IXPLORE: Bounded Ideal Point Estimation with Grid-Based Uncertainty Quantification

# IXPLORE: Bounded Ideal Point Estimation with Grid-Based Uncertainty Quantification
Source: [https://arxiv.org/html/2609.06018](https://arxiv.org/html/2609.06018)
Affiliation:Correspondence to[fynn\.bachmann@uzh\.ch](mailto:[email protected])

###### Abstract

Ideal point estimation is widely used to analyze and visualize political data\. However, selecting the corresponding spatial model involves various trade\-offs: while model\-based approaches such as Item Response Theory \(IRT\) are based on utility functions rather than optimized for predictive accuracy, most Machine Learning \(ML\) alternatives struggle to generalize beyond training data when embedding sparse test responses\. We introduceIXPLORE, a bounded ideal point estimation algorithm that combines a predictive fit objective with a sparsity\-aware likelihood function\. On five benchmark datasets spanning surveys, roll calls, and deliberation, this approach surpasses model\-based and ML\-based algorithms on reconstruction and imputation error—especially for users with sparse responses\. Furthermore, we show that non\-linear feature transforms can further reduce the reconstruction error while remaining visually interpretable\. To quantify uncertainty,IXPLOREapplies grid\-based posterior inference on a bounded 2D latent space\. Available as a Python package on PyPI,IXPLOREoffers a flexible framework for constructing bounded, interpretable political maps with fast inference and strong imputation performance\.

##### Keywords

Ideal Point Estimation, Spatial Models, Item Response Theory, Machine Learning, Bounded Latent Space, Political Data Analysis, Imputation, Uncertainty Quantification

##### Data availability

## 1Introduction

Spatial voting models continue to play an important role in the analysis of political data\. They are used to measure polarization on online platforms\([Barberá, 2015](https://arxiv.org/html/2609.06018#bib.bib7)\), reveal how legislators are more extreme than the citizens they represent\([Bafumi and Herron, 2010](https://arxiv.org/html/2609.06018#bib.bib5);[Leimgruber et al\., 2010](https://arxiv.org/html/2609.06018#bib.bib29)\), and model electoral competition\([Vaeth, 2025](https://arxiv.org/html/2609.06018#bib.bib48)\)\. Furthermore, the visualization of spatial models contributes to political education through low\-dimensional opinion landscapes\. For example, the political compass embeds users in a political spectrum\([Petrik, 2010](https://arxiv.org/html/2609.06018#bib.bib36)\), roll\-call analyses track the distribution of political candidates over time\([Boche et al\., 2018](https://arxiv.org/html/2609.06018#bib.bib8)\), and electoral geographers cluster municipalities by their voting behavior\([Leuthold et al\., 2007](https://arxiv.org/html/2609.06018#bib.bib30)\)\.

However, the data\-driven computation of such political maps comes with various trade\-offs\. While methods based on Machine Learning \(ML\) are optimized to minimize the reconstruction error of the response matrix, they have no native mechanism for incomplete response vectors and must impute missing entries before a user can be embedded\([Bachmann et al\., 2024](https://arxiv.org/html/2609.06018#bib.bib3)\)\. In contrast, model\-based algorithms, for example ideal point estimation using Item Response Theory \(IRT\), accept sparse responses by design, but their utility\-based objective is not targeted for predictive performance\([Clinton and Jackman, 2009](https://arxiv.org/html/2609.06018#bib.bib11)\)\. Both properties are, however, crucial when embedding users with sparse responses into a low\-dimensional space that should carry predictive information\. This is the case for wiki surveys that detect clusters to structure online deliberation\([Small et al\., 2021](https://arxiv.org/html/2609.06018#bib.bib45)\), Voting Advice Applications \(VAAs\) that embed voters into the candidates’ political map\([Thomeczek and others, 2025](https://arxiv.org/html/2609.06018#bib.bib47);[Otjes and Louwerse, 2014](https://arxiv.org/html/2609.06018#bib.bib35)\), and adaptive questionnaires that must predict users’ answers to the remaining questions\([Bachmann et al\., 2025](https://arxiv.org/html/2609.06018#bib.bib1);[Montgomery and Cutler, 2013](https://arxiv.org/html/2609.06018#bib.bib34)\)\.

We present theIXPLOREalgorithm to address current limitations of data\-driven political maps under sparse responses\. To unite the strengths of model\-based and ML\-based models,IXPLOREcombines a predictive fit objective with a sparsity\-aware likelihood function\. Users’ positions and model parameters are iteratively refined to minimize the reconstruction error, while missing responses are ignored in the likelihood function\. To quantify uncertainty,IXPLOREapplies a Bayesian framework with grid\-based posterior inference on a bounded latent space\. This ensures that all users land within the visualization after model training\. The grid therefore serves both as an approximation for posterior inference and as the map on which users are displayed\.

We demonstrate the predictive performance ofIXPLOREon five real\-world datasets spanning VAAs, roll calls, deliberation, and social surveys\. Under our evaluation framework,IXPLOREoutperforms model\-based algorithms such as IDEAL\([Clinton et al\., 2004](https://arxiv.org/html/2609.06018#bib.bib12)\)and LSIRM\([Lee et al\., 2025](https://arxiv.org/html/2609.06018#bib.bib28)\), as well as ML\-based algorithms such as Principal Component Analysis \(PCA\) and Variational Auto\-Encoders \(VAEs\)\. In particular, we find that a two\-dimensionalIXPLOREmodel reaches the same train reconstruction error as vanilla PCA withd=21d=21latent dimensions, whileIXPLORE’s imputation accuracy is never matched, regardless of PCA’s number of latent dimensions\. Beyond the core algorithm, we evaluate non\-linear feature transforms as an extension to classical ideal point estimation\. We find that the corresponding curved decision boundaries can further reduce reconstruction error, although this flexibility introduces a trade\-off between predictive accuracy and the visual interpretability expected from classical ideal point models\.

Based on these results, we conclude thatIXPLOREis well suited for applications involving sparse user responses and interpretable, bounded political maps\. Its fast inference makes it particularly relevant for online applications such as VAAs and deliberation platforms\. TheIXPLOREimplementation is available on[GitHub](https://github.com/fsvbach/ixplore)111[https://github\.com/fsvbach/ixplore](https://github.com/fsvbach/ixplore)and[PyPI](https://pypi.org/project/ixplore/)222[https://pypi\.org/project/ixplore](https://pypi.org/project/ixplore)as open\-source software\.

## 2Related Work

This paper builds on three streams of literature: spatial models of political ideology, visualizations of the political spectrum, and the evaluation of ideal point estimation algorithms under high sparsity\.

### 2\.1Spatial Models and Ideal Point Estimation

Spatial models represent political actors as points in a low\-dimensional space, where smaller distances indicate greater agreement\([Stokes, 1963](https://arxiv.org/html/2609.06018#bib.bib46);[Hinich and Pollard, 1981](https://arxiv.org/html/2609.06018#bib.bib20)\)\. To estimate such*ideal points*for legislators programmatically from their roll\-call votes,[Poole and Rosenthal \(1985\)](https://arxiv.org/html/2609.06018#bib.bib39)developed NOMINATE\. This model assumes a Gaussian utility function according to which legislators prefer votes closer to their ideal points\.[Clinton et al\. \(2004\)](https://arxiv.org/html/2609.06018#bib.bib12)extended this approach by introducing a Bayesian framework with a quadratic utility function\. Their method IDEAL infers actor positions and item parameters via Markov\-Chain Monte Carlo \(MCMC\) sampling\. Scaling this approach to larger datasets,[Imai et al\. \(2016\)](https://arxiv.org/html/2609.06018#bib.bib21)replaced the sampler with an expectation\-maximization \(EM\) algorithm alternating closed\-form updates of user positions and item parameters\. Their implementation emIRT estimates ideal points about an order of magnitude faster than NOMINATE, but its binary outcome model is currently restricted to one latent dimension\. As a more flexible and even faster alternative,[Potthoff \(2018\)](https://arxiv.org/html/2609.06018#bib.bib40)proposed to estimate ideal points by applying PCA to the response matrix and decoding each item with a logistic regression\. While this approach allows higher latent dimensions and works for non\-binary data, it must fill any missing responses before the embedding can be computed\. Moreover, the PCA approach replaces the utility function with the objective of minimizing the reconstruction error of the response matrix, which marks a shift from traditional IRT models to ML\-based algorithms\.IXPLORErefines this approach by iteratively optimizing the embedding and item parameters according to a sparsity\-aware likelihood function\. The corresponding posterior distribution over users’ latent positions furthermore addresses*uncertainty quantification*, which[Potthoff \(2018\)](https://arxiv.org/html/2609.06018#bib.bib40)still lists as a limitation\.

### 2\.2Visualization of Political Maps

In many applications, spatial models locate users in the political landscape\. For example, the political compass places users in a political spectrum after they have answered a short set of questions\([Petrik, 2010](https://arxiv.org/html/2609.06018#bib.bib36)\)\. Similar visualizations have been adopted in VAAs to support political education\([Otjes and Louwerse, 2014](https://arxiv.org/html/2609.06018#bib.bib35)\)\. For instance, voters are often positioned relative to parties in political maps\([Costa Lobo et al\., 2010](https://arxiv.org/html/2609.06018#bib.bib13);[Verboom et al\., 2025](https://arxiv.org/html/2609.06018#bib.bib50)\), which has been shown to increase their electoral participation\([Ioannidis, 2025](https://arxiv.org/html/2609.06018#bib.bib22)\)\. Often, however, designers of such maps manually define the dimensions of the model\([Otjes and Louwerse, 2014](https://arxiv.org/html/2609.06018#bib.bib35)\)\. Addressing this limitation,[Germann et al\. \(2015\)](https://arxiv.org/html/2609.06018#bib.bib18)propose dynamic scale validation, which derives the dimensions of the map from voters’ responses\. In follow\-up work, they show that this approach improves measurement quality across various European VAA deployments\([Germann and Mendez, 2016](https://arxiv.org/html/2609.06018#bib.bib17)\)\. Increasingly, such data\-driven methods are also used to visualize election outcomes: electoral geographers obtain political landscapes by factor analysis of referendum results\([Leuthold et al\., 2007](https://arxiv.org/html/2609.06018#bib.bib30)\), and multi\-dimensional scaling has been used to cluster voting districts in German federal elections\([Bachmann et al\., 2023](https://arxiv.org/html/2609.06018#bib.bib4)\)\. For legislative data, theVoteviewproject curates the roll\-call record of the U\.S\. Congress and displays legislators through NOMINATE scores\([Boche et al\., 2018](https://arxiv.org/html/2609.06018#bib.bib8)\)\. Recently, online deliberation platforms have adopted data\-driven political maps to visualize the opinion landscape of a conversation:Polisuses PCA to project participants onto a two\-dimensional latent space and outlines opinion clusters\([Small et al\., 2021](https://arxiv.org/html/2609.06018#bib.bib45)\)\.IXPLOREextends this line of work by proposing a sparsity\-robust method to embed users with predictive and interpretable per\-item decision boundaries\.

