27.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.28.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.29.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.30.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.31.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.32.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.35.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.36.2.1.1),[Appendix K](https://arxiv.org/html/2610.12586#A11.p1.1)\.
- Xueet al\.\(2024\)S\. Xue, X\. Shi, Z\. Chu, Y\. Wang, H\. Hao, F\. Zhou, C\. Jiang, C\. Pan, J\. Y\. Zhang, Q\. Wen, J\. Zhou, and H\. MeiEasyTPP: towards open benchmarking temporal point processes\.InInternational Conference on Learning Representations \(ICLR\),Cited by:[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.10.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.11.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.33.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.7.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.8.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.9.2.1.1),[Appendix K](https://arxiv.org/html/2610.12586#A11.p1.1),[Appendix L](https://arxiv.org/html/2610.12586#A12.p1.1),[Appendix B](https://arxiv.org/html/2610.12586#A2.SS0.SSS0.Px1.p1.1),[item 2](https://arxiv.org/html/2610.12586#S1.I1.i2.p1.1),[§1](https://arxiv.org/html/2610.12586#S1.p2.1),[§2\.2](https://arxiv.org/html/2610.12586#S2.SS2.p1.1),[§2\.3](https://arxiv.org/html/2610.12586#S2.SS3.SSS0.Px1.p1.1)\.
- Yanget al\.\(2022\)C\. Yang, H\. Mei, and J\. EisnerTransformer embeddings of irregularly spaced events and their participants\.InProceedings of the Tenth International Conference on Learning Representations \(ICLR\),Cited by:[§2\.3](https://arxiv.org/html/2610.12586#S2.SS3.SSS0.Px1.p2.1),[§4\.2](https://arxiv.org/html/2610.12586#S4.SS2.SSS0.Px2.p1.1)\.
- Zenget al\.\(2024\)M\. Zeng, F\. Regol, and M\. CoatesInteracting diffusion processes for event sequence forecasting\.InInternational Conference on Machine Learning,pp\. 58407–58430\.Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px2.p1.1),[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px3.p1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.10.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.11.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.12.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.33.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.7.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.8.2.1.1),[Table 16](https://arxiv.org/html/2610.12586#A11.T16.3.9.2.1.1),[Appendix K](https://arxiv.org/html/2610.12586#A11.p1.1),[§I\.3](https://arxiv.org/html/2610.12586#A9.SS3.SSS0.Px13.p1.1),[§2\.3](https://arxiv.org/html/2610.12586#S2.SS3.SSS0.Px1.p2.1),[Table 1](https://arxiv.org/html/2610.12586#S2.T1.5.2.1.1),[§3\.4](https://arxiv.org/html/2610.12586#S3.SS4.p1.1),[§4\.2](https://arxiv.org/html/2610.12586#S4.SS2.SSS0.Px2.p1.1)\.
- Zhanget al\.\(2024\)H\. Zhang, J\. Zhang, B\. Srinivasan, Z\. Shen, X\. Qin, C\. Faloutsos, H\. Rangwala, and G\. KarypisMixed\-type tabular data synthesis with score\-based diffusion in latent space\.InThe Twelfth International Conference on Learning Representations,Cited by:[Appendix A](https://arxiv.org/html/2610.12586#A1.SS0.SSS0.Px1.p1.1),[§D\.3](https://arxiv.org/html/2610.12586#A4.SS3.p1.1),[§E\.1](https://arxiv.org/html/2610.12586#A5.SS1.p1.1),[§I\.3](https://arxiv.org/html/2610.12586#A9.SS3.SSS0.Px11.p1.1),[§2\.3](https://arxiv.org/html/2610.12586#S2.SS3.SSS0.Px2.p2.1),[Table 1](https://arxiv.org/html/2610.12586#S2.T1.5.3.1),[§3\.3](https://arxiv.org/html/2610.12586#S3.SS3.p1.1),[§4\.2](https://arxiv.org/html/2610.12586#S4.SS2.p2.1)\.
- Zhaoet al\.\(2023\)S\. Zhao, R\. Wu, J\. Tao, M\. Qu, M\. Zhao, C\. Fan, and H\. ZhaoperCLTV: a general system for personalized customer lifetime value prediction in online games\.ACM Transactions on Information Systems41\(1\),pp\. 1–29\.Cited by:[§1](https://arxiv.org/html/2610.12586#S1.p1.1)\.

## Appendix Contents

## Appendix AGenerative Models for Event Data: Extended Discussion

This appendix expands the discussion of the related generative models summarized in Table[1](https://arxiv.org/html/2610.12586#S2.T1)and referenced in §[2\.1](https://arxiv.org/html/2610.12586#S2.SS1), grouped by the three lines of work they belong to\.

#### Tabular data\.

Event sequences can be viewed as time\-ordered sequences of tabular rows, so diffusion models for heterogeneous tabular data are directly relevant\. TabSyn\[[Zhang et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib4)\]is a latent diffusion model in which aβ\\beta\-VAE\[[Higgins et al\., 2017](https://arxiv.org/html/2610.12586#bib.bib14)\]encodes mixed\-type features into a latent space, and Gaussian diffusion\[[Karras et al\., 2022](https://arxiv.org/html/2610.12586#bib.bib17)\]generates new latents that are decoded back to rows\. TabDiff\[[Shi et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib6)\]is the data\-diffusion counterpart: it couples masked discrete diffusion\[[Shi et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib18),[Sahoo et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib5)\]for categorical features and Gaussian diffusion\[[Karras et al\., 2022](https://arxiv.org/html/2610.12586#bib.bib17)\]for numerical features, generating rows directly\. Neither model, however, natively targets sequences\.

#### Temporal point processes\.

Temporal point processes \(TPP\) are the classical formalism for random event times, and most prior generative work on event sequences falls under TPP\. In*unmarked*TPP, each event is characterized by its time only\. Add\-and\-Thin\[[Lüdke et al\., 2023](https://arxiv.org/html/2610.12586#bib.bib7)\]introduces a diffusion\-inspired noising and denoising process tailored for point processes, and its successor EdiTPP\[[Lüdke et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib8)\]casts sequence generation as edit\-based flow matching\[[Havasi et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib13)\]\.*Spatio\-temporal*point processes \(STPP\) extend events with numerical coordinates; Point Set Diffusion\[[Lüdke et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib9)\]generalizes Add\-and\-Thin to this setting\.*Marked*TPP \(MTPP\) adds a single categorical mark per event and is the most studied variant\. ADiff4TPP\[[Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\]uses aβ\\beta\-VAE to embed \(time, mark\) pairs and applies flow matching with non\-uniform noise scales over the sequence\. CDiff\[[Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1)\]combines uniform categorical diffusion\[[Hoogeboom et al\., 2021](https://arxiv.org/html/2610.12586#bib.bib15)\]for marks with Gaussian diffusion for inter\-arrival times and is trained to predict multiple future events jointly\. DEF\[[Karpukhin and Savchenko, 2026](https://arxiv.org/html/2610.12586#bib.bib3)\]is a recent non\-diffusion method that predicts a fixed set of candidate future events and scores them, keeping the top\-scoring ones\.

#### General event sequences\.

The MTPP methods above naturally extend to heterogeneous events: ADiff4TPP\[[Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\], CDiff\[[Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1)\], and DEF\[[Karpukhin and Savchenko, 2026](https://arxiv.org/html/2610.12586#bib.bib3)\]all generalize to additional numerical and categorical features\. TabDiT\[[Garuti et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib11)\]is a latent\-diffusion approach closely related to ADiff4TPP, but it targets attribute\-conditional generation rather than history\-conditional \(forecasting\)\. UniTTab\[[Luetto et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib10)\]is a transformer model tailored for heterogeneous event sequences, but its authors do not study generative modeling — they use the transformer only to extract sequence representations for downstream discriminative tasks\.

## Appendix BBenchmarks for Event Sequences: Extended Discussion

This appendix expands the discussion from §[2\.2](https://arxiv.org/html/2610.12586#S2.SS2)of the three prior benchmarks closest to our work — EasyTPP, HoTPP, and EBES — and details why each falls short of the heterogeneous, realism\-aware forecasting setting we target\.

#### EasyTPP\.

EasyTPP\[[Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19)\]is the first open benchmarking effort for neural TPPs\. It provides a unified interface to several widely used TPP datasets, along with reference implementations of popular neural TPP models\. Evaluation covers both next\-event prediction, reported as time mean absolute error and mark classification accuracy, and long\-horizon prediction via optimal transport distance on multi\-step forecasts\. Two properties of the benchmark limit its scope relative to our setting\. First, events are restricted to a single categorical mark, i\.e\., EasyTPP operates strictly within the MTPP formalism — there is no facility for additional numerical or categorical features, which rules out the domains \(finance, retail\) where heterogeneous events are the norm\. Second, all reported metrics quantify closeness of the prediction to the ground truth; the benchmark does not assess whether generated sequences are distributionally realistic, and in particular provides no way to detect mode collapse or distributional mismatch\.

#### HoTPP\.

HoTPP\[[Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20)\]explicitly targets long\-horizon forecasting within the MTPP formalism\. Its authors point out that next\-event metrics correlate poorly with long\-horizon quality, and that the classical optimal\-transport distance for multi\-step evaluation is sensitive to its hand\-chosen hyperparameters\. As a remedy, they propose Temporal mean Average Precision \(T\-mAP\), a metric inspired by object detection that evaluates predicted event\-label distributions over the full horizon\. It also reports a single accuracy\-style metric per configuration, without a dedicated account of distributional realism\.

#### EBES\.

EBES\[[Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22)\]is the closest prior benchmark in terms of data structure: it explicitly targets heterogeneous event sequences with irregular time stamps and a mix of numerical and categorical features, assembles a large collection of real\-world datasets, and implements representative sequence models\. Its scope, however, is*discriminative*: EBES evaluates sequence\-level classification, where each sequence carries a single target label\. It does not address forecasting or any form of generative modeling\.

## Appendix CTemporal Matching Score: full definition

This appendix gives the full construction summarized in equation \([1](https://arxiv.org/html/2610.12586#S4.E1)\)\. We define the per\-feature paired scores, their permutation\-dependent versions, the matched per\-feature variants, and the resulting TMS, and we derive the per\-pair decomposition that makes optimization a Hungarian assignment\.

### C\.1Notation

Let𝒯=\{\(S,S^\)\}\\mathcal\{T\}=\\\{\(S,\\hat\{S\}\)\\\}denote the evaluation set of sequence\-prediction pairs, all sharing forecast\-horizon lengthLL, and letℋ=\{H\+1,…,H\+L\}\\mathcal\{H\}=\\\{H\{\+\}1,\\ldots,H\{\+\}L\\\}be the horizon index set\. LetΔ​ti=ti−ti−1\\Delta t\_\{i\}=t\_\{i\}\-t\_\{i\-1\}, letximx\_\{i\}^\{m\}be themm\-th ofMMnumerical features of theii\-th event, and letkink\_\{i\}^\{n\}be thenn\-th ofNNcategorical features\. Hats denote predicted quantities,med⁡\[⋅\]\\mathrm\{med\}\\left\[\\cdot\\right\]denotes the median, and𝔼\(S,S^\)​\[⋅\]\\mathbb\{E\}\_\{\(S,\\hat\{S\}\)\}\[\\cdot\]is the uniform average over𝒯\\mathcal\{T\}\.

### C\.2PairedR1R\_\{1\}for numerical features and time

PairedR1R\_\{1\}answers: at each horizon step, how close are predicted magnitudes and timestamps to the ground truth, relative to a naive baseline? It compares mean absolute error to a simple baseline built only from the past \(constant history median for features, uniform spacing from history for time\), so the score is a unitless skill measure: higher means the model beats that naive reference\.