### 2\.3Sparse Responses and Model Evaluation

Missing responses are a recurring obstacle for most ML\-based embedding methods: PCA\([Potthoff, 2018](https://arxiv.org/html/2609.06018#bib.bib40)\),tt\-SNE\([van der Maaten and Hinton, 2008](https://arxiv.org/html/2609.06018#bib.bib49)\), UMAP\([McInnes et al\., 2018](https://arxiv.org/html/2609.06018#bib.bib33)\), and VAEs\([Kingma and Welling, 2014](https://arxiv.org/html/2609.06018#bib.bib26)\)have no native mechanism for incomplete response vectors\. Often, missing entries are filled by iterative imputation, where a fitted model’s reconstruction alternates with re\-estimation of the missing values, using either PCA\([Grung and Manne, 1998](https://arxiv.org/html/2609.06018#bib.bib19)\)or VAEs\([McCoy et al\., 2018](https://arxiv.org/html/2609.06018#bib.bib31)\)\. However, such preprocessing steps become restrictive when responses arrive incrementally and predictions must be made on the fly\. This is, for example, the case in computerized adaptive testing\([Montgomery and Cutler, 2013](https://arxiv.org/html/2609.06018#bib.bib34)\)\. Such surveys tailor the questionnaire to each respondent, selecting the next item based on existing responses333See[Sigfrid \(2024\)](https://arxiv.org/html/2609.06018#bib.bib44)and[Bachmann et al\. \(2026\)](https://arxiv.org/html/2609.06018#bib.bib2)for current applications in VAAs\.\. The online deliberation platformPolisfaces the same regime, as participants vote on only a fraction of the statements before being placed in the opinion landscape\([Small et al\., 2021](https://arxiv.org/html/2609.06018#bib.bib45)\)\. In this sparse\-response setting,[Bachmann et al\. \(2024\)](https://arxiv.org/html/2609.06018#bib.bib3)benchmark seven spatial models on VAA data and find that the model\-based algorithms IDEAL and NOMINATE generalize best to held\-out responses444See[Carroll et al\. \(2009\)](https://arxiv.org/html/2609.06018#bib.bib10)for a detailed comparison of NOMINATE and IDEAL\.\. However,[Clinton and Jackman \(2009\)](https://arxiv.org/html/2609.06018#bib.bib11)note that IRT models are not designed for out\-of\-sample predictions, since a new respondent has no estimated parameters\. Building on the framework of[Bachmann et al\. \(2024\)](https://arxiv.org/html/2609.06018#bib.bib3), we evaluateIXPLOREand compare its predictive performance to ten other algorithms across five datasets\.

## 3The IXPLORE Model

IXPLOREtargets questionnaire data where users’ responses to all items are numerical indications of agreement \(e\.g\., binary, ordinal, or continuous\)\. Two coupled inference tasks arise from such an item\-response matrix:*embedding*, i\.e\., placing each usernnat a latent positionxnx\_\{n\}that summarizes their response profile, and*imputation*, i\.e\., sampling the user’s missing responses based on their latent position\. Like model\-based algorithms,IXPLOREjointly embeds users and items in a shared latent space\. In contrast to existing algorithms, however, this latent space is bounded and discretized\. This yields a full posterior distribution over each user’s position without any assumptions on its shape\. From this posterior distribution,IXPLOREpredicts the user’s responses through per\-item logistic regressions\. The two sets of parameters—users’ posteriors and the logistic decision boundaries—are fit by alternating updates: the user step recomputes posteriors from the current item parameters, and the item step refits each logistic regression from the updated user embedding\. In this section, we present the statistical framework and technical implementation of the algorithm\.[AppendixA](https://arxiv.org/html/2609.06018#A1)provides further details on the model’s design choices\.

### 3\.1Statistical Framework

We observe a response matrix𝐘∈\[0,1\]N×K\\mathbf\{Y\}\\in\[0,1\]^\{N\\times K\}, whereNNis the number of users andKKis the number of items\. Each entryYn​kY\_\{nk\}is a response from usernnto itemkk\. Missing entries are flagged by an observation maskMn​k∈\{0,1\}M\_\{nk\}\\in\\\{0,1\\\}, indicating whether the response is observed\. We write𝒪n=\{k:Mn​k=1\}\\mathcal\{O\}\_\{n\}=\\\{k:M\_\{nk\}=1\\\}as the set of answered items for usernn, andYnobs=\{Yn​k:k∈𝒪n\}Y\_\{n\}^\{\\text\{obs\}\}=\\\{Y\_\{nk\}:k\\in\\mathcal\{O\}\_\{n\}\\\}as their observed responses\. Symmetrically,ℛk=\{n:Mn​k=1\}\\mathcal\{R\}\_\{k\}=\\\{n:M\_\{nk\}=1\\\}denotes the set of users who answered itemkk\.

##### Prior\.

For a user without any observed answers, we assume a Gaussian prior with isotropic covariance, truncated to the bounded latent space\[−L,L\]2\[\-L,L\]^\{2\}

p⁡\(x\)∝\{𝒩⁡\(x,0,τ2​I\)if​x∈\[−L,L\]2,0otherwise,p\(x\)\\;\\propto\\;\\begin\{cases\}\\mathcal\{N\}\(x;\\,0,\\,\\tau^\{2\}I\)&\\text\{if \}x\\in\[\-L,L\]^\{2\},\\\\\[4\.0pt\] 0&\\text\{otherwise,\}\\end\{cases\}\(1\)whereIIis the identity matrix and the marginal varianceτ2\\tau^\{2\}controls how tightly the prior concentrates around the center\. Alternatively, a log\-barrier prior

p⁡\(x\)∝\(1−\(x1/L\)2\)α​\(1−\(x2/L\)2\)αp\(x\)\\;\\propto\\;\\bigl\(1\-\(x\_\{1\}/L\)^\{2\}\\bigr\)^\{\\alpha\}\\,\\bigl\(1\-\(x\_\{2\}/L\)^\{2\}\\bigr\)^\{\\alpha\}\(2\)can be chosen; this prior is flatter in the interior and vanishes at the boundary\. Therefore, the boundary is modeled implicitly by the prior rather than imposed as a support constraint\. A comparison of the two priors is given in[SectionA\.1](https://arxiv.org/html/2609.06018#A1.SS1)\.

##### Response function\.

Given usernn’s latent positionxnx\_\{n\}, the probability of a positive response for itemkkis modeled by a logistic regression

yk​\(xn\):=p⁡\(Yn​k=1∣xn\)=σ⁡\(βk⊤​φ​\(xn\)\+αk\),σ⁡\(z\)=11\+e−z,y\_\{k\}\(x\_\{n\}\):=p\(Y\_\{nk\}=1\\mid x\_\{n\}\)=\\sigma\\bigl\(\\beta\_\{k\}^\{\\top\}\\varphi\(x\_\{n\}\)\+\\alpha\_\{k\}\\bigr\),\\qquad\\sigma\(z\)=\\frac\{1\}\{1\+e^\{\-z\}\},\(3\)with discriminationβk∈ℝD\\beta\_\{k\}\\in\\mathbb\{R\}^\{D\}, interceptαk∈ℝ\\alpha\_\{k\}\\in\\mathbb\{R\}, and an optional feature transformφ:ℝ2→ℝD\\varphi:\\mathbb\{R\}^\{2\}\\to\\mathbb\{R\}^\{D\}\. Withφ\\varphias the identity transform, this response function resembles that of the IDEAL model, whose quadratic spatial utility is also linear in the decision boundary\([Clinton et al\., 2004](https://arxiv.org/html/2609.06018#bib.bib12)\);IXPLOREonly replaces the probit link with the logistic sigmoid\. Optional polynomial or Random Fourier Feature \(RFF\) transforms yield curved decision boundaries\([Rahimi and Recht, 2007](https://arxiv.org/html/2609.06018#bib.bib41)\), departing from the Euclidean geometry assumed by the standard spatial model\([Eguia, 2013](https://arxiv.org/html/2609.06018#bib.bib14)\)\. We writeθk=\(βk,αk\)\\theta\_\{k\}=\(\\beta\_\{k\},\\alpha\_\{k\}\)for the parameters of itemkkandθ=\{θk\}k=1K\\theta=\\\{\\theta\_\{k\}\\\}\_\{k=1\}^\{K\}for all item parameters\.

##### User likelihood\.

As the responses across items are assumed conditionally independent givenxnx\_\{n\}, the likelihood factorizes into the per\-item Bernoulli terms of the response function,

p⁡\(Ynobs∣xn,θ\)=∏k∈𝒪nyk​\(xn\)Yn​k​\(1−yk​\(xn\)\)1−Yn​k\.p\(Y\_\{n\}^\{\\text\{obs\}\}\\mid x\_\{n\},\\theta\)=\\prod\_\{k\\in\\mathcal\{O\}\_\{n\}\}y\_\{k\}\(x\_\{n\}\)^\{Y\_\{nk\}\}\\,\\bigl\(1\-y\_\{k\}\(x\_\{n\}\)\\bigr\)^\{1\-Y\_\{nk\}\}\.\(4\)Because the product runs only over the observed items𝒪n\\mathcal\{O\}\_\{n\}, missing responses require no imputation\. For numerical stability,IXPLOREevaluates this likelihood in log space, where the product becomes a sum of per\-item terms\. In this form,IXPLOREadditionally allows optional user\-item confidence weightsWn​k≥0W\_\{nk\}\\geq 0

log⁡p⁡\(Ynobs∣xn,θ\)=∑k∈𝒪nWn​k​ℓn​k​\(xn\),whereℓn​k​\(xn\)=Yn​k​log⁡yk​\(xn\)\+\(1−Yn​k\)​log⁡\(1−yk​\(xn\)\)\.\\begin\{split\}\\log p\(Y\_\{n\}^\{\\text\{obs\}\}\\mid x\_\{n\},\\theta\)&=\\sum\_\{k\\in\\mathcal\{O\}\_\{n\}\}W\_\{nk\}\\,\\ell\_\{nk\}\(x\_\{n\}\),\\\\ \\text\{where\}\\quad\\ell\_\{nk\}\(x\_\{n\}\)&=Y\_\{nk\}\\log y\_\{k\}\(x\_\{n\}\)\+\(1\-Y\_\{nk\}\)\\log\\bigl\(1\-y\_\{k\}\(x\_\{n\}\)\\bigr\)\.\\end\{split\}\(5\)With the defaultWn​k=1W\_\{nk\}=1, this reduces to the log\-likelihood of[Eq\.4](https://arxiv.org/html/2609.06018#S3.E4); otherwise, a weightWn​kW\_\{nk\}can be interpreted as usernnrespondingWn​kW\_\{nk\}times to itemkk\. An optional weight\-scaling step rescales each user’s sum of weights toKK

Wn​k⟶Wn​k⋅K∑k′∈𝒪nWn​k′\.W\_\{nk\}\\;\\longrightarrow\\;\\frac\{W\_\{nk\}\\cdot K\}\{\\sum\_\{k^\{\\prime\}\\in\\mathcal\{O\}\_\{n\}\}W\_\{nk^\{\\prime\}\}\}\.\(6\)This gives users with sparse responses the same total weight as users with dense ones\.[AppendixD](https://arxiv.org/html/2609.06018#A4)provides further details on the use of confidence weights\.

##### Posterior\.

By Bayes’ theorem, the posterior over users’ latent positions combines the priorp⁡\(xn\)p\(x\_\{n\}\)with the likelihoodp⁡\(Ynobs∣xn,θ\)p\(Y\_\{n\}^\{\\text\{obs\}\}\\mid x\_\{n\},\\theta\)of their observed responses

p⁡\(xn∣Ynobs,θ\)∝p⁡\(Ynobs∣xn,θ\)⋅p⁡\(xn\)\.p\(x\_\{n\}\\mid Y\_\{n\}^\{\\text\{obs\}\},\\theta\)\\;\\propto\\;p\(Y\_\{n\}^\{\\text\{obs\}\}\\mid x\_\{n\},\\theta\)\\cdot p\(x\_\{n\}\)\.\(7\)For an unanswered itemkkof usernn, the predictive distribution follows from marginalizing out the latent position

p⁡\(Yn​k=1∣Ynobs,θ\)\\displaystyle p\(Y\_\{nk\}=1\\mid Y\_\{n\}^\{\\text\{obs\}\},\\theta\)=∫p\(Yn​k=1∣xn,Ynobs,θ\)⋅p\(xn∣Ynobs,θ\)dxn\\displaystyle=\\int p\(Y\_\{nk\}=1\\mid x\_\{n\},Y\_\{n\}^\{\\text\{obs\}\},\\theta\)\\,\\cdot\\,p\(x\_\{n\}\\mid Y\_\{n\}^\{\\text\{obs\}\},\\theta\)\\,\\mathop\{\}\\\!\\mathrm\{d\}x\_\{n\}=∫yk\(xn\)⋅p\(xn∣Ynobs,θ\)dxn,\\displaystyle=\\int y\_\{k\}\(x\_\{n\}\)\\,\\cdot\\,p\(x\_\{n\}\\mid Y\_\{n\}^\{\\text\{obs\}\},\\theta\)\\,\\mathop\{\}\\\!\\mathrm\{d\}x\_\{n\},\(8\)where we use the product rule and the conditional independence of responses givenxnx\_\{n\}\. Since the posterior appears as the second factor in the integral, both inference tasks reduce to the same object: the*embedding*is the posterior mean

x~n=𝔼\[xn∣Ynobs,θ\]=∫xnp\(xn∣Ynobs,θ\)dxn,\\tilde\{x\}\_\{n\}\\;=\\;\\mathbb\{E\}\\left\[\\,x\_\{n\}\\mid Y\_\{n\}^\{\\text\{obs\}\},\\theta\\,\\right\]\\;=\\;\\int x\_\{n\}\\,p\(x\_\{n\}\\mid Y\_\{n\}^\{\\text\{obs\}\},\\theta\)\\,\\mathop\{\}\\\!\\mathrm\{d\}x\_\{n\},\(9\)and the*imputation*is the posterior predictive from[Eq\.8](https://arxiv.org/html/2609.06018#S3.E8)\. Alternatively, imputation can evaluate the response function at the point estimateyk​\(x~n\)y\_\{k\}\(\\tilde\{x\}\_\{n\}\)rather than integrating over the posterior; we compare both variants in[SectionA\.2](https://arxiv.org/html/2609.06018#A1.SS2)\.