The paired score for a numerical featuremmand for time is the mean absolute prediction error rescaled by the mean absolute error of a horizon\-naive forecast built from the history,

R1m=def1−𝔼\(S,S^\)​\[∑i∈ℋ\|x^im−xim\|\]𝔼\(S,S^\)\[∑i∈ℋ\|xim−med\[x1:Hm\]\|\],R\_\{1\}^\{m\}\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}1\-\\frac\{\\mathbb\{E\}\_\{\(S,\\hat\{S\}\)\}\\\!\\left\[\\sum\_\{i\\in\\mathcal\{H\}\}\|\\hat\{x\}\_\{i\}^\{m\}\-x\_\{i\}^\{m\}\|\\right\]\}\{\\mathbb\{E\}\_\{\(S,\\hat\{S\}\)\}\\\!\\left\[\\sum\_\{i\\in\\mathcal\{H\}\}\|x\_\{i\}^\{m\}\-\\mathrm\{med\}\\left\[x\_\{1:H\}^\{m\}\\right\]\|\\right\]\},R1t=def1−𝔼\(S,S^\)​\[∑i∈ℋ\|t^i−ti\|\]𝔼\(S,S^\)\[∑i∈ℋ\|ti−\(tH\+\(i−H\)med\[Δt1:H\]\)\|\]\.R\_\{1\}^\{t\}\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}1\-\\frac\{\\mathbb\{E\}\_\{\(S,\\hat\{S\}\)\}\\\!\\left\[\\sum\_\{i\\in\\mathcal\{H\}\}\|\\hat\{t\}\_\{i\}\-t\_\{i\}\|\\right\]\}\{\\mathbb\{E\}\_\{\(S,\\hat\{S\}\)\}\\\!\\left\[\\sum\_\{i\\in\\mathcal\{H\}\}\\left\|t\_\{i\}\-\\bigl\(t\_\{H\}\+\(i\-H\)\\mathrm\{med\}\\left\[\\Delta t\_\{1:H\}\\right\]\\bigr\)\\right\|\\right\]\}\.The numerical baseline is the constant history median; the time baseline is a uniform\-spacing forecast from the last historical event using the median historical inter\-event gap\. This closely resembles the Mean Absolute Scaled Error of[Hyndman and Koehler \[2006\]](https://arxiv.org/html/2610.12586#bib.bib23), with a horizon\-naive baseline in place of the in\-sample one\-step\-ahead naive forecast, expressed as a skill score:R1=1R\_\{1\}=1corresponds to zero prediction error,R1=0R\_\{1\}=0means the model’s average error equals the baseline’s, andR1<0R\_\{1\}<0means the model is worse than the baseline on average — with no lower bound\. Heterogeneous features become commensurable through this baseline rescaling, soR1R\_\{1\}values can be averaged across features that live on incomparable units\.

### C\.3PairedF1F\_\{1\}for categorical features

PairedF1F\_\{1\}answers: when we compare predicted and true labels at the same horizon index, how well does the model recover each class, balancing false positives and false negatives? It is macro\-F1F\_\{1\}over classes that actually appear in the pair, so rare or absent labels do not dominate the average\.

For a categorical featurenn, let

TPcn=def∑i∈ℋ𝟏\[k^in=c\]1\[kin=c\],nc,predn=def∑i∈ℋ𝟏\[k^in=c\],nc,truen=def∑i∈ℋ𝟏\[kin=c\],\\mathrm\{TP\}\_\{c\}^\{n\}\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\sum\_\{i\\in\\mathcal\{H\}\}\\mathbf\{1\}\[\\hat\{k\}\_\{i\}^\{n\}=c\]\\,\\mathbf\{1\}\[k\_\{i\}^\{n\}=c\],\\quad n\_\{c,\\mathrm\{pred\}\}^\{n\}\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\sum\_\{i\\in\\mathcal\{H\}\}\\mathbf\{1\}\[\\hat\{k\}\_\{i\}^\{n\}=c\],\\quad n\_\{c,\\mathrm\{true\}\}^\{n\}\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\sum\_\{i\\in\\mathcal\{H\}\}\\mathbf\{1\}\[k\_\{i\}^\{n\}=c\],and let𝒞\(S,S^\)n=def\{c:nc,predn\+nc,truen\>0\}\\mathcal\{C\}\_\{\(S,\\hat\{S\}\)\}^\{n\}\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\\{c:n\_\{c,\\mathrm\{pred\}\}^\{n\}\+n\_\{c,\\mathrm\{true\}\}^\{n\}\>0\\\}be the set of class labels appearing in either continuation\. Using the standard identity2​Pc​Rc/\(Pc\+Rc\)=2​TPc/\(nc,pred\+nc,true\)2\\,P\_\{c\}R\_\{c\}/\(P\_\{c\}\+R\_\{c\}\)=2\\,\\mathrm\{TP\}\_\{c\}/\(n\_\{c,\\mathrm\{pred\}\}\+n\_\{c,\\mathrm\{true\}\}\), the paired score is the macro\-F1F\_\{1\}at matched horizon positions, normalized by the number of present classes per pair and averaged over𝒯\\mathcal\{T\},

F1n=def𝔼\(S,S^\)​\[1\|𝒞\(S,S^\)n\|​∑c∈𝒞\(S,S^\)n2​TPcnnc,predn\+nc,truen\]\.F\_\{1\}^\{n\}\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\mathbb\{E\}\_\{\(S,\\hat\{S\}\)\}\\\!\\left\[\\frac\{1\}\{\|\\mathcal\{C\}\_\{\(S,\\hat\{S\}\)\}^\{n\}\|\}\\sum\_\{c\\in\\mathcal\{C\}\_\{\(S,\\hat\{S\}\)\}^\{n\}\}\\frac\{2\\,\\mathrm\{TP\}\_\{c\}^\{n\}\}\{n\_\{c,\\mathrm\{pred\}\}^\{n\}\+n\_\{c,\\mathrm\{true\}\}^\{n\}\}\\right\]\.The bincountsnc,prednn\_\{c,\\mathrm\{pred\}\}^\{n\},nc,truenn\_\{c,\\mathrm\{true\}\}^\{n\}and the present\-class set𝒞\(S,S^\)n\\mathcal\{C\}\_\{\(S,\\hat\{S\}\)\}^\{n\}are invariant under permutations of horizon positions; onlyTPcn\\mathrm\{TP\}\_\{c\}^\{n\}depends on the alignment\. Restricting the average to present classes coincides with standard macro\-F1F\_\{1\}when all global labels appear, but avoids inflating the normalizer in pairs where most classes are absent — a common situation for high\-cardinality features\. In particularF1n∈\[0,1\]F\_\{1\}^\{n\}\\in\[0,1\]\.

### C\.4Permutation\-dependent scores

Strict index\-by\-index comparison can punish a model that gets the right events but in a permuted order along the horizon\. Permutation\-dependent scores re\-label which true position is paired with each predicted position:σ\\sigmais a one\-to\-one alignment between predicted and true horizon slots, and the formulas below evaluateR1R\_\{1\}andF1F\_\{1\}after applying that alignment\.

We parameterize each per\-feature score by a per\-pair permutationσ\(S,S^\):ℋ→ℋ\\sigma\_\{\(S,\\hat\{S\}\)\}\\colon\\mathcal\{H\}\\to\\mathcal\{H\}that aligns predicted positions to true positions\. Substitutingxσ⁡\(i\)x\_\{\\sigma\(i\)\},tσ⁡\(i\)t\_\{\\sigma\(i\)\}, andkσ⁡\(i\)k\_\{\\sigma\(i\)\}forxix\_\{i\},tit\_\{i\}, andkik\_\{i\}in the paired definitions gives

R1m\(σ\)=def1−𝔼\(S,S^\)​\[∑i∈ℋ\|x^im−xσ⁡\(i\)m\|\]𝔼\(S,S^\)\[∑i∈ℋ\|xim−med\[x1:Hm\]\|\]R\_\{1\}^\{m\}\(\\sigma\)\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}1\-\\frac\{\\mathbb\{E\}\_\{\(S,\\hat\{S\}\)\}\\\!\\left\[\\sum\_\{i\\in\\mathcal\{H\}\}\|\\hat\{x\}\_\{i\}^\{m\}\-x\_\{\\sigma\(i\)\}^\{m\}\|\\right\]\}\{\\mathbb\{E\}\_\{\(S,\\hat\{S\}\)\}\\\!\\left\[\\sum\_\{i\\in\\mathcal\{H\}\}\|x\_\{i\}^\{m\}\-\\mathrm\{med\}\\left\[x\_\{1:H\}^\{m\}\\right\]\|\\right\]\}R1t\(σ\)=def1−𝔼\(S,S^\)​\[∑i∈ℋ\|t^i−tσ⁡\(i\)\|\]𝔼\(S,S^\)\[∑i∈ℋ\|ti−\(tH\+\(i−H\)med\[Δt1:H\]\)\|\]R\_\{1\}^\{t\}\(\\sigma\)\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}1\-\\frac\{\\mathbb\{E\}\_\{\(S,\\hat\{S\}\)\}\\\!\\left\[\\sum\_\{i\\in\\mathcal\{H\}\}\|\\hat\{t\}\_\{i\}\-t\_\{\\sigma\(i\)\}\|\\right\]\}\{\\mathbb\{E\}\_\{\(S,\\hat\{S\}\)\}\\\!\\left\[\\sum\_\{i\\in\\mathcal\{H\}\}\\left\|t\_\{i\}\-\\bigl\(t\_\{H\}\+\(i\-H\)\\mathrm\{med\}\\left\[\\Delta t\_\{1:H\}\\right\]\\bigr\)\\right\|\\right\]\}F1n\(σ\)=def𝔼\(S,S^\)\[1\|𝒞\(S,S^\)n\|∑c∈𝒞\(S,S^\)n2​TPcn​\(σ\)nc,predn\+nc,truen\],TPcn\(σ\)=def∑i∈ℋ𝟏\[k^in=c\]1\[kσ⁡\(i\)n=c\]\.F\_\{1\}^\{n\}\(\\sigma\)\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\mathbb\{E\}\_\{\(S,\\hat\{S\}\)\}\\\!\\left\[\\frac\{1\}\{\|\\mathcal\{C\}\_\{\(S,\\hat\{S\}\)\}^\{n\}\|\}\\sum\_\{c\\in\\mathcal\{C\}\_\{\(S,\\hat\{S\}\)\}^\{n\}\}\\frac\{2\\,\\mathrm\{TP\}\_\{c\}^\{n\}\(\\sigma\)\}\{n\_\{c,\\mathrm\{pred\}\}^\{n\}\+n\_\{c,\\mathrm\{true\}\}^\{n\}\}\\right\],\\quad\\mathrm\{TP\}\_\{c\}^\{n\}\(\\sigma\)\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\sum\_\{i\\in\\mathcal\{H\}\}\\mathbf\{1\}\[\\hat\{k\}\_\{i\}^\{n\}=c\]\\,\\mathbf\{1\}\[k\_\{\\sigma\(i\)\}^\{n\}=c\]\.The paired scores correspond toσ=id\\sigma=\\mathrm\{id\}\. Only the numerators depend onσ\\sigma: theR1R\_\{1\}baselines, the per\-class bincount denominators ofF1nF\_\{1\}^\{n\}, and the present\-class set𝒞\(S,S^\)n\\mathcal\{C\}\_\{\(S,\\hat\{S\}\)\}^\{n\}are all invariant under permutations of horizon positions\.

### C\.5Matched per\-feature scores

Matched per\-feature scores ask: if we could reorder the ground truth independently for time, for each numerical feature, and for each categorical feature, what is the best alignment for that feature alone? They are optimistic upper envelopes of the paired scores and isolate how informative each channel is under its own best reordering\.

The matched score of each feature is the supremum of itsσ\\sigma\-parameterized form over per\-pair permutations,

R1m,⋆=defmaxσ⁡R1m​\(σ\),R1t,⋆=defmaxσ⁡R1t​\(σ\),F1n,⋆=defmaxσ⁡F1n​\(σ\)\.R\_\{1\}^\{m,\\star\}\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\max\_\{\\sigma\}R\_\{1\}^\{m\}\(\\sigma\),\\quad R\_\{1\}^\{t,\\star\}\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\max\_\{\\sigma\}R\_\{1\}^\{t\}\(\\sigma\),\\quad F\_\{1\}^\{n,\\star\}\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\max\_\{\\sigma\}F\_\{1\}^\{n\}\(\\sigma\)\.Each maximum is attained by independently choosingσ\(S,S^\)\\sigma\_\{\(S,\\hat\{S\}\)\}for every pair, since the dataset\-level denominators areσ\\sigma\-independent and the numerators decompose into sums of per\-pair contributions \([C\.6](https://arxiv.org/html/2610.12586#A3.SS6)\)\. By construction every matched score is at least its paired counterpart\.

### C\.6Per\-pair decomposition and Hungarian optimization

For a fixed feature and pair\(S,S^\)\(S,\\hat\{S\}\), everyσ\\sigma\-dependent numerator above is a sum overi∈ℋi\\in\\mathcal\{H\}of a term that depends only on the assigned pair\(i,σ⁡\(i\)\)\(i,\\sigma\(i\)\)\. ForR1m​\(σ\)R\_\{1\}^\{m\}\(\\sigma\)andR1t​\(σ\)R\_\{1\}^\{t\}\(\\sigma\)this is immediate from their absolute\-value form\. ForF1n​\(σ\)F\_\{1\}^\{n\}\(\\sigma\), swapping the two summations and collapsing the indicator onccgives

∑c∈𝒞\(S,S^\)n2​TPcn​\(σ\)nc,predn\+nc,truen=∑i∈ℋ𝟏\[k^in=kσ⁡\(i\)n\]2nk^in,predn\+nk^in,truen,\\sum\_\{c\\in\\mathcal\{C\}\_\{\(S,\\hat\{S\}\)\}^\{n\}\}\\frac\{2\\,\\mathrm\{TP\}\_\{c\}^\{n\}\(\\sigma\)\}\{n\_\{c,\\mathrm\{pred\}\}^\{n\}\+n\_\{c,\\mathrm\{true\}\}^\{n\}\}=\\sum\_\{i\\in\\mathcal\{H\}\}\\mathbf\{1\}\[\\hat\{k\}\_\{i\}^\{n\}=k\_\{\\sigma\(i\)\}^\{n\}\]\\,\\frac\{2\}\{n\_\{\\hat\{k\}\_\{i\}^\{n\},\\mathrm\{pred\}\}^\{n\}\+n\_\{\\hat\{k\}\_\{i\}^\{n\},\\mathrm\{true\}\}^\{n\}\},which is again a per\-pair sum:nc,prednn\_\{c,\\mathrm\{pred\}\}^\{n\}andnc,truenn\_\{c,\\mathrm\{true\}\}^\{n\}areσ\\sigma\-independent, and on each matchk^in=kσ⁡\(i\)n\\hat\{k\}\_\{i\}^\{n\}=k\_\{\\sigma\(i\)\}^\{n\}\. Each per\-feature numerator is therefore∑iCd​\(i,σ⁡\(i\)\)\\sum\_\{i\}C\_\{d\}\(i,\\sigma\(i\)\)for a precomputableL×LL\\times Lcost matrixCdC\_\{d\}, and the per\-pair maximum is obtained exactly by the Hungarian algorithm\.

### C\.7TMS

TMS is the headline accuracy metric: it measures how well the full continuation \(time, all numerical features, all categorical features\) matches the truth when a single reordering of the horizon is chosen for everyone at once\.

TMS applies the same matching to the uniform average of all per\-feature scores under a single shared permutation,

TMS⁡\(𝒯\)=defmaxσ⁡11\+M\+N​\(R1t​\(σ\)\+∑m=1MR1m​\(σ\)\+∑n=1NF1n​\(σ\)\)\.\\mathrm\{TMS\}\(\\mathcal\{T\}\)\\stackrel\{\{\\scriptstyle\\text\{def\}\}\}\{\{=\}\}\\max\_\{\\sigma\}\\,\\frac\{1\}\{1\+M\+N\}\\\!\\left\(R\_\{1\}^\{t\}\(\\sigma\)\+\\sum\_\{m=1\}^\{M\}R\_\{1\}^\{m\}\(\\sigma\)\+\\sum\_\{n=1\}^\{N\}F\_\{1\}^\{n\}\(\\sigma\)\\right\)\.The optimum is again attained per\-pair, by Hungarian assignment on the feature\-averaged cost matrixC¯=11\+M\+N​∑dCd\\bar\{C\}=\\tfrac\{1\}\{1\+M\+N\}\\sum\_\{d\}C\_\{d\}\. Because the sameσ\\sigmaenters every term, TMS is a single matched score rather than an average of independently aligned per\-feature ones, and in general lies below the average of the matched per\-feature scoresR1m,⋆R\_\{1\}^\{m,\\star\},R1t,⋆R\_\{1\}^\{t,\\star\},F1n,⋆F\_\{1\}^\{n,\\star\}defined in[C\.5](https://arxiv.org/html/2610.12586#A3.SS5)\.

## Appendix DDistributional Realism Metrics

### D\.1Technical Details of cDFS

For each test history, we first take the two continuation tables produced by evaluation:orig\(real\) andgen\(generated\)\. We then build one binary detection dataset by labelingorigwithy=0y\{=\}0andgenwithy=1y\{=\}1, and only after that, splitting this combined table into train/validation/test with proportions0\.7/0\.15/0\.150\.7/0\.15/0\.15\. In all experiments, conditioning equals target length \(H′=LH^\{\\prime\}=L\), so each sequence is truncated to its last2​L2Levents \(LLcondition \+LLtarget\); train is used for fitting, validation for early stopping, and test for reporting metrics\. Features are preprocessed with the same schema as the source dataset \(time feature processing, numerical/categorical transforms, and optional log transform for amount\), then fed into a 2\-layer GRU \(hidden size 512, dropout 0\.1\) and classified by a linear head\. We run three seeds and report mean held\-outMulticlassAUROC\. The reported cDFS is

cDFS=2​\(1−AUROC¯\),\\mathrm\{cDFS\}=2\(1\-\\overline\{\\mathrm\{AUROC\}\}\),with higher values indicating better fooling \(harder real/generated discrimination given the same history\)\.

### D\.2Robustness of cDFS to the discriminator choice

cDFS relies on a learned discriminator, so we check how the metric depends on the discriminator’s architecture and capacity\. We recompute cDFS under eight configurations,\{GRU,Transformer\}×\{128\-1L,512\-1L,512\-2L,1024\-3L\}\\\{\\text\{GRU\},\\text\{Transformer\}\\\}\\times\\\{\\text\{128\-1L\},\\text\{512\-1L\},\\text\{512\-2L\},\\text\{1024\-3L\}\\\}\(hidden width–number of layers\), on all three datasets \(Tables[5](https://arxiv.org/html/2610.12586#A4.T5)–[7](https://arxiv.org/html/2610.12586#A4.T7)\)\. In all 24 dataset–configuration pairs, the ground truth scores≈1\{\\approx\}1and LDM is the most realistic model\. The values of intermediate methods do move \(e\.g\., HistSampler on Age: 0\.20–0\.83\)\. However, the relative ranking stays stable: Spearman’sρ\\rhobetween the rankings under each configuration and under GRU\-512\-2L, used in the paper, computed on the reported values \(ties at three decimals\),ranges from 0\.91 to 1\.00; values below 0\.95 come from reorderings among point predictors with cDFS at or below 0\.05, except one case\.

That case is Repeat on Age under GRU\-1024\-3L: 0\.705 instead of 0\.065 \(ρ=0\.91\\rho=0\.91, or 0\.98 without Repeat\), high on both axes of the trade\-off plot\. The detector identifies Repeat by the verbatim copy of the history \(Appendix[D\.3](https://arxiv.org/html/2610.12586#A4.SS3)\), and this pattern is easy to fit: all other configurations except Transformer\-1024\-3L separate Repeat almost perfectly on their training set \(train AUROC≥0\.97\\geq 0\.97\)\. GRU\-1024\-3L on Age reaches only 0\.73 even on the training set, so the high cDFS of Repeat reflects a detector that failed to fit, not realistic continuations\. The same holds for Transformer\-1024\-3L \(train AUROC 0\.76–0\.90 on Repeat\), which underfits on all datasets and raises almost all methods at once\. A larger discriminator is thus not a stronger one\.

GRU\-512\-2L, the configuration we use throughout the paper, catches the repetition pattern on all three datasets, which makes it both cheaper and more reliable than the largest configurations\.

MethodGRU128\-1LGRU512\-1LGRU512\-2L \(default\)GRU1024\-3LTr\.128\-1LTr\.512\-1LTr\.512\-2LTr\.1024\-3LGT1\.0001\.0001\.0000\.9961\.0001\.0001\.0001\.000Hist\.0\.1980\.2340\.2340\.2250\.7270\.7560\.7270\.831Mode0\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.000Rep\.0\.0310\.0370\.0650\.7050\.0080\.0030\.0020\.334AR0\.0000\.0000\.0000\.0000\.0000\.0000\.0010\.007MT0\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.069LDM0\.9910\.9750\.9660\.9760\.9960\.9930\.9820\.998DDM0\.1760\.2030\.2020\.2190\.4700\.5210\.4940\.709DEF0\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.036LDM\-WB0\.1870\.2090\.2240\.2350\.1480\.1660\.0970\.448Spearmanρ\\rhovs\. default1\.001\.001\.000\.910\.990\.990\.970\.96

Table 5:Age: cDFS under eight discriminator configurations, horizonL=32L=32\. Spearmanρ\\rhocompares the model ranking under each configuration with the ranking under GRU\-512\-2L\.MethodGRU128\-1LGRU512\-1LGRU512\-2L \(default\)GRU1024\-3LTr\.128\-1LTr\.512\-1LTr\.512\-2LTr\.1024\-3LGT1\.0001\.0001\.0001\.0001\.0001\.0001\.0001\.000Hist\.0\.4270\.4930\.4770\.4700\.6120\.6730\.5350\.871Mode0\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.000Rep\.0\.1770\.0670\.0500\.0540\.0070\.0080\.0010\.197AR0\.0010\.0020\.0020\.0010\.0010\.0000\.0000\.001MT0\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.000LDM0\.9010\.9300\.8740\.9360\.8340\.8490\.8310\.884DDM0\.0030\.0020\.0020\.0030\.0040\.0070\.0030\.007DEF0\.0030\.0020\.0020\.0010\.0010\.0020\.0010\.211LDM\-WB0\.2350\.2850\.2580\.2520\.1400\.2140\.1200\.341Spearmanρ\\rhovs\. default0\.991\.001\.000\.990\.990\.970\.930\.96