### 3\.2Implementation

The posterior in[Eq\.7](https://arxiv.org/html/2609.06018#S3.E7)has no analytical solution, since the Bernoulli likelihood factors are non\-conjugate with the Gaussian prior\. Standard approximations rely on MCMC to estimate parameters\([Clinton et al\., 2004](https://arxiv.org/html/2609.06018#bib.bib12)\); however, because the latent space is two\-dimensional and bounded,IXPLOREcan approximate the posterior on a fine grid instead\. The latent space\[−L,L\]2\[\-L,L\]^\{2\}is discretized into the gridXgrid=\{g1,…,gG\}X\_\{\\text\{grid\}\}=\\\{g\_\{1\},\\ldots,g\_\{G\}\\\}ofG=R2G=R^\{2\}points, withRRequispaced coordinates per axis ranging from−L\-LtoLL\. Evaluating Bayes’ rule pointwise gives the discrete posterior

p⁡\(xn=g∣Ynobs,θ\)=p⁡\(g\)​p​\(Ynobs∣g,θ\)∑g′∈Xgridp⁡\(g′\)​p​\(Ynobs∣g′,θ\),p\(x\_\{n\}=g\\mid Y\_\{n\}^\{\\text\{obs\}\},\\theta\)\\;=\\;\\frac\{p\(g\)\\,p\(Y\_\{n\}^\{\\text\{obs\}\}\\mid g,\\theta\)\}\{\\sum\_\{g^\{\\prime\}\\in X\_\{\\text\{grid\}\}\}p\(g^\{\\prime\}\)\\,p\(Y\_\{n\}^\{\\text\{obs\}\}\\mid g^\{\\prime\},\\theta\)\},\(10\)whichIXPLOREcomputes in log\-space for numerical stability\. Because the Bernoulli log\-likelihood admits a vectorized implementation with two dense matrix products, evaluating the grid posterior for all users is a tractable per\-iteration cost \(see[SectionA\.2\.1](https://arxiv.org/html/2609.06018#A1.SS2.SSS1)\)\.

##### Initialization\.

Apart from random initialization,IXPLOREsupports initializing the embedding with PCA, which has been shown to be effective for other non\-linear dimensionality reduction algorithms\([Kobak and Berens, 2019](https://arxiv.org/html/2609.06018#bib.bib27)\)\. When performing PCA on the response matrix𝐘\\mathbf\{Y\}, missing values are filled by iterative imputation\([Grung and Manne, 1998](https://arxiv.org/html/2609.06018#bib.bib19)\): column means serve as initial values, after which the missing entries are repeatedly replaced by their PCA reconstruction\. To fit the resulting PCA embedding on the grid, the two principal components of the imputed matrix are centered and isotropically rescaled\. Conditional on the initialized embedding,IXPLOREfits per\-item logistic regressions to obtain the initial parametersθ\\theta\.

##### Iterative optimization\.

We alternate two updates for a fixed number of iterationsTT, mirroring the structure of the EM ideal point estimation of[Imai et al\. \(2016\)](https://arxiv.org/html/2609.06018#bib.bib21): the*user step*recomputes the grid posteriorp⁡\(xn∣Ynobs,θ\)p\(x\_\{n\}\\mid Y\_\{n\}^\{\\text\{obs\}\},\\theta\)for every user from the current item parameters and replaces their embedding with the posterior mean from[Eq\.9](https://arxiv.org/html/2609.06018#S3.E9)\. The*item step*re\-fits each per\-item logistic regression on the updated embedding by maximizing the regularized log\-likelihood, whose per\-response terms are given in[Eq\.5](https://arxiv.org/html/2609.06018#S3.E5),

θ^k=arg⁡max⁡∑n∈ℛkθk⁡log⁡p⁡\(Yn​k∣x~n,θk\)−λ2​‖θk‖2,\\hat\{\\theta\}\_\{k\}\\;=\\;\\arg\\max\_\{\\theta\_\{k\}\}\\sum\_\{n\\in\\mathcal\{R\}\_\{k\}\}\\log p\(Y\_\{nk\}\\mid\\tilde\{x\}\_\{n\},\\theta\_\{k\}\)\\;\-\\;\\tfrac\{\\lambda\}\{2\}\\,\\\|\\theta\_\{k\}\\\|^\{2\},\(11\)with a weak L2 regularizerλ\\lambdaand Newton\-Raphson optimization\([McCullagh, 1989](https://arxiv.org/html/2609.06018#bib.bib32)\)\. Since the Bernoulli log\-likelihood can be evaluated for all response valuesYn​k∈\[0,1\]Y\_\{nk\}\\in\[0,1\], no thresholding to\{0,1\}\\\{0,1\\\}is necessary\. The full procedure is summarized in[Algorithm1](https://arxiv.org/html/2609.06018#alg1)\.

Algorithm 1IXPLORE’s training procedure\.1:response matrix

𝐘\\mathbf\{Y\}and optional confidence weights

𝐖\\mathbf\{W\}\.

2:Configuration:grid resolution

RR, number of iterations

TT, prior

p⁡\(x\)p\(x\), regularizer

λ\\lambda\.

3:

\{x~n\(0\)\}←\\\{\\tilde\{x\}\_\{n\}^\{\(0\)\}\\\}\\leftarrowinitialize embedding \(random or PCA\)\.

4:

\{θk\(0\)\}←\\\{\\theta\_\{k\}^\{\(0\)\}\\\}\\leftarrowper\-item logistic regression on

\{x~n\(0\)\}\\\{\\tilde\{x\}\_\{n\}^\{\(0\)\}\\\}\.

5:for

t=1,…,Tt=1,\\ldots,Tdo

6:for

n=1,…,Nn=1,\\ldots,Ndo⊳\\trianglerightuser step

7:compute the log\-likelihood of

YnobsY\_\{n\}^\{\\text\{obs\}\}given

\{θk\(t−1\)\}\\\{\\theta\_\{k\}^\{\(t\-1\)\}\\\}\.

8:

x~n\(t\)←\\tilde\{x\}\_\{n\}^\{\(t\)\}\\leftarrowposterior mean combining log\-likelihood with log\-prior\.

9:endfor

10:for

k=1,…,Kk=1,\\ldots,Kdo⊳\\trianglerightitem step

11:

θk\(t\)←\\theta\_\{k\}^\{\(t\)\}\\leftarrowregularized logistic regression of item

kk’s responses on

\{x~n\(t\)\}\\\{\\tilde\{x\}\_\{n\}^\{\(t\)\}\\\}\.

12:endfor

13:endfor

14:returnembedding

\{x~n\(T\)\}\\\{\\tilde\{x\}\_\{n\}^\{\(T\)\}\\\}and item parameters

\{θk\(T\)\}\\\{\\theta\_\{k\}^\{\(T\)\}\\\}\.

##### Inference\.

For a new user with a partial response vectorYobsY^\{\\text\{obs\}\}, embedding and imputation are achieved in a single user step with fixed item parametersθ\\thetaby evaluating the grid posterior from[Eq\.10](https://arxiv.org/html/2609.06018#S3.E10)\. The user’s ideal point is then given by the posterior mean\. In contrast to ML methods like PCA\([Potthoff, 2018](https://arxiv.org/html/2609.06018#bib.bib40)\),IXPLOREdoes not need to impute unanswered items before inference\. Instead, it can be used for*multiple imputation*\([Rubin, 1987](https://arxiv.org/html/2609.06018#bib.bib42)\)by sampling latent positions from the user’s posterior distribution and computing the corresponding synthetic response vectors\. A comparison of sampling mechanisms is given in[SectionA\.2\.2](https://arxiv.org/html/2609.06018#A1.SS2.SSS2)\.

## 4Evaluation Framework

To benchmark the performance ofIXPLORE, we use the evaluation framework of[Bachmann et al\. \(2024\)](https://arxiv.org/html/2609.06018#bib.bib3)on five different datasets\. This section describes the datasets, summarizes the evaluation metrics, and presents the experimental setup together with the research questions for each experiment\.

### 4\.1Datasets

We consider five response matrices drawn from voting\-advice, deliberation, roll\-call, and social\-survey settings, summarized in[Table1](https://arxiv.org/html/2609.06018#S4.T1)\. To probe imputation behavior as a function of how many answers per user are observed, we additionally mask responses at sparsity levelsu,v∈\{0\.0,0\.3,0\.6,0\.9\}u,v\\in\\\{0\.0,0\.3,0\.6,0\.9\\\}, whereuudenotes the train\-user sparsity andvvthe test\-user sparsity\.

Table 1:Datasets used in the experiments\. All response values are rescaled to\[0,1\]\[0,1\]\. Where no separate test set is available, we hold out a test set from the train data\.##### Smartvote \(2019\)\.

The VAASmartvotecollects political candidates’ responses to 75 questions related to topics in the 2019 Swiss federal election\([Politools, 2019](https://arxiv.org/html/2609.06018#bib.bib37)\)\. Each candidate answers all 75 items, so no entries are missing\.[Bachmann et al\. \(2024\)](https://arxiv.org/html/2609.06018#bib.bib3)use a subset of those candidates in their analysis, yielding 1,912 candidates across eight major parties\. We use this subset, mapped on a 0–1 scale, to compareIXPLORE’s performance against an existing baseline on complete data\.

##### Smartvote \(2023\)\.

TheSmartvotedata from the 2023 Swiss federal election contains both candidate and voter responses on the same 75\-item instrument\([Politools, 2023](https://arxiv.org/html/2609.06018#bib.bib38)\)\. We use a subset provided by[Bachmann et al\. \(2026\)](https://arxiv.org/html/2609.06018#bib.bib2), who generate a representative sample of voters from the complete data, where no entries are missing\. This representative candidate/voter division gives a natural train/test split: item parameters are estimated from the candidates, and voters serve as held\-out users for inference\.

##### Voteview \(U\.S\. Senate, S117\)\.

Voteviewcurates the complete roll\-call record of the U\.S\. Congress—one row per legislator, one column per recorded vote—and is the canonical input for spatial models of congressional ideology\([Boche et al\., 2018](https://arxiv.org/html/2609.06018#bib.bib8)\)\. We retain the 102 voting senators from the 117th Senate, drop 272 cloture votes, and remove 84 near\-unanimous votes \(more than 90% on either side\); cast codes are mapped to a binary scale with abstentions treated as missing\. The roll\-call data serves as a binary setting with highKKand lowNNthat complements theSmartvotedata\.