Table 6:Gender: cDFS under eight discriminator configurations, horizonL=32L=32\. Spearmanρ\\rhocompares the model ranking under each configuration with the ranking under GRU\-512\-2L\.MethodGRU128\-1LGRU512\-1LGRU512\-2L \(default\)GRU1024\-3LTr\.128\-1LTr\.512\-1LTr\.512\-2LTr\.1024\-3LGT1\.0001\.0001\.0001\.0001\.0001\.0001\.0001\.000Hist\.0\.5730\.5560\.5440\.5500\.3380\.3130\.1470\.874Mode0\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.000Rep\.0\.0150\.0110\.0090\.0060\.0000\.0020\.0000\.494AR0\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.000MT0\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.000LDM0\.8670\.8660\.8550\.8320\.8100\.8360\.8110\.929DDM0\.0040\.0040\.0040\.0050\.0050\.0080\.0050\.021DEF0\.0000\.0000\.0000\.0000\.0000\.0000\.0000\.203LDM\-WB0\.1610\.1600\.1750\.1780\.1110\.1780\.0500\.606Spearmanρ\\rhovs\. default1\.001\.001\.001\.000\.930\.990\.930\.95

Table 7:Alphabattle: cDFS under eight discriminator configurations, horizonL=32L=32\. Spearmanρ\\rhocompares the model ranking under each configuration with the ranking under GRU\-512\-2L\.
### D\.3Shape and Trend scores

As auxiliary distributional diagnostics, we report Shape and Trend from[Zhang et al\. \[2024\]](https://arxiv.org/html/2610.12586#bib.bib4), computed via thesdmetricsQualityReporton the pool of predicted and real continuations in the evaluation set\. Shape aggregates Kolmogorov–Smirnov distances \(for numerical features\) and total\-variation distances \(for categorical features\) between the marginal distributions of each generated and real feature\. Trend aggregates disagreements between pairwise feature dependencies \(correlations for numerical pairs, contingency\-table distances for categorical pairs\)\. Both scores are bounded in\[0,1\]\[0,1\], with11indicating perfect agreement\. In contrast to cDFS, Shape and Trend ignore event ordering and history, so they quantify only marginal and low\-order joint fidelity of features pooled across sequences\. We include them as a sanity check on per\-feature distribution quality rather than as a primary realism measure\. Table[8](https://arxiv.org/html/2610.12586#A4.T8)reports Shape and Trend alongside cDFS for all models at horizonL=32L=32\. Within the cluster that cDFS maps to near zero, the discriminator\-free Shape and Trend scores still separate models by a large margin \(e\.g\., on Age: 0\.79/0\.40 for Mode vs\. 0\.90/0\.61 for DEF vs\. 0\.99/0\.97 for Repeat\), providing a graded signal where the detector saturates\. Conversely, Repeat shows that near\-perfect Shape and Trend do not imply realism: the detector identifies its continuations almost solely through the verbatim repetition of past events\. The relationship between cDFS and the two scores is analyzed in Appendix[E\.1](https://arxiv.org/html/2610.12586#A5.SS1)\.

cDFS↑\\uparrowShape↑\\uparrowTrend↑\\uparrowMethodAgeGenderAlph\.AgeGenderAlph\.AgeGenderAlph\.Hist\.0\.2340\.4760\.5440\.9870\.9620\.9840\.9610\.7850\.949Mode0\.0000\.0000\.0000\.7920\.8060\.8010\.4020\.7000\.567Rep\.0\.0650\.0500\.0090\.9940\.9780\.9910\.9730\.8790\.955AR0\.0000\.0020\.0000\.8520\.8510\.8400\.5380\.6420\.620MT0\.0000\.0000\.0000\.8530\.8110\.8550\.4010\.2720\.712LDM1\.0000\.9070\.8560\.9890\.9600\.9770\.9500\.9260\.914DDM0\.2020\.0020\.0040\.9660\.6950\.7400\.8880\.5670\.603DEF0\.0000\.0020\.0000\.9020\.8760\.8840\.6070\.7040\.563LDM\-WB0\.2240\.2560\.1770\.9520\.9380\.9430\.8010\.8910\.868DDM\-WB0\.1020\.0010\.0010\.9410\.7090\.7330\.7110\.6130\.579

Table 8:Shape and Trend scores alongside cDFS at horizonL=32L=32on the three transaction datasets\. Bold marks cDFS\>0\.07\>0\.07and the corresponding Shape and Trend cells\.

## Appendix EAdditional Experiments

### E\.1Replacing cDFS with simpler diagnostics\.

![Refer to caption](https://arxiv.org/html/2610.12586v1/figures/plot_shape_trend_32.png)Figure 3:Relationship between cDFS and the auxiliary fidelity scores — Shape score \(left\) and Trend score \(right\) — across all three datasets pooled\. HorizonL=32L=32\. Higher is better on both axes\. The figure shows that high Shape and Trend scores do not, by themselves, imply a high cDFS: many baselines achieve competitive Shape or Trend values while collapsing along the cDFS axis, suggesting that cDFS is a more holistic criterion\.cDFS is computationally heavier than per\-feature distributional scores — it requires training and evaluating a discriminator — so it is natural to ask whether the cheaper Shape and Trend statistics from[Zhang et al\. \[2024\]](https://arxiv.org/html/2610.12586#bib.bib4)suffice as a substitute\. Figure[3](https://arxiv.org/html/2610.12586#A5.F3)pools predictions from all three datasets atL=32L=32and plots cDFS against each of the two scores\. The relationship is strikingly asymmetric: a high Shape or Trend score implies nothing about cDFS, but a high cDFS does require high Shape and Trend\. In other words, Shape and Trend agreement is necessary but far from sufficient for distributional realism\. The reason is structural\. Shape and Trend pool features across all sequence positions, ignoring event order, conditioning on history, and joint dependencies of more than two features\. A model that produces marginally and pairwise plausible rows but mis\-orders them in time, or generates implausible continuations of a given history, can score well on Shape and Trend while being trivially detected by cDFS\. Shape and Trend are useful auxiliary diagnostics, but cDFS captures sequential and conditional structure that they miss, and is therefore the appropriate primary realism criterion for event\-sequence forecasting\.

### E\.2Wasserstein BaryBooster Independent Evaluation

In the main text \(§[6](https://arxiv.org/html/2610.12586#S6)\), we show that BaryBoosted diffusion models achieve SoTA TMS values\. We conclude that diffusion models can produce accurate predictions with the proposed technique\. However, the algorithm is designed to optimize the same metric used for its evaluation, potentially leading to biased results\. In this section, we support the broad claim of high prediction accuracy by reporting independent metrics that BaryBooster does not optimize directly\.

Table[9](https://arxiv.org/html/2610.12586#A5.T9)shows that the application of Wasserstein BaryBooster on top of LDM improves all prediction accuracy metrics consistently across datasets\. Despite its poor realism, DDM also shows improved accuracy after BaryBooster application, with a few exceptions\. These observations support our broad claim of improved prediction accuracy\.

Table 9:TMS, R1 \(amount\), R1 \(time\), F1 \(mcc\) at horizonL=32L=32\. Across all shown methods, the best is inbold, the second best isunderlinedand the third best isdash\-underlined\.MethodTMS ↑R1 \(amount\) ↑R1 \(time\) ↑F1 \(mcc\) ↑AlphabattleGenderAgeAlphabattleGenderAgeAlphabattleGenderAgeAlphabattleGenderAgeHist\.0\.3820\.2210\.3660\.4980\.1170\.245\-0\.049\-0\.325\-0\.1790\.4240\.3970\.331Mode0\.2030\.0470\.2630\.0000\.0000\.0000\.0000\.0000\.0000\.0810\.0880\.052Rep\.0\.4400\.3610\.4540\.5240\.1820\.2360\.1470\.1370\.1280\.4440\.4250\.374AR0\.2860\.1880\.3540\.043\-0\.0180\.0820\.0640\.0960\.1830\.2870\.3240\.208MT0\.2860\.1530\.360\-0\.0000\.0160\.020\-0\.042\-0\.0430\.1850\.3690\.3430\.270LDM0\.4050\.3180\.4260\.4790\.1800\.2160\.1480\.1440\.0740\.4000\.3640\.323DDM0\.2510\.1070\.441\-0\.0150\.0590\.3010\.1250\.0710\.2030\.2750\.2380\.299DEF0\.4240\.4130\.5080\.3640\.2870\.3310\.2940\.3240\.3530\.4400\.4130\.360LDM\-WB \(ours\)0\.4950\.4360\.5220\.5830\.3340\.4380\.3370\.3310\.3310\.4590\.4480\.371DDM\-WB \(ours\)0\.2740\.1360\.4800\.0860\.0700\.3730\.0060\.0470\.2400\.2850\.2640\.358

### E\.3TMS and matched submetrics

![Refer to caption](https://arxiv.org/html/2610.12586v1/figures/plot_r1_f1_compare_32.png)Figure 4:Relationship between TMS and matched per\-feature submetrics at horizonL=32L=32: matchedR1R\_\{1\}for amount \(left\), matchedR1R\_\{1\}for time \(middle\), and matchedF1F\_\{1\}for MCC \(right\)\. Higher is better on both axes\. Across all three views, TMS grows consistently with each matched submetric\.Figure[4](https://arxiv.org/html/2610.12586#A5.F4)shows a clear positive correlation between TMS and matchedR1R\_\{1\}for amount, matchedR1R\_\{1\}for time, and matchedF1F\_\{1\}for MCC\. For clarity, these matched submetrics are*separately computed*feature\-wise metrics \(R1m,⋆R\_\{1\}^\{m,\\star\},R1t,⋆R\_\{1\}^\{t,\\star\},F1n,⋆F\_\{1\}^\{n,\\star\}; Appendix[C\.5](https://arxiv.org/html/2610.12586#A3.SS5)\), each using its own best permutation, rather than direct components of TMS\. TMS instead uses a*single shared*permutation across all features \(Appendix[C\.7](https://arxiv.org/html/2610.12586#A3.SS7)\); the observed correlation therefore indicates that TMS aligns well with common, easy\-to\-interpret per\-feature metrics and that TMS gains reflect real quality improvements across forecasting dimensions\.

### E\.4Other Horizons

In this section, we present experimental results with horizonsL∈\{4,16,64\}L\\in\\\{4,16,64\\\}\(Figures[5](https://arxiv.org/html/2610.12586#A5.F5),[6](https://arxiv.org/html/2610.12586#A5.F6), and[7](https://arxiv.org/html/2610.12586#A5.F7)\)\.

![Refer to caption](https://arxiv.org/html/2610.12586v1/figures/plot_main_4.png)Figure 5:Trade\-off between TMS and cDFS for each dataset on horizonL=4L=4\. The solid line shows the Pareto front over baseline methods\.![Refer to caption](https://arxiv.org/html/2610.12586v1/figures/plot_main_16.png)Figure 6:Trade\-off between TMS and cDFS for each dataset on horizonL=16L=16\. The solid line shows the Pareto front over baseline methods\.![Refer to caption](https://arxiv.org/html/2610.12586v1/figures/plot_main_64.png)Figure 7:Trade\-off between TMS and cDFS for each dataset on horizonL=64L=64\. The solid line shows the Pareto front over baseline methods\.AtL=64L=64, Repeat receives a high cDFS on Gender and Age \(0\.73 and 0\.83\)\. As in Appendix[D\.2](https://arxiv.org/html/2610.12586#A4.SS2), we attribute this to the detector not catching the verbatim copy of the history at this horizon, not to realistic continuations, so the cDFS of Repeat atL=64L=64should be read with caution\. The rest of the picture matchesL=32L=32: LDM is the most realistic model, and LDM\-WB has the highest TMS on all datasets\.

### E\.5Statistical Significance

Table 10:Age, horizonL=32L=32: 3\-seed mean±\\pmstd vs\. reported in main results \(single reported values shown without uncertainty\)\. Higher is better\. Within each column, the best is inbold, second best isunderlinedMethodTMS ↑cDFS ↑3\-seedreported3\-seedreportedLDM0\.430±\\pm0\.0020\.4260\.983±\\pm0\.0061\.000DEF0\.505±\\pm0\.0040\.5080\.001±\\pm0\.0000\.000LDM\-WB0\.523±\\pm0\.0010\.5220\.202±\\pm0\.0050\.224In this experiment, we show that the metrics are stable across runs and justify our choice not to run each experiment multiple times\. We report mean and standard deviations over three random seeds in Table[10](https://arxiv.org/html/2610.12586#A5.T10)for the spotlight methods\. We draw our conclusions on metric gaps that are much larger than standard deviations across runs, so we do not question the statistical significance\. One entry of Table[10](https://arxiv.org/html/2610.12586#A5.T10)deserves a comment: the reported cDFS of LDM\-WB \(0\.224\) lies several across\-seed standard deviations away from the 3\-seed mean \(0\.202±0\.0050\.202\\pm 0\.005\)\. The across\-seed deviation, however, is only one component of the variance: each cDFS value is itself the mean of a 3\-fold cross\-validation of the detector, with an internal standard deviation of≈0\.02\{\\approx\}0\.02\(0\.024 for LDM\-WB\), so the gap is within one standard deviation of the full estimate\. The detector is inherently noisier in the intermediate score range, while the discriminator\-free Shape and Trend scores are stable there\.

### E\.6Qualitative Samples

Figure[8](https://arxiv.org/html/2610.12586#A5.F8)shows generated continuations for one Age test client at horizonL=32L=32\. All methods are conditioned on the identical history, so the panels differ only in what each model generates\.

Several failure modes are directly visible\. MT and AR keep a plausible mix of categories, but their numerical features collapse: the amounts sit at a nearly constant value and the inter\-arrival times form an almost uniform grid — the conditional\-mean pathology of Figure[1](https://arxiv.org/html/2610.12586#S1.F1)\. Mode degenerates into a single repeated event, and Repeat copies the history verbatim\. LDM is the only model whose continuation reproduces both the irregular timing and the amount variability of the real one; HistSampler matches the marginals but places events at implausible times\.

Figure 8:Generated continuations for one Age test client at horizonL=32L=32\. Each event is a stem at its timestamp; height is the log\-transformed amount and color encodes the category \(the nine most frequent categories receive their own color, the rest share grey\)\. The vertical rule separates the observed history, identical across all panels, from the 32 generated events\.*Original*is the ground\-truth continuation\.
### E\.7Comparison of LDM with ADiff4TPP

Table 11:ADiff4TPP and LDM at horizonL=32L=32on Age dataset, with and without Wasserstein BaryBooster\.MethodTMS ↑cDFS ↑ADiff4TPP0\.4020\.997LDM0\.4261\.000ADiff4TPP\-WB0\.5120\.108LDM\-WB0\.5220\.224Table[11](https://arxiv.org/html/2610.12586#A5.T11)shows that our implementation of the latent diffusion model performs on par with or slightly better than ADiff4TPP\[[Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\], so we do not include it as a separate model in the comparison\.

## Appendix FWasserstein BaryBooster: optimization details

We solve equation \([2](https://arxiv.org/html/2610.12586#S5.E2)\) by alternating minimization \(Algorithm[1](https://arxiv.org/html/2610.12586#alg1)\), similar to consensus decoding for event sequences\[[Mei et al\., 2019](https://arxiv.org/html/2610.12586#bib.bib31)\]\. WithS^\\hat\{S\}fixed, the optimal alignment to eachS^n\\hat\{S\}\_\{n\}is recovered exactly by Hungarian matching \(Appendix[C\.6](https://arxiv.org/html/2610.12586#A3.SS6)\)\. With the alignments fixed, each feature ofS^\\hat\{S\}admits a closed\-form coordinate\-wise update: time and numerical features are set to the median of the aligned values \(the Fréchet mean under the absolute\-deviation cost of[C\.2](https://arxiv.org/html/2610.12586#A3.SS2)\); categorical features are set to the mode \(the Fréchet mean under the0/10/1cost of[C\.3](https://arxiv.org/html/2610.12586#A3.SS3)\)\. We initializeS^\\hat\{S\}with the medoid of\{S^n\}n=1N\\\{\\hat\{S\}\_\{n\}\\\}\_\{n=1\}^\{N\}, the sample minimizing the sum of pairwise TMS costs to the others\. For categorical features, the matching uses the0/10/1cost as a surrogate of the F1 term, so the procedure optimizes this surrogate objective and finds a local solution\. Each iteration weakly decreases the objective, and the alignment set is finite, so the procedure terminates in finitely many steps\.

Algorithm 1Wasserstein BaryBooster1:base generator

GG, history

SS, sample count

NN, max iterations

TT
2:sample

S^n∼G\(⋅∣S\)\\hat\{S\}\_\{n\}\\sim G\(\\cdot\\mid S\)for

n=1,…,Nn=1,\\ldots,N
3:

n⋆←arg⁡min⁡∑m=1Nn∈\[N\]⁡dTMS​\(S^n,S^m\)n^\{\\star\}\\leftarrow\\arg\\min\_\{n\\in\[N\]\}\\sum\_\{m=1\}^\{N\}d\_\{\\mathrm\{TMS\}\}\\\!\\left\(\\hat\{S\}\_\{n\},\\hat\{S\}\_\{m\}\\right\);

S^←S^n⋆\\hat\{S\}\\leftarrow\\hat\{S\}\_\{n^\{\\star\}\}⊳\\trianglerightmedoid init

4:for

t=1,…,Tt=1,\\ldots,Tdo

5:for

n=1,…,Nn=1,\\ldots,Ndo

6:

σn←arg⁡min⁡∑i∈ℋσ⁡C¯\(S^,S^n\)​\(i,σ⁡\(i\)\)\\sigma\_\{n\}\\leftarrow\\arg\\min\_\{\\sigma\}\\sum\_\{i\\in\\mathcal\{H\}\}\\bar\{C\}\_\{\(\\hat\{S\},\\,\\hat\{S\}\_\{n\}\)\}\(i,\\sigma\(i\)\)⊳\\trianglerightHungarian

7:endfor

8:if

t\>1t\>1and

σn=σnprev\\sigma\_\{n\}=\\sigma\_\{n\}^\{\\mathrm\{prev\}\}for all

n∈\[N\]n\\in\[N\]then

9:break

10:endif

11:

σnprev←σn\\sigma\_\{n\}^\{\\mathrm\{prev\}\}\\leftarrow\\sigma\_\{n\}for all

n∈\[N\]n\\in\[N\]
12:for

i∈ℋi\\in\\mathcal\{H\}do

13:

t^i←med\{tσn​\(i\)\(n\):n=1,…,N\}\\hat\{t\}\_\{i\}\\leftarrow\\mathrm\{med\}\\bigl\\\{t^\{\(n\)\}\_\{\\sigma\_\{n\}\(i\)\}:n=1,\\ldots,N\\bigr\\\}
14:for

m=1,…,Mm=1,\\ldots,Mdo

15:

x^im←med\{xσn​\(i\)m,\(n\):n=1,…,N\}\\hat\{x\}\_\{i\}^\{m\}\\leftarrow\\mathrm\{med\}\\bigl\\\{x^\{m,\(n\)\}\_\{\\sigma\_\{n\}\(i\)\}:n=1,\\ldots,N\\bigr\\\}
16:endfor

17:for

q=1,…,Ncatq=1,\\ldots,N\_\{\\mathrm\{cat\}\}do

18:

k^iq←mode\{kσn​\(i\)q,\(n\):n=1,…,N\}\\hat\{k\}\_\{i\}^\{q\}\\leftarrow\\operatorname\{mode\}\\bigl\\\{k^\{q,\(n\)\}\_\{\\sigma\_\{n\}\(i\)\}:n=1,\\ldots,N\\bigr\\\}
19:endfor

20:endfor

21:endfor

22:return

S^\\hat\{S\}

### F\.1Effect of the sample sizeNN

Table[12](https://arxiv.org/html/2610.12586#A6.T12)reports LDM\-WB at horizonL=32L=32as a function of the number of aggregated samplesNN;N=1N=1corresponds to the base LDM without boosting\. AlreadyN=5N=5recovers about75%75\\%of the full TMS gain at one sixth of the sampling cost,N=10N=10recovers about90%90\\%, and the curve saturates afterN≈10N\\approx 10–1515\.

AgeGenderAlphabattleNNTMS↑\\uparrowcDFS↑\\uparrowTMS↑\\uparrowcDFS↑\\uparrowTMS↑\\uparrowcDFS↑\\uparrow10\.4280\.9660\.3190\.8740\.4040\.85550\.4990\.2570\.4070\.1370\.4720\.133100\.5100\.2710\.4250\.3000\.4850\.210150\.5170\.2290\.4300\.2710\.4910\.188300\.5220\.2240\.4360\.2580\.4950\.177Table 12:Wasserstein BaryBooster on LDM as a function of the sample sizeNNat horizonL=32L=32;N=1N\{=\}1is the base LDM without boosting\.

## Appendix GWasserstein BaryBooster on remaining models

Figure[9](https://arxiv.org/html/2610.12586#A7.F9)confirms that BaryBooster is most effective when the base generator is diffusion\-based: the boost is large for diffusion models, absent for DEF, and small for the other non\-diffusion methods\.

Figure 9:TMS change at horizonL=32L=32after applying Wasserstein BaryBooster to different base models\. The largest improvement is observed for diffusion\-based models, while DEF shows no visible gain and the remaining methods improve only slightly\.
## Appendix HPreprocessing Details

Across all datasets, we applylog1pto the transaction amount\. We also split the data into three subsets: “train” for neural network training, “val” for early stopping, and “test” for reporting the metrics\. The test split is out\-of\-sample, and the validation split is randomly sampled from the training data\. During training, we apply a diff transform to the time feature, but we restore it back to absolute time for metrics calculation\.

#### Split strategy\.

Ageuses a stratified user split by bins with a20%20\\%test fraction\.Genderuses a random user split with a20%20\\%test fraction\.Alphabattleuses separate raw train/test populations, with the test population strictly later than the train population\.

#### Dataset\-specific filtering\.

Genderkeeps only sequences with length in\[20,1500\)\[20,1500\), maximumhour\_diffat most100100, and medianhour\_diffat most1010\.Alphabattlekeeps only sequences with length in\[128,1200\]\[128,1200\], maximumhour\_diffbelow800800, medianhour\_diffbelow7070, drops clients with rarepayment\_system,mcc\_category, andcard\_typevalues, restricts tocurrency = 1, and is subsampled to 100K users for computational efficiency\.

#### History–continuation cutoffs\.

During training and validation we sample a cutoff time uniformly at random over an interval of admissible times for each sequence; events strictly before the cutoff form the history, and the nextLLevents form the target continuation\. The same cutoff protocol applies to training, validation, and evaluation; during training a new cutoff is sampled every epoch, whereas validation and test use fixed cutoffs per sequence for the entire study\. For validation and test we use one fixed cutoff per sequence onAgeandAlphabattle\. OnGender\(our smallest dataset\) we instead use three fixed cutoffs per sequence, spreading evaluation contexts to reduce metric variance\. After sampling cutoffs, we keep only sequences with at leastLLevents on the history side\. Using random cutoffs during training lets the model forecast from diverse time periods rather than repeatedly from the same calendar segment\.

## Appendix IImplementation Details

This appendix records implementation and training details omitted from §[3](https://arxiv.org/html/2610.12586#S3), including optimization and validation protocols, shared backbones, diffusion schedules, and non\-learned baselines\.