##### Polis \(vTaiwan\)\.

Polisis an online deliberation platform on which participants vote*agree*,*disagree*, or*pass*on user\-generated statements\([Small et al\., 2021](https://arxiv.org/html/2609.06018#bib.bib45)\)\. We use the public vTaiwan dataset, in which 1,921 participants discussed the regulation of taxi services in Taiwan\. The original ternary codes are mapped to\{0,0\.5,1\}\\\{0,0\.5,1\\\}\. The resulting response matrix is sparse by design: each participant only rates a subset of the statements, leaving 86\.9% of entries missing\. The Polis data serves as a stress test for models to fit with a large fraction of missing data\.

##### European Values Study \(2017\)\.

The European Values Study \(EVS\) surveys attitudes toward family, work, religion, politics, and society across European countries\([EVS, 2017](https://arxiv.org/html/2609.06018#bib.bib15)\)\. We restrict our analysis to the numerical subset of 32 items of the EVS 2017 dataset, drawn from the Joint EVS/WVS 2017–2021 release\([EVS/WVS, 2024](https://arxiv.org/html/2609.06018#bib.bib16)\)\. After removing respondents with any unanswered item and restricting to countries with more than 40 respondents, we retain 35,116 respondents across 34 countries\. Each item is rescaled per question to\[0,1\]\[0,1\]\. This dataset tests how well the algorithms perform with large numbers of users\.

### 4\.2Metrics

The four\-cell evaluation framework of[Bachmann et al\. \(2024\)](https://arxiv.org/html/2609.06018#bib.bib3)operates on two splits of the response matrix: first, it splits the data into train and test users; second, it holds out a fraction of existing responses and distinguishes between reconstructing the observed and imputing the withheld answers\. Therefore,*train reconstruction*evaluates how well the fitted model reconstructs the observed answers of train users\.*Train imputation*evaluates how well the fitted model predicts the withheld answers of train users\.*Test reconstruction*evaluates how well the fitted model generalizes to held\-out users, measuring the reconstruction of their observed answers\. Finally,*test imputation*evaluates how well held\-out users’ remaining questions can be imputed based on the embedding of their observed answers\.

##### Prediction metrics\.

Within each of the four evaluation cells, we report two metrics that emphasize complementary aspects of the prediction: The mean absolute error \(MAE\) treats the prediction as a continuous agreement\-probability and sums the absolute differences between predicted and true values\. The accuracy \(ACC\) considers a binary classification and counts a prediction as correct whenever the predicted and true responses fall on the same side of 0\.5\. While MAE rewards nuanced model predictions, ACC is more commonly used to assess model fit in ideal point estimation\([Bafumi et al\., 2005](https://arxiv.org/html/2609.06018#bib.bib6)\)\. For ACC, we exclude neutral ground\-truth values at exactlyy=0\.5y=0\.5\.

##### Latent\-geometry diagnostics\.

We additionally report two diagnostics of the latent embedding\. We define the boundary fraction \(BF\) as the share of users within 10% of the grid boundary; larger values indicate clusters at the boundary\. By distortion \(DIS\), we refer to the expected displacement of a latent point under the model parameters\. This metric is obtained by generating uniformly distributed users at positionsxnx\_\{n\}and re\-embedding them based on their synthetic responsesyk​\(xn\)y\_\{k\}\(x\_\{n\}\)using the model’s item parameters\. The distortion is then computed as the Euclidean distance between the original positionxnx\_\{n\}and the resulting posterior meanx~n\\tilde\{x\}\_\{n\}\. To approximate the distortion, we use a30×3030\\times 30grid spanning the latent space\.

### 4\.3Experimental Setup

We run two experiments to evaluate the performance ofIXPLORE\. The first experiment comparesIXPLOREagainst ten state\-of\-the\-art reference algorithms\. The second experiment analyzesIXPLORE’s configuration, varying three parameters of its optimization procedure \(prior variance, number of iterations, and feature transforms\)\.

#### 4\.3\.1Reference Algorithms

The core question of this report is whetherIXPLOREreconstructs and imputes responses at least as well as established latent\-space models, and whether it generalizes from train users to held\-out test users under increasing sparsity\. We compareIXPLOREagainst ten reference algorithms: First, we consider PCA with linear and logistic decoding\([Potthoff, 2018](https://arxiv.org/html/2609.06018#bib.bib40)\), as well as kernel PCA paired with a per\-item support vector machine\([Schölkopf and Smola, 2002](https://arxiv.org/html/2609.06018#bib.bib43)\)\. Second, we consider the multi\-dimensional scaling variantstt\-SNE\([van der Maaten and Hinton, 2008](https://arxiv.org/html/2609.06018#bib.bib49)\)and UMAP\([McInnes et al\., 2018](https://arxiv.org/html/2609.06018#bib.bib33)\), both with logistic decoding\. Third, we consider two VAEs\([Kingma and Welling, 2014](https://arxiv.org/html/2609.06018#bib.bib26)\), in a 2\-layer and a logistic\-decoder variant as previously used as a reference by[Bachmann et al\. \(2024\)](https://arxiv.org/html/2609.06018#bib.bib3)\. Fourth, we use the IRT models IDEAL\([Clinton et al\., 2004](https://arxiv.org/html/2609.06018#bib.bib12)\), its expectation\-maximization counterpart emIRT\([Imai et al\., 2016](https://arxiv.org/html/2609.06018#bib.bib21)\), and LSIRM\([Lee et al\., 2025](https://arxiv.org/html/2609.06018#bib.bib28)\)\. We do not include NOMINATE\([Poole and Rosenthal, 1985](https://arxiv.org/html/2609.06018#bib.bib39)\), because its estimation does not handle response matrices with many missing values well, which rules it out for the sparsity levels and thePolisdata considered here\. The exact configuration of each reference algorithm is given in[AppendixB](https://arxiv.org/html/2609.06018#A2)\.IXPLOREitself is run both with the original and binarized response matrix atT=10T=10iterations and otherwise default parameters \(see[Table3](https://arxiv.org/html/2609.06018#A1.T3)in[AppendixA](https://arxiv.org/html/2609.06018#A1)\)\.

##### Predictive performance\.

Each model is fitted on the train users at each train sparsityu∈\{0\.0,0\.3,0\.6,0\.9\}u\\in\\\{0\.0,0\.3,0\.6,0\.9\\\}over five random masking seeds, recording train reconstruction and imputation at every sparsity and seed\. The test users are embedded by the model fitted with unmasked train data \(u=0u=0\) and evaluated across the test\-sparsity levelsv∈\{0\.0,0\.3,0\.6,0\.9\}v\\in\\\{0\.0,0\.3,0\.6,0\.9\\\}, again over five seeds\. We report each metric both at every individual sparsity level and as a single aggregate over all sparsities\. While the aggregate can hide differences that appear only at specific sparsity levels, it yields one number per evaluation cell, dataset, and metric, which makes the comparison across all five datasets easier to interpret\.

##### Latent dimensionality\.

All reference algorithms embed into a two\-dimensional space to matchIXPLORE’s 2D grid \(except emIRT, whose current implementation is constrained to one dimension\)\. To separate the choice of algorithm from the choice of latent dimensionality, we further ask how many dimensions a PCA baseline needs to matchIXPLORE’s predictive performance\. To this end, we sweep PCA with linear and logistic decoding overd∈\{1,…,74\}d\\in\\\{1,\\dots,74\\\}dimensions, using the same sparsity levels and seeds, and read off, per evaluation cell, the smallestddat which each variant reaches the reconstruction and imputation error ofIXPLORE\.

#### 4\.3\.2Configuration Analysis

The second objective of this report is to analyze how the configuration ofIXPLOREshapes its performance and latent geometry\. For these experiments, we focus on theSmartvote2023 data\.

##### Prior regularization\.

The prior variance sets the diagonal of the Gaussian prior on the latent space in[Eq\.1](https://arxiv.org/html/2609.06018#S3.E1), with a smallerτ2\\tau^\{2\}concentrating the prior near the origin and a largerτ2\\tau^\{2\}flattening it toward a uniform prior\. We ask whether this prior variance affects the reconstruction and imputation error, as well as the latent\-geometry diagnostics\. For all values ofτ2∈\{0\.05,0\.1,0\.25,0\.5,1\.0,106\}\\tau^\{2\}\\in\\\{0\.05,\\,0\.1,\\,0\.25,\\,0\.5,\\,1\.0,\\,10^\{6\}\\\}, we train anIXPLOREmodel forT=20T=20iterations with PCA initialization and report the prediction metrics and latent\-geometry diagnostics aggregated over the sparsity levelsu∈\{0\.0,0\.3,0\.6,0\.9\}u\\in\\\{0\.0,0\.3,0\.6,0\.9\\\}\.

##### Iterative optimization\.

After initialization, each iteration first refits the per\-user posteriors against the current item models via[Eq\.9](https://arxiv.org/html/2609.06018#S3.E9), then refits the item models against the new posterior point estimates via[Eq\.11](https://arxiv.org/html/2609.06018#S3.E11)\. We ask \(i\) whether the iterative optimization of posteriors and item models improves the reconstruction and imputation error, and \(ii\) whether PCA initialization reduces the number of iterations until convergence\. We vary the iteration countT∈\{0,1,2,5,10,20\}T\\in\\\{0,1,2,5,10,20\\\}and the initialization scheme \(PCA vs\. random\) over the standard sparsity levels and seeds withτ2=0\.25\\tau^\{2\}=0\.25fixed\. The prediction metrics and latent\-geometry diagnostics are tracked as a function ofTT, and we compare how many iterations the PCA and random initializations each need to reach the same reconstruction and imputation error\.

##### Feature transforms\.

In the response function of[Eq\.3](https://arxiv.org/html/2609.06018#S3.E3), the optional feature transformφ\\varphimaps each 2D latent position into a higher dimensional vector before the per\-item logistic regression\. We ask whether non\-linear feature transforms improve reconstruction and imputation error while keeping decision boundaries visually interpretable\. We compare the default identity transform with linear decision boundaries; a polynomial transform with squared coordinates,φ⁡\(x\)=\(x1,x2,x12,x22\)\\varphi\(x\)=\(x\_\{1\},x\_\{2\},x\_\{1\}^\{2\},x\_\{2\}^\{2\}\); and an RFF transform that projects the position into a high\-dimensional cosine basis to approximate an RBF kernel\([Rahimi and Recht, 2007](https://arxiv.org/html/2609.06018#bib.bib41)\)\. We evaluate the three feature transforms over the standard sparsity levels and seeds at the default configuration \(PCA initialization,τ2=0\.25\\tau^\{2\}=0\.25,T=10T=10\)\.

Figure 1:IXPLOREand five reference algorithms onSmartvote2023\. Reconstruction and imputation error are reported as a function of sparsity in each of the four evaluation cells\. Three points of interest are highlighted: the train reconstruction error atu=0u=0corresponds to embedding candidates with full data; the test reconstruction error \(e\.g\., atv=0\.6v=0\.6\) corresponds to embedding voters in the same space; the test imputation error \(e\.g\., atv=0\.6v=0\.6\) corresponds to predicting voters’ remaining responses\.

## 5Results

We report our findings in two sections corresponding to the experiments described in[Section4\.3](https://arxiv.org/html/2609.06018#S4.SS3)\. All data and code to reproduce the results are available in our repository on GitHub\.555[https://github\.com/fsvbach/ixplore\-paper](https://github.com/fsvbach/ixplore-paper)

### 5\.1Reference Algorithms

[Fig\.1](https://arxiv.org/html/2609.06018#S4.F1)comparesIXPLOREagainst reference algorithms on theSmartvote2023 data\. As an overview, we show a selection of five reference algorithms that cover the variance across methods\. Similar figures for the other datasets are provided in[AppendixC](https://arxiv.org/html/2609.06018#A3)\([Figs\.11](https://arxiv.org/html/2609.06018#A3.F11)to[15](https://arxiv.org/html/2609.06018#A3.F15)\)\.