### I\.1Comparison of LDM and ADiff4TPP\[[Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\]

Table 13:High\-level comparison between ADiff4TPP and our latent diffusion model \(LDM\)\.AspectADiff4TPP\[[Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\]Our LDMDiffusion formulationFlow matchingEDMBackboneDiTDiTData–latent mappingβ\\beta\-VAEβ\\beta\-VAEIn the original ADiff4TPP setup, forecasting is done with an*unconditional*model\. For an apples\-to\-apples empirical comparison, we train*conditional*variants for both our LDM and our re\-implementation of ADiff4TPP\. Additionally, we apply classifier\-free guidance \(CFG\) to our LDM at inference time \(§[I\.3](https://arxiv.org/html/2610.12586#A9.SS3)\)\.

### I\.2Schematic Illustration of Methods

\(a\)Latent Diffusion Model \(LDM\)\(b\)Data Diffusion Model \(DDM\)\(c\)DEF
Figure 10:Schematic illustration of the considered methods\.Figure[10](https://arxiv.org/html/2610.12586#A9.F10)illustrates the three adapted methods: LDM, DDM, and DEF\.

### I\.3Shared protocol and infrastructure

#### Classifier\-free guidance \(CFG\)\.

CFG is a well\-established and popular tool\[[Ho and Salimans, 2022](https://arxiv.org/html/2610.12586#bib.bib21)\]which allows adjusting \(conditional\) diffusion models ondiversity\-fidelitycoordinate plane, enabling practitioners to balance sample diversity against generation fidelity by modulating the strength \(w≥0w\\geq 0\) of the conditioning signal during inference\. Technically speaking, CFG prescribes simultaneous training of conditional and unconditional denoising models \(parameterized as a single model, unconditional training appears by substituting the actual class with the “empty” class∅\\emptysetwith probabilitypuncondp\_\{\\text\{uncond\}\}\) and subsequentguidanceof the usual conditional model with extra conditioning at inference stage:

DenoiserCFG​\(data,condition\)=\(1\+w\)​Denoiser​\(data,condition\)−w​Denoiser​\(data,∅\)\.\\displaystyle\\text\{Denoiser\}\_\{\\text\{CFG\}\}\(\\text\{data\},\\text\{condition\}\)=\(1\+w\)\\text\{Denoiser\}\(\\text\{data\},\\text\{condition\}\)\-w\\text\{Denoiser\}\(\\text\{data\},\\emptyset\)\.This technique was found to be promising for event sequence generation\[[Garuti et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib11)\], and we adopt it for our LDM and DDM models\.

#### Framework and reproducibility\.

All neural models are implemented in PyTorch\. Unless stated otherwise below, training and evaluation follow a single shared protocol\. Randomness is controlled with a single global seed\.

#### Optimization\.

We optimize with AdamW \(weight decay0\.010\.01\), clip gradients at global norm11, and search the learning rate on a logarithmic grid from10−510^\{\-5\}to3×10−33\\times 10^\{\-3\}\. Training may run for up to one million iterations with early stopping when validation loss plateaus \(patience2020epochs, or1010forAlphabattle\)\. For latent diffusion only, we apply exponential smoothing to the validation loss before comparing it to the running optimum\.

#### Weight tracking and batching\.

We maintain an exponential moving average of model parameters with decay0\.99990\.9999, applied every ten steps after70,00070\{,\}000training steps\. Unless noted, the batch size is128128;DEFandARuse3232because of memory limits\.

#### Sequences, time, and splits\.

OnGenderandAlphabattle, sequences are truncated or padded to at most512512events per forward pass; onAgethe cap is10241024\. Timestamps are represented as successive inter\-event differences during training and mapped back to absolute time before metric computation\. After removing the held\-out test users \(Appendix[H](https://arxiv.org/html/2610.12586#A8)\), we randomly assign15%15\\%of the remaining users to validation for model selection and early stopping\.

#### Compute and inference calibration\.

Each model is trained on a single NVIDIA A100 80GB GPU\. Hyperparameter grids, detector ablations, and other horizons require additional compute beyond the final runs in Appendix[J](https://arxiv.org/html/2610.12586#A10)\. Validation loss often correlates only weakly with forecast metrics, so we select hyperparameters on the validation set by a common rule: take the configuration with the highest cDFS, or the highest TMS when cDFS differs only slightly or is low for all candidates, which is the case for most models \(§[6](https://arxiv.org/html/2610.12586#S6)\)\. We apply this rule first to the learning rate, then to the sampling knob at inference: temperature forARandMT, and classifier\-free guidance scale forLDMandDDM\. Exceptions: forARandLDM, the learning rate is chosen mainly by validation loss; for the guidance scale ofLDM, we take the highest TMS that keeps cDFS close to its best value; for the VAE ofLDM, we take the checkpoint with the lowest KL among the two with the best reconstruction\.

#### Classifier\-free guidance \(LDMandDDM\)\.

Both diffusion models use classifier\-free guidance\[[Ho and Salimans, 2022](https://arxiv.org/html/2610.12586#bib.bib21)\]\. During training, conditioning is dropped with probabilitypuncond=0\.1p\_\{\\mathrm\{uncond\}\}=0\.1so the score network sees unconditional targets as well as conditional ones\. At inference we grid\-search the guidance scale over\{0,1,1\.2,1\.3,1\.4,1\.5,2,3,4,5\}\\\{0,1,1\.2,1\.3,1\.4,1\.5,2,3,4,5\\\}as described above\.

#### Feature representations and losses\.

Categorical attributes use embedding vectors of dimension88; numerical attributes are projected linearly to dimension88\. Reconstruction losses combine mean absolute error on numerical features with cross\-entropy on categorical features\.

#### Shared transformer hyperparameters\.

AR,MT,LDM, andDDMuse the same transformer trunk:66layers,88attention heads, hidden size512512, dropout probability0\.10\.1, and feed\-forward expansion factor44\.

#### GPT\-style baselines\.

ARfollows a GPT\-2\-style causal decoder; our code is adapted from thenanoGPTreference implementation\.555[https://github\.com/karpathy/nanoGPT/blob/master/model\.py](https://github.com/karpathy/nanoGPT/blob/master/model.py)Unlike the other sequence models,ARis trained without sampling a random history endpoint; random cutoffs are used only at evaluation\.MTreuses the same trunk but is a*non\-autoregressive*multi\-step head: it pools the final hidden state and applies a linear projection to obtainLLparallel next\-event predictions in one pass \(no autoregressive roll\-out over future time steps\)\.

#### β\\beta\-VAE encoder \(TabSyn\-style\)\.

Events are encoded independently with the variational architecture from TabSyn\[[Zhang et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib4)\]:33layers,11attention head per feature, feature embedding dimension88, and feed\-forward expansion factor3232\. The KL weight follows a scheduledβ\\beta: initial value0\.010\.01, multiplicative decay factor0\.70\.7, patience five validation checks, and floor10−510^\{\-5\}\.

#### DEF\.

DEF\[[Karpukhin and Savchenko, 2026](https://arxiv.org/html/2610.12586#bib.bib3)\]; we adapt the publicly released implementation\.666[https://github\.com/ivan\-chai/hotpp\-benchmark](https://github.com/ivan-chai/hotpp-benchmark)The history encoder is a single\-layer GRU with hidden width512512, following the reference training setup\. We use a loss subset ratio of0\.050\.05, matching weights\{\_presence:4\}\\\{\\texttt\{\\\_presence\}:4\\\}, andkfactor=3k\_\{\\mathrm\{factor\}\}=3as recommended in the original work\. Where the baseline method applies a presence head to keep events inside a prescribed forecast horizon, we instead use its scores to retain the top\-LLcandidates\. Our implementation predicts the full heterogeneous feature vector at each retained event, rather than only the time and a single mark\.

#### DDM\.

We run the reverse process for500500diffusion steps\. The history is encoded with a one\-layer GRU \(hidden width512512\)\. We follow the original implementation of[Zeng et al\. \[2024\]](https://arxiv.org/html/2610.12586#bib.bib1)except that we disable the multiple random time draws used in the original training recipe and keep a single sampling schedule\. The original work also aggregates five forecasts at inference, averaging inter\-arrival times and taking a majority vote over marks\. We drop this step so thatDDMmeasures the generative model alone, and treat sample aggregation as a separate axis evaluated through the Wasserstein BaryBooster \(§[5](https://arxiv.org/html/2610.12586#S5)\);DDM\-WBis the aggregated variant\. Averaging and majority voting also do not extend to the numerical features of general event sequences, whereas the barycentric aggregation does\.

#### LDM\.

We use5050diffusion steps in the latent space\. The denoiser follows a Diffusion Transformer \(DiT\) layout as in ADiff4TPP\[[Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\], trained with the elucidated diffusion objective of[Karras et al\. \[2022\]](https://arxiv.org/html/2610.12586#bib.bib17)\. A one\-layer GRU \(hidden width512512\) produces the history embedding consumed by the transformer\. Inside the denoiser, we further condition on theβ\\beta\-VAE latents of the lastLLhistory events by prepending them to the noised latent sequence being denoised\.

## Appendix JRuntime and memory consumption

This appendix summarizes training time, inference time, and post\-evaluation GPU memory usage for the main learned models at horizonL=64L=64\. Memory is reported in GB\. Inference times were measured under a single controlled protocol: the full test split, one job per GPU\. Training times were logged on a shared cluster where a GPU could host several concurrent experiments; they should be read as upper bounds, with the relative comparison remaining fair since all models were trained under the same scheduling conditions\.

DatasetModelTrain \(min\)Inference \(min\)Memory \(GB\)AgeLDM525\.61\.21\.71AgeDDM776\.98\.01\.62AgeDEF281\.00\.1371\.12AgeMT111\.90\.083\.94AgeAR83\.12\.47\.43AgeLDM\-WB \(N=30N\{=\}30\)–52\.41\.71GenderLDM219\.70\.91\.08GenderDDM94\.67\.01\.22GenderDEF84\.30\.0836\.84GenderMT14\.40\.051\.93GenderAR14\.71\.03\.94GenderLDM\-WB \(N=30N\{=\}30\)–44\.01\.08AlphabattleLDM1844\.35\.43\.35AlphabattleDDM807\.434\.01\.25AlphabattleDEF473\.00\.436\.68AlphabattleMT116\.30\.21\.94AlphabattleAR83\.65\.53\.95AlphabattleLDM\-WB \(N=30N\{=\}30\)–243\.73\.35Table 14:Runtime and memory summary for Age, Gender, and Alphabattle at horizonL=64L=64\.Table[15](https://arxiv.org/html/2610.12586#A10.T15)decomposes the BaryBooster wall time \(N=30N=30, 10 test batches per dataset\) into the base\-model sampling and the stages of the algorithm itself \(medoid selection, Hungarian matching, aggregation\)\. Sampling dominates at both horizons \(≈85%\{\\approx\}85\\%atL=32L=32,≈62%\{\\approx\}62\\%atL=64L=64\); the share of the matching grows with the horizon, since the Hungarian step scales faster with the continuation length than sampling does\. Note that theNNcontinuations are sampled independently, so the dominant stage can be distributed across several GPUs, reducing the time of this stage almost proportionally\.

DatasetHorizonTotal \(s\)SamplingMedoidMatchingAggregationAge32310\.0264\.9 \(85%\)17\.826\.20\.3Gender32307\.2261\.8 \(85%\)18\.026\.40\.3Alphabattle32349\.9301\.6 \(86%\)18\.229\.00\.3Age64788\.7484\.8 \(61%\)108\.0194\.80\.2Gender64786\.3480\.6 \(61%\)107\.5197\.30\.2Alphabattle64861\.5547\.2 \(64%\)105\.6207\.70\.3Table 15:Decomposition of the BaryBooster wall time into base\-model sampling and the stages of the aggregation algorithm\. Seconds per 10 test batches,N=30N=30\.
## Appendix KPublic datasets considered

We surveyed datasets used by the closest prior benchmarks and adjacent generative\-modeling papers for event sequences — EasyTPP\[[Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19)\], HoTPP\[[Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20)\], CDiff\[[Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1)\], ADiff4TPP\[[Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\], TabDiT\[[Garuti et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib11)\], EBES\[[Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22)\], and[Surkov et al\. \[2025\]](https://arxiv.org/html/2610.12586#bib.bib32)— and considered each public corpus against two quantitative criteria induced by our protocol and five qualitative ones induced by the task definition\.

#### Quantitative thresholds\.

We drop any candidate with strictly fewer than5,0005\{,\}000sequences as*too small*— a fixed cutoff that is independent of our benchmark construction \(Table[2](https://arxiv.org/html/2610.12586#S4.T2)\)\. Our forecast horizons go up toL=32L=32in the main results andL=64L=64in the long\-horizon study, so a usable corpus must contain at leasthistory\+L≥64\\text\{history\}\+L\\geq 64events on average; we drop any candidate with*mean sequence length*below6464as*too short*\.

#### Qualitative criteria\.

\(i\) Repetitive identical events\.On corpora dominated by long runs of literally identical events, the trivialRepeatbaseline scores high on TMS by construction and a discriminator cannot meaningfully separate continuations, collapsing both axes of our evaluation\. This applies to*Taobao*\(in the EBES and[Surkov et al\. \[2025\]](https://arxiv.org/html/2610.12586#bib.bib32)preprocessing\),*Retail / X5 RetailHero*, and*Megamarket*\.\(ii\) Recommendation\-scale cardinality\.Datasets such as*Megamarket*and*Zvuk*have item or artist vocabularies in the104​–​10510^\{4\}\\text\{\-\-\}10^\{5\}range and live in the recommender\-systems literature, where the standard scores \(HitRate@KK, NDCG@KK, MRR\) and modeling assumptions differ substantially from the generative\-forecasting setting we target\.\(iii\) Order semantics broken at source\.The Multimodal Banking Dataset \(*MBD*\) is released with shuffled event order and is therefore incompatible with any temporal\-order\-aware metric or model\.\(iv\) Continuous time series mislabeled as event sequences\.*PhysioNet 2012*is essentially an irregularly\-sampled multivariate clinical time series with mostly missing values; it is a missing\-data interpolation problem, not a heterogeneous event\-sequence forecasting one\.\(v\) Large number of events with identical timestamps\.In several corpora, many events share the same timestamp within a sequence, weakening temporal ordering signal for point\-process style forecasting\. This is prominent in*MIMIC\-IV*\[[Johnson et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib26)\],*Retail / X5 RetailHero*,*Taobao*\(User Behavior\),*Zvuk*, and*Megamarket*\.

#### Catalog\.

Table[16](https://arxiv.org/html/2610.12586#A11.T16)consolidates the corpora we encountered\. For datasets that appear in multiple papers under different names but share a single underlying source — HoTPP*Transactions*, EBES*AGE*, TabDiT*Age1*, and[Surkov et al\. \[2025\]](https://arxiv.org/html/2610.12586#bib.bib32)*AGE*are the same Sberbank Sirius bank\-transaction corpus;*Alphabattle\-2\.0*in[Surkov et al\. \[2025\]](https://arxiv.org/html/2610.12586#bib.bib32)is the same source as ourAlphabattle, with us using the official test split of the Alfa\-Bank release — we report a single canonical row\. Counts and mean lengths are taken from each paper’s own statistics tables \(HoTPP Table 1; EBES Table 1; TabDiT Appendix E Table 9;[Surkov et al\. \[2025\]](https://arxiv.org/html/2610.12586#bib.bib32)Table 1\) and from the EasyTPP HuggingFace mirrors when paper\-side numbers are ambiguous \(e\.g\., when EasyTPP reports event\-token counts rather than sequence counts\)\.

After applying these filters, the only public corpora that satisfy both quantitative thresholds \(≥5,000\\geq\\\!5\{,\}000sequences and mean length≥64\\geq 64\) are bank\-transaction event sequences with heterogeneous fields, and have unshuffled temporal order are the Sberbank Sirius corpus \(= ourAge\) and the Alfa\-Bank Alphabattle release \(= ourAlphabattle\)\. Even though lowering the size bar to5​k5\\mathrm\{k\}leaves many more corpora above the sequence\-count cutoff, nearly all of them still fail the mean\-length requirement or the qualitative criteria above; our three benchmark datasets \(Gender,Age,Alphabattle\) each exceed5​k5\\mathrm\{k\}sequences \(Table[2](https://arxiv.org/html/2610.12586#S4.T2)\)\.

DatasetReported by\# seq\.Mean len\.Reason*Our benchmark \(Table[2](https://arxiv.org/html/2610.12586#S4.T2)\)*Genderours14k \(11k/2\.8k\)404we useAgeours30k \(24k/6k\)881we useAlphabattleours \(Alfa\-Bank, official test split\)135k \(100k/35k\)367we use*Generative\-modeling papers for event sequences*Retweet\[[Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19),[Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20),[Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1),[Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\]24k41–56LAmazon\[[Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19),[Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20),[Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1),[Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\]9k44LStackOverflow\[[Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19),[Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20),[Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1),[Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\]2k64S, LTaxi\[[Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19),[Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1),[Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\]2k37–39S, LTaobao \(TPP cut\)\[[Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19),[Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1),[Mukherjee et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib12)\]2–5k58–150S, RMOOC\[[Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1)\]4kn/aSMIMIC\-IV\[[Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20),[Johnson et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib26)\]120k19\.7L, IAge2\[[Garuti et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib11)\]43k84XLeaving\[[Garuti et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib11)\]5k98SPKDD’99 Financial\[[Garuti et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib11)\]4\.5k235SRossmann\[[Garuti et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib11)\]1\.1k61S, LAirbnb\[[Garuti et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib11)\]10k19LLarge\-Scale Bank Data\[[Garuti et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib11)\]110k873P*Discriminative event\-sequence benchmarks \(EBES\) and adjacent*ElectricDevices\[[Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22)\]17k96TArabicDigits\[[Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22)\]9k40L, TPhysioNet 2012\[[Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22)\]8k75TMIMIC\-III\[[Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22)\]56k58L†, TPendulum \(synthetic\)\[[Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22)\]100k32L, YBPI17\[[Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22)\]43k13LRetail / X5 RetailHero\[[Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22),[Surkov et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib32)\]40–399k112–114R, ITaobao \(User Behav\.\)\[[Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22),[Surkov et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib32)\]10k535R, IMBD\[[Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22),[Surkov et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib32)\]1\.5–9\.2M21–313OMegamarket\[[Surkov et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib32)\]2\.73M653C, R, IZvuk\[[Surkov et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib32)\]380k1020C, IShakespeare\[[Surkov et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib32)\]5k106THawkes\-1D\[[Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19),[Zeng et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib1)\]77k79Y*Same source as our datasets \(already covered above\)*Transactions / AGE / Age1\[[Karpukhin et al\., 2026](https://arxiv.org/html/2610.12586#bib.bib20),[Osin et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib22),[Garuti et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib11),[Surkov et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib32)\]24–50k875–882=AgeAlphabattle\-2\.0\[[Surkov et al\., 2025](https://arxiv.org/html/2610.12586#bib.bib32)\]1M213=Alphabattle
Reason key\.S= too small \(<5​k<\\\!5\\mathrm\{k\}sequences\);L= too short \(mean length<64<\\\!64\);R= long runs of repetitive identical events \(trivializesRepeatand cDFS\);I= large number of events with identical timestamps;C= recommender\-systems scale cardinality \(item/artist IDs104−10510^\{4\}\\\!\-\\\!10^\{5\}\);O= released with shuffled event order;T= continuous / regular time series, not heterogeneous EvS \(or text\);P= not publicly released;Y= synthetic;X= redundant lineage with ourAge;†\\dagger= borderline \(within∼10%\\sim\\\!10\\%of the cutoff\)\.

Table 16:Public event\-sequence datasets considered and reasons for exclusion\.

## Appendix LTransfer to non\-transactional TPP corpora

To probe how the conclusions transfer beyond card transactions, we evaluate the benchmark on two public TPP corpora, Retweet and Amazon\[[Xue et al\., 2024](https://arxiv.org/html/2610.12586#bib.bib19)\]\. Both fail our selection criteria \(Appendix[K](https://arxiv.org/html/2610.12586#A11)\): they are much smaller, with far shorter sequences, and each event carries only a timestamp and a categorical mark\. This makes them a stress test rather than an extension of the benchmark: if the findings survive even there, they are not an artifact of the transaction domain\. Since these corpora carry no amount feature, TMS is computed without the amount term\. Table[17](https://arxiv.org/html/2610.12586#A12.T17)reports the full pool of methods at horizonL=32L=32, with Age alongside for reference\.

The main findings carry over despite the harder training conditions: the accuracy–realism trade\-off holds, LDM remains the most realistic model on both corpora, point predictors stay far below LDM on cDFS, and LDM\-WB attains the best TMS on both datasets while retaining a large part of the realism \(cDFS 0\.672 on Retweet and 0\.243 on Amazon vs\. 0\.017 and 0\.002 for DEF\)\.

MethodTMS↑\\uparrowcDFS↑\\uparrowAgeRetweetAmazonAgeRetweetAmazonHist\.0\.3660\.3910\.3430\.2340\.0350\.082Mode0\.2630\.1560\.0360\.0000\.0090\.002Rep\.0\.4540\.4350\.4150\.0650\.0530\.070AR0\.3540\.5300\.1510\.0000\.0410\.110MT0\.3600\.6040\.1960\.0000\.4410\.140LDM0\.4260\.5790\.3571\.0001\.0000\.608DDM0\.4410\.5650\.3480\.2020\.1920\.557DEF0\.5080\.6530\.4300\.0000\.0170\.002LDM\-WB0\.5220\.6630\.4380\.2240\.6720\.243DDM\-WB0\.4800\.5660\.4120\.1020\.1440\.428Table 17:TMS and cDFS at horizonL=32L=32on the two TPP corpora, with Age for reference\. Within each column:best,second,third\.
## NeurIPS Paper Checklist