##### Reconstruction error\.

Panel A shows the reconstruction error across different sparsity values\. Most algorithms reconstruct the full data atu=0u=0well, starting from an MAE between 0\.181 \(IXPLORE\) and 0\.210 \(UMAP\)\. With increasing sparsity, the range of the reconstruction error grows:IXPLOREfurther reduces the MAE to 0\.141, while the MAE of LSIRM and UMAP increases to around 0\.238\. For the test data, all reconstruction errors are significantly higher due to the distribution shift from candidates \(train\) to voters \(test\)\. The algorithms fall into two categories: model\-based algorithms \(IDEAL, LSIRM, andIXPLORE\) improve performance with increasing sparsity, while the ML\-based algorithms \(UMAP, VAE, and PCA\) struggle to reconstruct test data at higher sparsity\. Atv=0\.9v=0\.9, the difference in MAE grows to a range from 0\.206 \(LSIRM\) to 0\.318 \(UMAP\)\.

##### Imputation error\.

Panel B shows the corresponding imputation error atv∈\{0\.3,0\.6,0\.9\}v\\in\\\{0\.3,0\.6,0\.9\\\}\. At low sparsity, the imputation errors are comparable across algorithms\. Atu=0\.9u=0\.9, the separation between algorithms becomes more visible: LSIRM and UMAP increase the imputation error up to 0\.281, whileIXPLORE’s error remains around 0\.223\. For the test data, the distinction between model\-based and ML\-based algorithms is less visible than before\. Notably, for both training and testing,IXPLOREreaches the lowest imputation error across all sparsities\.

Table 2:Performance comparison on Smartvote \(2023\)\. Each cell is the mean over all sparsity levels\. Reference algorithms are listed in the top block;IXPLOREis run both with the continuous input and the binarized input∗\. The best\-performing algorithm is shown inbold, the runner\-up initalics\.
#### 5\.1\.1Predictive Performance

[Table2](https://arxiv.org/html/2609.06018#S5.T2)aggregates MAE and ACC per evaluation cell across all sparsity levels and random seeds for theSmartvote2023 data\. For the other datasets, the corresponding tables are provided in[AppendixC](https://arxiv.org/html/2609.06018#A3)\([Table6](https://arxiv.org/html/2609.06018#A3.T6)–[Table9](https://arxiv.org/html/2609.06018#A3.T9)\)\.

##### Comparison across metrics\.

OnSmartvote2023,IXPLOREattains the lowest aggregated MAE in three of the four evaluation cells: train reconstruction \(0\.168\), train imputation \(0\.200\), and test imputation \(0\.261\)\. On test reconstruction it is the runner\-up at 0\.232 behind LSIRM \(0\.230\)\. On ACC, theIXPLOREvariant trained on binarized data classifies the train data 1 percentage point better than standardIXPLORE\. LSIRM generalizes best to test data, leadingIXPLOREby 1\.3 percentage points on test reconstruction\. For test imputation, bothIXPLOREvariants classify the held\-out responses equally well, ahead of the runner\-up IDEAL by 1 percentage point\.

##### Comparison across datasets\.

The observed ranking is stable across all five datasets: In every evaluation cell,IXPLOREachieves either the lowest or second\-lowest MAE\. In particular,IXPLOREreaches the lowest train reconstruction MAE everywhere \(0\.032 onVoteviewup to 0\.173 on EVS\) and the lowest test imputation MAE on every dataset exceptVoteview, where it ties IDEAL \(0\.083\)\. The margin to the runner\-up is widest onSmartvote2023 andPolisand narrowest onVoteview, where the IRT models IDEAL and emIRT remain competitive\. On EVS,IXPLOREgeneralizes best to test data, with a train reconstruction MAE of 0\.173 and a test reconstruction MAE of 0\.180\. However, on this larger dataset, IDEAL and LSIRM could not be run due to computational constraints\.

#### 5\.1\.2Latent Dimensionality

Figure 2:Reconstruction and imputation error of PCA onSmartvote2023 as a function of the latent dimensionalitydd, averaged over all sparsity levels\. The circles mark each curve’s minimum\. The horizontal red lines indicate theIXPLOREperformance withτ2=0\.25\\tau^\{2\}=0\.25, PCA initialization, andT=10T=10iterations\. Logistic PCA exceedsIXPLORE’s training reconstruction error ford≥3d\\geq 3\(Linear PCA ford≥21d\\geq 21\); neither variant reachesIXPLORE’s training or test imputation error at anyd≤74d\\leq 74\.[Fig\.2](https://arxiv.org/html/2609.06018#S5.F2)shows the reconstruction and imputation error of the two PCA variants as a function of the latent dimensionalitydd\. The training and test reconstruction error decreases monotonically withddfor both variants\. The logistic variant reaches the same train reconstruction error asIXPLOREatd=3d=3\(d=9d=9for test\) and the linear variant atd=21d=21\(d=15d=15for test\)\. PCA’s train and test imputation errors, in contrast, attain minima at small dimensionalities: the logistic variant decreases untild=3d=3with a train MAE of 0\.203; the linear variant decreases untild=4d=4with a train MAE of 0\.269\. For test imputation error, both minima are reached slightly later atd=5d=5\(logistic\) andd=7d=7\(linear\)\. At none of these minima does either variant reachIXPLORE’s training \(0\.200\) or test \(0\.261\) imputation error\.

### 5\.2Configuration Analysis

Figure 3:IXPLOREembeddings of the fullSmartvote2023 candidate data atT=20T=20for three different priors\. The reconstruction error stays stable, while the boundary fraction \(share of points within 10% of the grid boundary\) increases withτ2\\tau^\{2\}\.[Fig\.3](https://arxiv.org/html/2609.06018#S5.F3)shows theIXPLOREembeddings of theSmartvote2023 data for three different prior variances\. The tightest prior withτ2=0\.05\\tau^\{2\}=0\.05collapses the candidates into a tight cluster around the origin \(Panel A\), while the flat prior withτ2=106\\tau^\{2\}=10^\{6\}leaves 17\.6% of candidates within 10% of the grid boundary \(Panel C\)\. The prior atτ2=0\.25\\tau^\{2\}=0\.25produces a visible party structure without large boundary clusters \(Panel B\)\. Notably, there is no difference in predictive performance across priors, as all embeddings reach a reconstruction MAE of around 0\.18\.

Figure 4:Iteration trajectoriesT∈\{0,1,2,5,10,20\}T\\in\\\{0,1,2,5,10,20\\\}for each priorτ2\\tau^\{2\}with PCA initialization\. The marker size grows withTTwhile the first and last iteration are connected with an arrow pointing fromT=0T=0toT=20T=20\. \(A\) The boundary fraction and train reconstruction error atu=0u=0decrease over the course of iterations\. \(B\) The distortion of the latent space remains stable over the course of iterations, while the test imputation error decreases \(averaged overv∈\{0\.3,0\.6,0\.9\}v\\in\\\{0\.3,0\.6,0\.9\\\}\)\.#### 5\.2\.1Prior Regularization

[Fig\.4](https://arxiv.org/html/2609.06018#S5.F4)traces each prior’s iteration trajectory fromT=0T=0toT=20T=20in a more detailed analysis\. Panel A shows the boundary fraction and train reconstruction error atu=0u=0\. AtT=0T=0the PCA initialization places every prior at the same point\. AsTTincreases, each prior moves toward lower reconstruction error and boundary fraction at the bottom left\. Smaller priors reduce the boundary fraction much faster than larger priors, while all priors converge to a similar reconstruction error\. Panel B shows the distortion and test imputation error aggregates across sparsity levelsv∈\{0\.3,0\.6,0\.9\}v\\in\\\{0\.3,0\.6,0\.9\\\}\. While large priors already achieve minimal imputation error after initialization, smaller priors reach a similar performance only after several iterations\. The distortion of the embedding, however, is stable across iterations\. Larger priors have vanishing distortion independent of the iterations, while smaller priors first slightly decrease and then increase distortion withTT\.

Figure 5:Effect of initialization across iterations onSmartvote2023 atτ2=0\.25\\tau^\{2\}=0\.25\. \(A\) The reconstruction MAE is reported atu=0u=0\. \(B\) The imputation MAE is averaged overv∈\{0\.3,0\.6,0\.9\}v\\in\\\{0\.3,0\.6,0\.9\\\}\.
#### 5\.2\.2Iterative Optimization

[Fig\.5](https://arxiv.org/html/2609.06018#S5.F5)shows the convergence of the MAE withτ2=0\.25\\tau^\{2\}=0\.25for random and PCA initialization\. Random initialization starts with a reconstruction MAE above 0\.330 and then quickly decreases to 0\.183 after two iterations\. For the imputation MAE, convergence takes roughly ten iterations\. InitializingIXPLOREwith a PCA embedding significantly reduces the number of iterations needed for convergence\. With PCA initialization,IXPLOREconverges in less than two iterations, with the test\-data MAE improving only minimally thereafter\. Both random and PCA initialization, however, converge to the same reconstruction and imputation errors\.

#### 5\.2\.3Feature Transforms

Figure 6:Reconstruction and imputation error onSmartvote2023 data for the threeIXPLOREfeature transforms \(colored\) against the reference algorithms \(gray\)\.[Fig\.6](https://arxiv.org/html/2609.06018#S5.F6)reports reconstruction and imputation error for the three feature transforms compared to the reference algorithms\. Outperforming the results of the linear baseline, both the polynomial and RFF transforms can significantly improve the train reconstruction error across all sparsities\. The RFF transform, in particular, reduces the MAE from 0\.166 to 0\.144 on average\. The train imputation performance at high sparsities, however, is best for the linear model\. Atu=0\.9u=0\.9, both polynomial and RFF transforms exceed the MAE of the linear model by 0\.015 and 0\.021\. For test data, there is no visible difference across feature transforms\.

Figure 7:Predicted probability maps for three different feature transformations of theIXPLOREmodel initialized withτ2=0\.25\\tau^\{2\}=0\.25afterT=10T=10iterations on theSmartvote2023 data\. In each panel, the item with the largest accuracy spread across feature transforms is chosen \(item 32263\)\. The dots correspond to the 1,029 candidates colored by their response to this item \(orange/blue==no/yes\)\.In[Fig\.7](https://arxiv.org/html/2609.06018#S5.F7), we further inspect the*visual interpretability*of the three feature transforms by plotting the predicted probability contours for item 32263666Question:“Should it be possible to hold a referendum on federal spending above a certain amount?”\. This item was chosen as an example as it maximizes the range of predictive accuracy across feature transforms: the RFF transform reaches the highest accuracy with 80\.3%, while the polynomial transform reaches 67% and the linear baseline stays at 56%\. The RFF surface in Panel A, however, splits the latent space into three distinct areas of agreement, which might make it harder to interpret\. The polynomial transform in Panel B, in contrast, follows the candidate clusters with a single, smooth decision boundary on the 2D latent space, which might be more desirable for visual interpretability\. The linear transform in Panel C fails to capture the non\-linear structure of the data\.

## 6Discussion

Our experimental results show thatIXPLOREmatches or exceeds state\-of\-the\-art spatial models in terms of reconstruction and imputation quality\. Aggregated across sparsities,*IXPLOREattains the lowest or second\-lowest MAE in every evaluation cell across the five datasets*\. We discuss what drives this performance, howIXPLORE’s configuration shapes the embedding geometry, and what these findings imply for spatial modeling of political data\.

### 6\.1Predictive Performance

Regardless of the number of latent dimensions, PCA cannot matchIXPLORE’s imputation error on sparse users in theSmartvote2023 data \(see[Fig\.2](https://arxiv.org/html/2609.06018#S5.F2)\)\. This performance gap cannot be explained by the iterative optimization alone, which does*not*dramatically increaseIXPLORE’s predictive performance over the PCA initialization\. Instead, we attribute the improved performance toIXPLORE’s inference mechanism: while ML\-based models—such as PCA—minimize the reconstruction error, they must impute missing responses*before*embedding a new user\.IXPLOREavoids this step through a sparsity\-aware likelihood function inspired by IRT models\. For sparse test users in particular, this framework generalizes well and outperforms even IDEAL, which was previously found to excel at modeling theSmartvotedata\([Bachmann et al\., 2024](https://arxiv.org/html/2609.06018#bib.bib3)\)\. This performance gain may stem from the fact that IRT\-based models—such as IDEAL—are derived from a spatial\-utility function rather than optimized to predict held\-out responses\([Clinton and Jackman, 2009](https://arxiv.org/html/2609.06018#bib.bib11)\)\.