1. 1\.Claims
2. Question: Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
3. Answer:\[Yes\]
4. Justification: The abstract and Introduction state the same scope, the benchmark setup, the empirical trade\-off, and Wasserstein BaryBooster; the contribution list points to Sections[3\.1](https://arxiv.org/html/2610.12586#S3.SS1)–[3\.5](https://arxiv.org/html/2610.12586#S3.SS5),[4](https://arxiv.org/html/2610.12586#S4),[6](https://arxiv.org/html/2610.12586#S6), and[5](https://arxiv.org/html/2610.12586#S5)where these are developed\.
5. Guidelines: - •The answer\[N/A\]means that the abstract and introduction do not include the claims made in the paper\. - •The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations\. A\[No\]or\[N/A\]answer to this question will not be perceived well by the reviewers\. - •The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings\. - •It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper\.
6. 2\.Limitations
7. Question: Does the paper discuss the limitations of the work performed by the authors?
8. Answer:\[Yes\]
9. Justification: See §[7](https://arxiv.org/html/2610.12586#S7.SS0.SSS0.Px2)
10. Guidelines: - •The answer\[N/A\]means that the paper has no limitation while the answer\[No\]means that the paper has limitations, but those are not discussed in the paper\. - •The authors are encouraged to create a separate “Limitations” section in their paper\. - •The paper should point out any strong assumptions and how robust the results are to violations of these assumptions \(e\.g\., independence assumptions, noiseless settings, model well\-specification, asymptotic approximations only holding locally\)\. The authors should reflect on how these assumptions might be violated in practice and what the implications would be\. - •The authors should reflect on the scope of the claims made, e\.g\., if the approach was only tested on a few datasets or with a few runs\. In general, empirical results often depend on implicit assumptions, which should be articulated\. - •The authors should reflect on the factors that influence the performance of the approach\. For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting\. Or a speech\-to\-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon\. - •The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size\. - •If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness\. - •While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren’t acknowledged in the paper\. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community\. Reviewers will be specifically instructed to not penalize honesty concerning limitations\.
11. 3\.Theory assumptions and proofs
12. Question: For each theoretical result, does the paper provide the full set of assumptions and a complete \(and correct\) proof?
13. Answer:\[N/A\]
14. Justification: The paper is empirical; it does not present numbered theoretical results with explicit assumption blocks and complete proofs\.
15. Guidelines: - •The answer\[N/A\]means that the paper does not include theoretical results\. - •All the theorems, formulas, and proofs in the paper should be numbered and cross\-referenced\. - •All assumptions should be clearly stated or referenced in the statement of any theorems\. - •The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition\. - •Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material\. - •Theorems and Lemmas that the proof relies upon should be properly referenced\.
16. 4\.Experimental result reproducibility
17. Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper \(regardless of whether the code and data are provided or not\)?
18. Answer:\[Yes\]
19. Justification: Methods, metrics \(including BaryBooster, Appendix[F](https://arxiv.org/html/2610.12586#A6)\), datasets, splits, and training/evaluation protocol are specified in the main text and Appendices[H](https://arxiv.org/html/2610.12586#A8)–[I](https://arxiv.org/html/2610.12586#A9); public data URLs appear in §[4\.1](https://arxiv.org/html/2610.12586#S4.SS1)\.
20. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •If the paper includes experiments, a\[No\]answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not\. - •If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable\. - •Depending on the contribution, reproducibility can be accomplished in various ways\. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model\. In general\. releasing code and data is often one good way to accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model \(e\.g\., in the case of a large language model\), releasing of a model checkpoint, or other means that are appropriate to the research performed\. - •While NeurIPS does not require releasing code, the conference does require all submissions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution\. For example 1. \(a\)If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm\. 2. \(b\)If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully\. 3. \(c\)If the contribution is a new model \(e\.g\., a large language model\), then there should either be a way to access this model for reproducing the results or a way to reproduce the model \(e\.g\., with an open\-source dataset or instructions for how to construct the dataset\)\. 4. \(d\)We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility\. In the case of closed\-source models, it may be that access to the model is limited in some way \(e\.g\., to registered users\), but it should be possible for other researchers to have some path to reproducing or verifying the results\.
21. 5\.Open access to data and code
22. Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?
23. Answer:\[Yes\]
24. Justification: The full source code, with documentation, is publicly available \(§[4](https://arxiv.org/html/2610.12586#S4)\) and enables reproduction of the experiments\.
25. Guidelines: - •The answer\[N/A\]means that paper does not include experiments requiring code\. - • - •While we encourage the release of code and data, we understand that this might not be possible, so\[No\]is an acceptable answer\. Papers cannot be rejected simply for not including code, unless this is central to the contribution \(e\.g\., for a new open\-source benchmark\)\. - •The instructions should contain the exact command and environment needed to run to reproduce the results\. See the NeurIPS code and data submission guidelines \([https://neurips\.cc/public/guides/CodeSubmissionPolicy](https://neurips.cc/public/guides/CodeSubmissionPolicy)\) for more details\. - •The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc\. - •The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines\. If only a subset of experiments are reproducible, they should state which ones are omitted from the script and why\. - •At submission time, to preserve anonymity, the authors should release anonymized versions \(if applicable\)\. - •Providing as much information as possible in supplemental material \(appended to the paper\) is recommended, but including URLs to data and code is permitted\.
26. 6\.Experimental setting/details
27. Question: Does the paper specify all the training and test details \(e\.g\., data splits, hyperparameters, how they were chosen, type of optimizer\) necessary to understand the results?
28. Answer:\[Yes\]
29. Justification: Core choices appear in §[6](https://arxiv.org/html/2610.12586#S6)and §[4](https://arxiv.org/html/2610.12586#S4); full optimization, architectures, splits, and schedules are in Appendix[I](https://arxiv.org/html/2610.12586#A9)and[H](https://arxiv.org/html/2610.12586#A8)\.
30. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them\. - •The full details can be provided either with the code, in appendix, or as supplemental material\.
31. 7\.Experiment statistical significance
32. Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
33. Answer:\[Yes\]
34. Justification: We address statistical significance in Appendix[E\.5](https://arxiv.org/html/2610.12586#A5.SS5)\.
35. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The authors should answer\[Yes\]if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper\. - •The factors of variability that the error bars are capturing should be clearly stated \(for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions\)\. - •The method for calculating the error bars should be explained \(closed form formula, call to a library function, bootstrap, etc\.\) - •The assumptions made should be given \(e\.g\., Normally distributed errors\)\. - •It should be clear whether the error bar is the standard deviation or the standard error of the mean\. - •It is OK to report 1\-sigma error bars, but one should state it\. The authors should preferably report a 2\-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified\. - •For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range \(e\.g\., negative error rates\)\. - •If error bars are reported in tables or plots, the authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text\.
36. 8\.Experiments compute resources
37. Question: For each experiment, does the paper provide sufficient information on the computer resources \(type of compute workers, memory, time of execution\) needed to reproduce the experiments?
38. Answer:\[Yes\]
39. Justification: Appendix[I\.3](https://arxiv.org/html/2610.12586#A9.SS3)states single NVIDIA A100 training; Appendix[J](https://arxiv.org/html/2610.12586#A10)tabulates train/inference wall times and GPU memory per model/dataset while noting further compute for grids and screening \(Appendix[I\.3](https://arxiv.org/html/2610.12586#A9.SS3)\)\.
40. Guidelines: - •The answer\[N/A\]means that the paper does not include experiments\. - •The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage\. - •The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute\. - •The paper should disclose whether the full research project required more compute than the experiments reported in the paper \(e\.g\., preliminary or failed experiments that didn’t make it into the paper\)\.
41. 9\.Code of ethics
43. Answer:\[Yes\]
44. Justification: Secondary analysis on public anonymized benchmark corpora cited in §[4\.1](https://arxiv.org/html/2610.12586#S4.SS1); no human\-subjects experimentation\.
45. Guidelines: - •The answer\[N/A\]means that the authors have not reviewed the NeurIPS Code of Ethics\. - •If the authors answer\[No\], they should explain the special circumstances that require a deviation from the Code of Ethics\. - •The authors should make sure to preserve anonymity \(e\.g\., if there is a special consideration due to laws or regulations in their jurisdiction\)\.
46. 10\.Broader impacts
47. Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
48. Answer:\[Yes\]
49. Justification: See §[7](https://arxiv.org/html/2610.12586#S7.SS0.SSS0.Px3)\.
50. Guidelines: - •The answer\[N/A\]means that there is no societal impact of the work performed\. - •If the authors answer\[N/A\]or\[No\], they should explain why their work has no societal impact or why the paper does not address societal impact\. - •Examples of negative societal impacts include potential malicious or unintended uses \(e\.g\., disinformation, generating fake profiles, surveillance\), fairness considerations \(e\.g\., deployment of technologies that could make decisions that unfairly impact specific groups\), privacy considerations, and security considerations\. - •The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments\. However, if there is a direct path to any negative applications, the authors should point it out\. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate Deepfakes for disinformation\. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster\. - •The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from \(intentional or unintentional\) misuse of the technology\. - •If there are negative societal impacts, the authors could also discuss possible mitigation strategies \(e\.g\., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML\)\.
51. 11\.Safeguards
52. Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse \(e\.g\., pre\-trained language models, image generators, or scraped datasets\)?
53. Answer:\[N/A\]
54. Justification: We study evaluation on curated public transactional benchmarks rather than scraped media; the released code exposes methodology but not frontier\-scale multimodal generators, so no gated\-release safeguards apply\.
55. Guidelines: - •The answer\[N/A\]means that the paper poses no such risks\. - •Released models that have a high risk for misuse or dual\-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters\. - •Datasets that have been scraped from the Internet could pose safety risks\. The authors should describe how they avoided releasing unsafe images\. - •We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort\.
56. 12\.Licenses for existing assets
57. Question: Are the creators or original owners of assets \(e\.g\., code, data, models\), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
58. Answer:\[Yes\]
59. Justification: Yes\. All datasets used have accompanying URLs and are explicitly cited; all codebases and models employed are properly licensed, and their licenses are respected and listed in the repository README\.
60. Guidelines: - •The answer\[N/A\]means that the paper does not use existing assets\. - •The authors should cite the original paper that produced the code package or dataset\. - •The authors should state which version of the asset is used and, if possible, include a URL\. - •The name of the license \(e\.g\., CC\-BY 4\.0\) should be included for each asset\. - •For scraped data from a particular source \(e\.g\., website\), the copyright and terms of service of that source should be provided\. - •If assets are released, the license, copyright information, and terms of use in the package should be provided\. For popular datasets,[paperswithcode\.com/datasets](https://paperswithcode.com/datasets)has curated licenses for some datasets\. Their licensing guide can help determine the license of a dataset\. - •For existing datasets that are re\-packaged, both the original license and the license of the derived asset \(if it has changed\) should be provided\. - •If this information is not available online, the authors are encouraged to reach out to the asset’s creators\.
61. 13\.New assets
62. Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
63. Answer:\[Yes\]
64. Justification: Benchmark metrics and BaryBooster are defined in the main text and Appendices[C](https://arxiv.org/html/2610.12586#A3)–[F](https://arxiv.org/html/2610.12586#A6); consolidated training documentation appears in Appendix[I](https://arxiv.org/html/2610.12586#A9)with the public code repository referenced in §[4](https://arxiv.org/html/2610.12586#S4)\.
65. Guidelines: - •The answer\[N/A\]means that the paper does not release new assets\. - •Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates\. This includes details about training, license, limitations, etc\. - •The paper should discuss whether and how consent was obtained from people whose asset is used\. - •At submission time, remember to anonymize your assets \(if applicable\)\. You can either create an anonymized URL or include an anonymized zip file\.
66. 14\.Crowdsourcing and research with human subjects
67. Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation \(if any\)?
68. Answer:\[N/A\]
69. Justification: No crowdsourcing or primary human\-subjects studies were conducted\.
70. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper\. - •According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector\.
71. 15\.Institutional review board \(IRB\) approvals or equivalent for research with human subjects
72. Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board \(IRB\) approvals \(or an equivalent approval/review based on the requirements of your country or institution\) were obtained?
73. Answer:\[N/A\]
74. Justification: No IRB\-governed human\-subjects component; only secondary use of public datasets\.
75. Guidelines: - •The answer\[N/A\]means that the paper does not involve crowdsourcing nor research with human subjects\. - •Depending on the country in which research is conducted, IRB approval \(or equivalent\) may be required for any human subjects research\. If you obtained IRB approval, you should clearly state this in the paper\. - •We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution\. - •For initial submissions, do not include any information that would break anonymity \(if applicable\), such as the institution conducting the review\.
76. 16\.Declaration of LLM usage
77. Question: Does the paper describe the usage of LLMs if it is an important, original, or non\-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does*not*impact the core methodology, scientific rigor, or originality of the research, declaration is not required\.
78. Answer:\[N/A\]
79. Justification: Large language models are not part of the proposed forecasting methods; any writing assistance is incidental to the technical contribution\.
80. Guidelines: - •The answer\[N/A\]means that the core method development in this research does not involve LLMs as any important, original, or non\-standard components\. - •Please refer to our LLM policy in the NeurIPS handbook for what should or should not be described\.