However, the results also showed some indications of overfitting\. While the generalization gap between train and test reconstruction is small forIXPLORE\(comparable to IRT baselines and much smaller than for ML baselines\), the gap between train reconstruction and train imputation is larger forIXPLOREon binary data\. On bothVoteviewandPolis, this reconstruction\-imputation gap exceeds the corresponding gap of IDEAL, especially at high sparsity\. This is likely driven by the weak L2 regularization used in our experiments forIXPLORE, which lets the per\-item logistic regression fit the few observed binary responses too tightly\. At the same time, this regularization choice also contributes toIXPLORE’s lowest train reconstruction on ordinal data, where the continuous response values bound the logistic gradient and provide implicit regularization\.

### 6\.2Latent Geometry

The choice of the prior variance does not affect the predictive quality ofIXPLORE\. This indicates that the prior acts purely as a geometric regularizer, controlling the trade\-off between boundary fraction and distortion of the embedding\. This regularization is particularly important forIXPLORE’s bounded latent space, as the Bernoulli likelihood pushes the posterior mass outwards for users with consistent response profiles\. A tight prior pulls the embedding back toward the origin, however, at the cost of a higher distortion\. Whether or not this distortion is problematic depends on the application: the higher the distortion, the more difficult it becomes for users to reach the extremes of the latent space, possibly undermining the credibility of the latent geometry\. An intermediate prior variance balances these forces, letting the embedding contract while keeping a low boundary fraction\. The optimal value forτ2\\tau^\{2\}, however, must be identified by visual inspection, which makes prior selection a target for automation in future work\.

### 6\.3Non\-Linear Feature Transforms

Non\-linear feature transforms can significantly improve the predictive performance ofIXPLORE\. The RFF transform, inspired by the RBF kernel\([Schölkopf and Smola, 2002](https://arxiv.org/html/2609.06018#bib.bib43);[Rahimi and Recht, 2007](https://arxiv.org/html/2609.06018#bib.bib41)\), outperforms both the linear and polynomial transforms in terms of reconstruction and imputation error\. Choosing such a non\-linear transform, however, departs from the tradition of utility functions in ideal point estimation\([Eguia, 2013](https://arxiv.org/html/2609.06018#bib.bib14)\)\. While the logistic regression of the linearIXPLOREvariant resembles the quadratic utility function of IRT models, the non\-linear transforms do not have a straightforward interpretation in terms of spatial utility\. In this trade\-off between accuracy and interpretability, the polynomial transform \([Fig\.7](https://arxiv.org/html/2609.06018#S5.F7)B\) may be a reasonable compromise, as its contour follows the candidate clusters with a single smooth decision boundary\.

### 6\.4Implications

IXPLOREis well suited for applications in which visualization is at the core of the analysis, since its bounded latent space keeps all users within a fixed frame\. It is particularly useful when predictive performance should be combined with visual interpretability, either through the per\-item logistic regression decision boundaries, or through the grid\-based uncertainty quantification777OnSmartvote2023,IXPLORElocates half of the held\-out users within their final1​σ1\\sigmaposterior ellipse after roughly a third of the items \(see[SectionA\.2\.3](https://arxiv.org/html/2609.06018#A1.SS2.SSS3)\)\.\.IXPLOREis also suited for online applications where inference must be fast: held\-out users are embedded through a single grid posterior evaluation against fixed item parameters\. Beyond these direct use cases,IXPLOREfills a methodological niche as one of few Python\-based IRT implementations, offering an alternative to the R\-based tooling that has historically dominated the field\. For developers of opinion landscapes and analysts of political response data,IXPLOREtherefore offers a new option in the algorithmic toolbox\.

## 7Limitations

There are several limitations to the current implementation ofIXPLORE, as well as to the experimental setup used to evaluate its performance\.

##### Implementation\.

WithRdR^\{d\}grid points for add\-dimensional latent space,IXPLORE’s grid\-based posterior inference does not scale beyond two dimensions\. Moreover, the grid\-based inference requires matrix multiplications overR2×NR^\{2\}\\times Nentries per iteration, which becomes prohibitive for very large user populations\. ExtendingIXPLOREto such regimes—through sparse or hierarchical grid approximations—is a natural direction for future work\. Furthermore, as with other dimensionality\-reduction algorithms, the axes of theIXPLOREembedding do not carry an inherent meaning\. When visualizing the embedding, this requires manual labeling of the axes\. In contrast to other algorithms,IXPLOREembeddings are not rotation invariant: the square grid breaks the rotational symmetry of the likelihood, so two runs with random initialization cannot be aligned by a post\-hoc rotation as in models such as IDEAL\. Future work could explore the use of language models to automatically label the axes based on the most influential items along each\.

##### Evaluation\.

The evaluation criteria chosen in this report almost exclusively reward predictive performance\. Yet for a political map intended for interpretation and visualization, reconstruction and imputation accuracy may not be the only objective\. Other criteria commonly used to evaluate ideal point estimation algorithms—such as the unidimensionality, reliability, and data quality of the derived scales\([Bruinsma, 2020](https://arxiv.org/html/2609.06018#bib.bib9)\)—are not considered in this report\. Furthermore, while the chosen datasets forIXPLORE’s evaluation cover a wide range of applications and datatypes, some of the reference algorithms only take binary values \(e\.g\., IDEAL\)\. This was shown to be an advantage when optimizing for accuracy but a disadvantage for the reconstruction error, as it prevents them from fitting the ordinal responses\. Finally, the sparsity in the data is induced synthetically: apart fromPolis, all datasets are complete, and we simulate sparse responders by masking answers uniformly at random, which does not reproduce the selective non\-response of real users in real\-world applications\. Within the scope of our evaluation, however,IXPLOREproves to be a robust and effective approach to latent space modeling\.

## 8Conclusion

We presentedIXPLORE, an ideal point estimation algorithm that places users on a bounded 2D latent space through grid\-based posterior inference\. Existing methods force a choice between model\-based algorithms, which are not optimized to predict held\-out responses, and ML\-based algorithms, which generalize poorly to sparse users\.IXPLOREcombines the strengths of both families with a sparsity\-aware likelihood function and a predictive fit objective\. Our experiments across five datasets showed that*this approach matches or surpasses both model\-based and ML\-based models on reconstruction and imputation error*\. This is particularly relevant for the sparse responses of online deliberation settings and the dense ordinal data of VAAs\.

Beyond predictive performance, we studied howIXPLORE’s configuration shapes the geometry of the latent space\. We found that the*prior variance trades boundary fraction against distortion without changing the reconstruction error*, while iterations further refine the embedding with slight improvement in imputation error\. When usingIXPLORE, we recommend PCA initialization for reproducibility and faster convergence\. As an extension to standard ideal point estimation, non\-linear feature transforms increase the flexibility ofIXPLORE: the RFF transform significantly lowered the train reconstruction error, while the polynomial transform kept the decision boundaries visually interpretable\.

Based on these findings, we conclude thatIXPLOREoffers a novel framework for constructing predictive political maps\. Several of its components—the bounded latent space, the grid\-based posterior, and the per\-item response surfaces—are*directly tailored to visualization settings*in which new users arrive with sparse answers\. In interactive applications such as adaptive questionnaires, VAAs, and deliberation platforms, where users must be embedded online and their remaining responses predicted,IXPLOREprovides a competitive alternative to both classical ideal point models and generic ML embeddings\. Thereby, our study contributes to visualizing political maps as instruments for democratic education\.

## Acknowledgements

I acknowledge the Swiss National Science Foundation \(SNSF\), which funds the research with grant ID CRSII5\-205975\. Furthermore, I am very grateful to the team fromPolitoolsfor providing theSmartvotedata\. Lastly, I thank my supervisors, collaborators, and students for their insights and support throughout the project\.

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## Appendix AModel Details

IXPLORE—short for*Iterative X\-Posterior LOgistic Regression Embedding*—exposes a small set of hyperparameters that control the prior, the optimization loop, and the per\-item logistic regression\. Their operating values are listed in[Table3](https://arxiv.org/html/2609.06018#A1.T3), alongside which sub\-paragraph of the configuration analysis \([Section5\.2](https://arxiv.org/html/2609.06018#S5.SS2)\) varies each one\. This section documents the design choices that affectIXPLORE’s behavior but are not central to the main text\.

Table 3:IXPLOREhyperparameters\. The defaults are used in the baseline comparison \([Section5\.1](https://arxiv.org/html/2609.06018#S5.SS1)\), while the configuration analysis \([Section5\.2](https://arxiv.org/html/2609.06018#S5.SS2)\) varies parameters as described\.### A\.1Gaussian Prior and Log\-Barrier

Figure 8:One\-dimensional slice of the two prior families atx2=0x\_\{2\}=0over the grid\[−L,L\]2\[\-L,L\]^\{2\}withL=1L=1\. \(A\) The Gaussian prior concentrates around the origin asτ2\\tau^\{2\}shrinks\. \(B\) The log\-barrier prior is flat in the interior and vanishes at the grid boundary for largerα\\alpha\. Both priors are symmetric inx1x\_\{1\}andx2x\_\{2\}\.IXPLOREsupports two prior families over the latent position, shown in[Fig\.8](https://arxiv.org/html/2609.06018#A1.F8)\. The Gaussian prior with varianceτ2\\tau^\{2\}is centrally peaked\. The log\-barrier prior with strengthα\\alphais flat in the interior and vanishes at the grid boundary\. We compare both onSmartvote2023 under the default configuration \(T=5T=5, candidates as training users\), sweeping the Gaussian varianceτ2\\tau^\{2\}and the log\-barrier strengthα\\alpha, and additionally embedding voters with between 5 and 75 observed answers\.

The reconstruction error is flat across every prior and strength \(MAE in 0\.1814–0\.1831\)\. The boundary fraction varies from 0% to 38%, and collapses to 0% under either a strong Gaussian \(τ2≤0\.1\\tau^\{2\}\\leq 0\.1\) or a strong log\-barrier \(α≥2\\alpha\\geq 2\)\. At a matched boundary fraction of 0%, the log\-barrier retains more embedding spread than the Gaussian \(spread 0\.43 atα=5\\alpha=5versus 0\.37 atτ2=0\.05\\tau^\{2\}=0\.05\)\. For voters embedded from 5 to 75 observed answers, the imputation MAE varies by at most 0\.01 across the Gaussian, log\-barrier, and uniform\-prior configurations\.

### A\.2Point Estimate from the Grid Posterior

To summarize each user’s grid posterior with a single 2D positionx~n\\tilde\{x\}\_\{n\}, three options are natural\. The*posterior mean*is the probability\-weighted average grid position of[Eq\.10](https://arxiv.org/html/2609.06018#S3.E10)

x~n=∑g∈Xgridg​p​\(xn=g∣Ynobs,θ\)\.\\tilde\{x\}\_\{n\}\\;=\\;\\sum\_\{g\\in X\_\{\\text\{grid\}\}\}g\\,p\(x\_\{n\}=g\\mid Y\_\{n\}^\{\\text\{obs\}\},\\theta\)\.The*MAP*is the single highest\-posterior grid point

x^n=arg⁡maxg∈Xgrid⁡p⁡\(xn=g∣Ynobs,θ\)\.\\hat\{x\}\_\{n\}\\;=\\;\\arg\\max\_\{g\\in X\_\{\\text\{grid\}\}\}p\(x\_\{n\}=g\\mid Y\_\{n\}^\{\\text\{obs\}\},\\theta\)\.*Full marginalization*avoids collapsing the posterior at all: predictions for new items integrate the response function against the posterior, the discrete counterpart of[Eq\.8](https://arxiv.org/html/2609.06018#S3.E8),

Y^n​k=∑g∈Xgridyk​\(g\)​p​\(xn=g∣Ynobs,θ\)\.\\hat\{Y\}\_\{nk\}\\;=\\;\\sum\_\{g\\in X\_\{\\text\{grid\}\}\}y\_\{k\}\(g\)\\,p\(x\_\{n\}=g\\mid Y\_\{n\}^\{\\text\{obs\}\},\\theta\)\.We compare the three onSmartvote2023 \(N=1029N\{=\}1029,K=75K\{=\}75\) at the default configuration \(τ2=0\.25\\tau^\{2\}=0\.25,T=10T=10\)\.

##### Predictive accuracy\.

The posterior mean yields the lowest held\-out MAE at every observation level \([Table4](https://arxiv.org/html/2609.06018#A1.T4)\)\. Its gap to the MAP is largest with few observed answers \(Δ​MAE=0\.0058\\Delta\\text\{MAE\}=0\.0058atnobs=5n\_\{\\text\{obs\}\}\{=\}5\) and narrows to 0\.0014 atnobs=70n\_\{\\text\{obs\}\}\{=\}70\. Full marginalization is consistently the worst of the three: the smoothing it provides does not compensate for predicting a probability that need not match any single grid point\.

##### Latent geometry\.

The MAP estimator is constrained to theG=R2G=R^\{2\}grid coordinates, which produces lattice artifacts in the scatter plot at any practical resolution and makes it pile up on the boundary as the prior weakens: the fraction of embeddings within 5% of the grid edge grows from 0\.000 atτ2=0\.05\\tau^\{2\}=0\.05to 0\.149 atτ2=106\\tau^\{2\}=10^\{6\}\([Table5](https://arxiv.org/html/2609.06018#A1.T5)\)\. The posterior mean, pulled inward by the probability\-weighted average, stays at or below 0\.007 across the whole range while the reconstruction quality is essentially identical between the two estimators: MAE stays near 0\.181 and ACC at 0\.835–0\.836 for everyτ2\\tau^\{2\}\.

Table 4:Held\-out MAE onSmartvote2023 by point estimator and number of observed answers, at the default configuration \(τ2=0\.25\\tau^\{2\}=0\.25,T=10T=10\)\. Best per row inbold\.Table 5:Boundary fraction \(share of embeddings within 5% of the grid edge\) and train reconstruction MAE / accuracy onSmartvote2023, per prior variance with a model retrained at eachτ2\\tau^\{2\}\(T=10T=10\), for the MAP and posterior\-mean estimators\.The posterior mean is therefore set as the default as it achieves the lowest held\-out MAE at every observation level, matches reconstruction quality with the MAP, and raises no boundary artifacts\.

#### A\.2\.1Vectorized Grid Evaluation

The dominant per\-iteration cost is evaluating the Bernoulli log\-likelihood at theGGgrid points for each ofNNusers againstKKitems, naively𝒪⁡\(N​K​G\)\\mathcal\{O\}\(NKG\)\. Defining the masked answer matricesY~n​k=Mn​k​Yn​k\\tilde\{Y\}\_\{nk\}=M\_\{nk\}Y\_\{nk\}andZ~n​k=Mn​k​\(1−Yn​k\)\\tilde\{Z\}\_\{nk\}=M\_\{nk\}\(1\-Y\_\{nk\}\), and the per\-grid\-cell log\-prediction tablesLPg​k=log⁡yk​\(g\)\\mathrm\{LP\}\_\{gk\}=\\log y\_\{k\}\(g\)andLNg​k=log⁡\(1−yk​\(g\)\)\\mathrm\{LN\}\_\{gk\}=\\log\(1\-y\_\{k\}\(g\)\), the full\(N,G\)\(N,G\)matrix of user log\-likelihoods is

L=Y~​LP⊤\+Z~​LN⊤,L\\;=\\;\\tilde\{Y\}\\,\\mathrm\{LP\}^\{\\top\}\\;\+\\;\\tilde\{Z\}\\,\\mathrm\{LN\}^\{\\top\},\(12\)two BLAS matrix products of shape\(N,K\)×\(K,G\)\(N,K\)\\times\(K,G\), which replaces the triple loop with two dense matrix multiplications\.

#### A\.2\.2Sampling Strategies

For applications that require synthetic answer vectors rather than point predictions—e\.g\., data augmentation or active\-learning ensembles—IXPLOREsupports two sampling strategies in addition to a uniform\-random baseline, following the principle of multiple imputation\([Rubin, 1987](https://arxiv.org/html/2609.06018#bib.bib42)\)\.*Posterior sampling*draws latent positions from the grid posteriorp⁡\(xn=g∣Ynobs,θ\)p\(x\_\{n\}=g\\mid Y\_\{n\}^\{\\text\{obs\}\},\\theta\)and predicts continuous answer probabilities from each sampled position via the fitted item models, so the output is continuous in\[0,1\]\[0,1\]and the sample variance reflects the posterior spread\.*Rasch sampling*first imputes all missing answers via posterior marginalization, then, for each item, evaluates normal category\-response functions centered at each discrete answer option \(e\.g\.\{0,0\.25,0\.5,0\.75,1\}\\\{0,0\.25,0\.5,0\.75,1\\\}for a 5\-point scale\) and samples a category by the Gumbel\-max trick, producing discrete output whose variance is set by the CRF bandwidth\.

We compare the two onSmartvote2023 under the same protocol as the point\-estimate study \([SectionA\.2](https://arxiv.org/html/2609.06018#A1.SS2)\): for each candidate and eachnobsn\_\{\\text\{obs\}\}, a random subset ofnobsn\_\{\\text\{obs\}\}answers is revealed; we measure \(i\) the MAE between the per\-item sample mean and the true answer on the remaining items, and \(ii\) the per\-item sample standard deviation on those items\.

Figure 9:Sampling strategies onSmartvote2023 as a function of the number of observed answers\. \(A\) The MAE is computed between each method’s per\-item sample mean and the true answer on the unobserved items, for Rasch, posterior, and uniform\-random sampling, with the deterministic posterior\-mean prediction for reference\. \(B\) The mean per\-item sample standard deviation on the unobserved items is reported; higher values indicate more diverse samples\.Both strategies track the deterministic posterior\-mean prediction closely on MAE at every sparsity level, while the uniform\-random baseline has substantially higher MAE\. They differ in sample diversity: posterior samples inherit their spread from the posterior, so variance is high when the user has answered few questions and shrinks as the posterior concentrates; Rasch samples have variance fixed by the CRF bandwidth, which makes them less diverse for sparse users but more interpretable as plausible discrete answers\.

Since both strategies preserve predictive accuracy while adding well\-calibrated diversity, neither is set as a default\. The choice depends on the downstream use: posterior sampling for continuous augmentation that reflects posterior uncertainty, Rasch sampling for discrete pseudo\-answers in the model’s native answer space\.

#### A\.2\.3Posterior Convergence on Test Users

We track at which itemkkeach test user’s running posterior mean first enters the1​σ1\\sigmaellipse of their final posterior, as a measure of how many answers suffice to localize the user up to posterior uncertainty \([Fig\.10](https://arxiv.org/html/2609.06018#A1.F10)\)\. Half of the held\-out voters are inside the final1​σ1\\sigmaellipse afterk=23k=23items and 95% byk=53k=53—roughly a third of the questionnaire suffices for the median user\. This rate is essentially insensitive to the prior: shrinkingτ2\\tau^\{2\}from10610^\{6\}\(effectively flat\) to 0\.05 leaves the CDF unchanged within one item across the whole range\.

Figure 10:Posterior trajectory and convergence onSmartvote2023\. \(A\) The test user whose final posterior mean lies furthest from the origin is shown, fitted atτ2=0\.25\\tau^\{2\}=0\.25\. The contour is the posterior density after the user has answeredk=15k=15ofK=75K=75items; black dots and the connecting line trace the running posterior mean at everykkin answer order; the dashed ellipse is the1​σ1\\sigmacontour of the moment\-matched Gaussian fit to the final posterior\. \(B\) The CDF of the smallestkkat which the running posterior mean enters the final1​σ1\\sigmaellipse is shown, evaluated over all 1,121 test users, for the tightest \(τ2=0\.05\\tau^\{2\}=0\.05\) and weakest \(τ2=106\\tau^\{2\}=10^\{6\}\) priors in our sweep\. Dotted lines mark the 50% and 95% crossings\.

## Appendix BReference Algorithms

All baselines embed into a 2\-dimensional latent space to be visualizable as a political spectrum \(emIRT excepted, see below\)\. When requiring binary input, each algorithm binarizes responses at threshold 0\.5 before fitting\. Neutral responses at exactly 0\.5 are treated as missing\. When an algorithm does not have native inference, a 2D embedding is fitted, and a per\-item binary logistic decoderY^n​k=σ⁡\(xn⊤​βk\+αk\)\\hat\{Y\}\_\{nk\}=\\sigma\(x\_\{n\}^\{\\top\}\\beta\_\{k\}\+\\alpha\_\{k\}\)predicts the answers from the embedded position\. Where necessary, sparse held\-out \(test\) users are imputed by the train\-column means\.

##### PCA\.

The pipeline of[Potthoff \(2018\)](https://arxiv.org/html/2609.06018#bib.bib40)represents each user by add\-dimensional PCA score\.*Linear PCA*skips the logistic decoder and instead reconstructsY^n​k=\(xn⊤​vk\+μk\)\\hat\{Y\}\_\{nk\}=\(x\_\{n\}^\{\\top\}v\_\{k\}\+\\mu\_\{k\}\)clipped to\[0,1\]\[0,1\], where\(vk,μk\)\(v\_\{k\},\\mu\_\{k\}\)are the rank\-ddPCA loadings and column mean, with missing training entries filled by column means before a single PCA fit\.*Logistic PCA*uses the shared logistic decoder, with missing entries handled by the iterative\-PCA imputation of[Grung and Manne \(1998\)](https://arxiv.org/html/2609.06018#bib.bib19)\(column\-mean initialization, then alternating rank\-ddreconstruction until convergence\)\. For the main analysis in[Section5\.1](https://arxiv.org/html/2609.06018#S5.SS1), we choosed=2d=2to matchIXPLORE’s latent space; for the dimensionality study of[Section5\.1\.2](https://arxiv.org/html/2609.06018#S5.SS1.SSS2), we sweepd∈\{1,…,74\}d\\in\\\{1,\.\.\.,74\\\}\.

##### Kernel PCA\.

The embedding stage is an RBF\-kernel PCA\([Schölkopf and Smola, 2002](https://arxiv.org/html/2609.06018#bib.bib43)\), the nonlinear counterpart of the linear PCA above, with missing entries mean\-imputed before the kernel is computed\. The kernel bandwidthγ\\gammais the only hyper\-parameter; its useful scale is dataset\-dependent, so it is selected on the training reactions alone: a random 15% of the observed training entries is held out, kernel PCA and the decoder are refitted for each candidateγ\\gammaon the remaining entries, and theγ\\gammawith the lowest mean absolute error on the held\-out entries is kept\. The candidate grid is expressed as the multipliers\{0\.1,0\.25,0\.5,1,2,4\}\\\{0\.1,0\.25,0\.5,1,2,4\\\}of the RBF median heuristic \(γ=1/median\\gamma=1/\\mathrm\{median\}pairwise squared distance\) so that it adapts to each dataset’s scale\. Held\-out users are projected withscikit\-learn’s kernel\-PCA out\-of\-sample transform against the fitted training points\.

##### t\-SNE\.

The embedding stage istt\-SNE \(perplexity 30\), using theopenTSNEimplementation so that held\-out users can be projected into the fitted space with its out\-of\-sample transform\.

##### UMAP\.

The embedding stage is UMAP \(15 neighbors, minimum distance 0\.1\); a fixed random state is used for reproducibility, which forces single\-threaded execution\. Held\-out users are projected with UMAP’s out\-of\-sample transform\.

##### Variational Auto\-Encoder\.

VAEs learn the embedding and the decoder jointly as one generative model\. We use the variational auto\-encoder\([Kingma and Welling, 2014](https://arxiv.org/html/2609.06018#bib.bib26)\)in the configuration of the VAE baseline of[Bachmann et al\. \(2024\)](https://arxiv.org/html/2609.06018#bib.bib3)\. The encoder is a single 64\-unit ReLU hidden layer followed by two heads producingμ⁡\(y\)\\mu\(y\)andlog⁡σ2​\(y\)\\log\\sigma^\{2\}\(y\)for a 2D Gaussian posterior over the latentzz\. We compare two decoders: a*logistic*decoderσ⁡\(W​z\+b\)\\sigma\(Wz\+b\)and a*2\-layer*decoder with a 64\-unit ReLU hidden layer and a sigmoid output\. Training minimizes a mask\-renormalized reconstruction loss plus a KL term,

ℒ=∑\(n,k\)∈𝒪\(Y^n​k−Yn​k\)2\|𝒪\|/\(N​K\)\+β⋅KL\(q\(z∣Y\)∥𝒩\(0,I\)\),\\mathcal\{L\}\\;=\\;\\frac\{\\sum\_\{\(n,k\)\\in\\mathcal\{O\}\}\(\\hat\{Y\}\_\{nk\}\-Y\_\{nk\}\)^\{2\}\}\{\|\\mathcal\{O\}\|/\(NK\)\}\\;\+\\;\\beta\\cdot\\mathrm\{KL\}\\left\(q\(z\\mid Y\)\\,\\\|\\,\\mathcal\{N\}\(0,I\)\\right\),\(13\)withβ=1\\beta=1for the 2\-layer decoder andβ=0\\beta=0for the logistic decoder, mean\-imputation of missing entries at the encoder input, and a binary observation mask on the reconstruction term\. We optimize with Adam \(lr=10−3\\text\{lr\}=10^\{\-3\}\) and early\-stop on a 10% held\-out validation split with patience 300\. Predictions for any cell\(n,k\)\(n,k\)are obtained by passing the user’s mean\-imputed answer vector through the encoder, taking the posterior mean as the latent code, and reading off the decoder output at indexkk\.

##### IDEAL\.

We use the Bayesian item\-response model of[Clinton et al\. \(2004\)](https://arxiv.org/html/2609.06018#bib.bib12)and[Jackman \(2001\)](https://arxiv.org/html/2609.06018#bib.bib23)with the probit likelihood

Pr⁡\(Yn​k=1∣xn,βk,αk\)=Φ⁡\(xn⊤​βk−αk\),\\Pr\(Y\_\{nk\}=1\\mid x\_\{n\},\\beta\_\{k\},\\alpha\_\{k\}\)\\;=\\;\\Phi\\bigl\(x\_\{n\}^\{\\top\}\\beta\_\{k\}\-\\alpha\_\{k\}\\bigr\),\(14\)as implemented in thepsclR package\([Jackman, 2024](https://arxiv.org/html/2609.06018#bib.bib24)\)\. Item parameters are fitted on the training matrix by MCMC \(10,000 iterations, 5,000 burn\-in, thinning 100\); the wrapper retains the posterior\-mean discriminationsβk\\beta\_\{k\}and difficultiesαk\\alpha\_\{k\}\. To embed a held\-out user, we run a second MCMC chain with item parameters held fixed via spike priors \(variance10610^\{6\}onβ\\beta\), so only the new user’s ideal point is updated\. Predictions for any cell\(n,k\)\(n,k\)areY^n​k=Φ⁡\(xn⊤​βk−αk\)\\hat\{Y\}\_\{nk\}=\\Phi\(x\_\{n\}^\{\\top\}\\beta\_\{k\}\-\\alpha\_\{k\}\)at the posterior\-mean estimate ofxnx\_\{n\}\.

##### emIRT\.

We use the fast expectation\-maximization estimator of[Imai et al\. \(2016\)](https://arxiv.org/html/2609.06018#bib.bib21)for the binary ideal point model, as implemented by thebinIRTroutine of the emIRT R package\. This is the EM counterpart of IDEAL: it maximizes the same probit likelihood

Pr⁡\(Yn​k=1∣xn,βk,αk\)=Φ⁡\(βk​xn−αk\),\\Pr\(Y\_\{nk\}=1\\mid x\_\{n\},\\beta\_\{k\},\\alpha\_\{k\}\)\\;=\\;\\Phi\\bigl\(\\beta\_\{k\}\\,x\_\{n\}\-\\alpha\_\{k\}\\bigr\),\(15\)but recovers point estimates by EM rather than MCMC, so it is included as a fast\-estimation reference on the same 0\.5\-binarized data as IDEAL\. The estimator is one\-dimensional only; we expose the single estimated ideal point as the first embedding coordinate and fix the second at 0, so emIRT appears as a horizontal line in the two\-dimensional embedding plots\. Item parameters\(βk,αk\)\(\\beta\_\{k\},\\alpha\_\{k\}\)are fitted on the training matrix by EM \(convergence threshold10−610^\{\-6\}\)\. To embed a held\-out user, the item parameters are held fixed and the user’s ideal point is obtained by maximizing their binary log\-likelihood, the same one\-dimensional counterpart of the procedure used for the other reference algorithms\. Predictions for any cell\(n,k\)\(n,k\)areY^n​k=Φ⁡\(βk​xn−αk\)\\hat\{Y\}\_\{nk\}=\\Phi\(\\beta\_\{k\}\\,x\_\{n\}\-\\alpha\_\{k\}\)at the estimatedxnx\_\{n\}\.

##### LSIRM\.

We use the latent\-space item\-response model of[Jeon et al\. \(2021\)](https://arxiv.org/html/2609.06018#bib.bib25), in the metric\-space formulation of[Lee et al\. \(2025\)](https://arxiv.org/html/2609.06018#bib.bib28), as implemented by thelsirm2plroutine of thelsirm12plR package\. Each usernnand itemkkreceive a 2D latent positionznz\_\{n\}andwkw\_\{k\}, and the response probability decreases with their Euclidean distance,

logit​Pr⁡\(Yn​k=1∣zn,wk,θn,αk,βk\)=θn​αk\+βk−γ⁡∥zn−wk∥,\\mathrm\{logit\}\\,\\Pr\(Y\_\{nk\}=1\\mid z\_\{n\},w\_\{k\},\\theta\_\{n\},\\alpha\_\{k\},\\beta\_\{k\}\)\\;=\\;\\theta\_\{n\}\\,\\alpha\_\{k\}\+\\beta\_\{k\}\-\\gamma\\,\\lVert z\_\{n\}\-w\_\{k\}\\rVert,\(16\)whereθn\\theta\_\{n\}is a per\-user random effect,αk\\alpha\_\{k\}andβk\\beta\_\{k\}are item discrimination and difficulty, andγ\\gammascales the distance term\. The user positionsznz\_\{n\}are the embedding\. Item parameters and positions are fitted on the training matrix by MCMC \(5,000 iterations, 1,000 burn\-in, thinning 5, missing entries handled under the missing\-at\-random model\); the wrapper retains the posterior\-mean estimates\. To embed a held\-out user, the item parameters\(wk,αk,βk,γ\)\(w\_\{k\},\\alpha\_\{k\},\\beta\_\{k\},\\gamma\)are held fixed and the user’s\(zn,θn\)\(z\_\{n\},\\theta\_\{n\}\)is the maximum\-a\-posteriori estimate under the same prior the training MCMC uses \(zn∼𝒩⁡\(0,I\)z\_\{n\}\\sim\\mathcal\{N\}\(0,I\),θn∼𝒩⁡\(0,1\)\\theta\_\{n\}\\sim\\mathcal\{N\}\(0,1\)\), so out\-of\-sample positions stay in the same regularized regime as the training posterior means rather than diverging under an unconstrained fit\. Predictions for any cell\(n,k\)\(n,k\)areY^n​k=σ⁡\(θn​αk\+βk−γ⁡∥zn−wk∥\)\\hat\{Y\}\_\{nk\}=\\sigma\\bigl\(\\theta\_\{n\}\\alpha\_\{k\}\+\\beta\_\{k\}\-\\gamma\\lVert z\_\{n\}\-w\_\{k\}\\rVert\\bigr\)at the estimated\(zn,θn\)\(z\_\{n\},\\theta\_\{n\}\)\.

## Appendix CAdditional Tables and Figures

Table 6:Performance comparison on Smartvote \(2019\)\. Each cell is the mean over all sparsity levels\. Reference algorithms are listed in the top block;IXPLOREis run both with the continuous input and the binarized input∗\. The best\-performing algorithm is shown inbold, the runner\-up initalics\.Table 7:Performance comparison on Polis \(vTaiwan\)\. Each cell is the mean over all sparsity levels\. Reference algorithms are listed in the top block;IXPLOREis run both with the continuous input and the binarized input∗\. The best\-performing algorithm is shown inbold, the runner\-up initalics\.Table 8:Performance comparison on Voteview \(S117\)\. Each cell is the mean over all sparsity levels\. Reference algorithms are listed in the top block;IXPLOREis run both with the continuous input and the binarized input∗\. The best\-performing algorithm is shown inbold, the runner\-up initalics\.Table 9:Performance comparison on European Values Study\. Each cell is the mean over all sparsity levels\. Reference algorithms are listed in the top block;IXPLOREis run both with the continuous input and the binarized input∗\. The best\-performing algorithm is shown inbold, the runner\-up initalics\.Figure 11:Reconstruction \(solid, circles\) and imputation \(dashed, squares\) MAE onSmartvote2019 as a function of train sparsityuuand test sparsityvv\. Error bars show the standard deviation across seeds\.Figure 12:Reconstruction \(solid, circles\) and imputation \(dashed, squares\) MAE onSmartvote2023 as a function of train sparsityuuand test sparsityvv\. Error bars show the standard deviation across seeds\.Figure 13:Reconstruction \(solid, circles\) and imputation \(dashed, squares\) MAE onVoteviewas a function of train sparsityuuand test sparsityvv\. Error bars show the standard deviation across seeds\.Figure 14:Reconstruction \(solid, circles\) and imputation \(dashed, squares\) MAE onPolisas a function of train sparsityuuand test sparsityvv\. Error bars show the standard deviation across seeds\.Figure 15:Reconstruction \(solid, circles\) and imputation \(dashed, squares\) MAE on EVS as a function of train sparsityuuand test sparsityvv\. Error bars show the standard deviation across seeds\.
## Appendix DConfidence Weights

Finally, we ask how users’ confidence weights, optionally scaled, affect the imputation error and the latent geometry\.IXPLOREmultiplies each answer’s contribution to the user log\-likelihood by a per\-\(user, item\) weightWn​kW\_\{nk\}, and optionally rescales each user’s weights so that sparse and dense responders carry the same total weight \([5](https://arxiv.org/html/2609.06018#S3.E5)\)\. We evaluate three weight schedules derived from theSmartvote2023 confidence matrix introduced in[Section4\.1](https://arxiv.org/html/2609.06018#S4.SS1)—*uniform*\(all 1s\),*provided*\(the matrix as\-is\), and*extreme*\(the matrix squared\)—each with weight scaling disabled and enabled\. The model is trained once with uniform weights at the default configuration and grid resolutionR=100R=100; the weights are then applied to the held\-out test users at inference time only, across the test\-sparsity levelsvv, so the underlying item parameters are held fixed across schedules\.

### D\.1Results

[Fig\.16](https://arxiv.org/html/2609.06018#A4.F16)shows three diagnostics as a function of test\-time sparsity\. The weight schedule itself has a negligible effect on the reconstruction and imputation error\. Across*uniform*,*provided*, and*extreme*weights the imputation MAE differs by at most 0\.0015 within either setting of weight\-scaling\.

Weight\-scaling, by contrast, changes the latent geometry\. Enabling it pushes sparse users toward the grid boundary, raising the boundary fraction atv=0\.9v=0\.9from 0\.0002 to 0\.093 \([Fig\.16](https://arxiv.org/html/2609.06018#A4.F16)B\)\. At the same time it reduces the distortion of the latent space, with the mean distortion dropping from 0\.119 to 0\.095 atv=0\.3v=0\.3and from 0\.357 to 0\.134 atv=0\.9v=0\.9\([Fig\.16](https://arxiv.org/html/2609.06018#A4.F16)C\)\. A grid\-level visualization of this effect is shown in[Fig\.17](https://arxiv.org/html/2609.06018#A4.F17)\.

Figure 16:Latent\-geometry diagnostics onSmartvote2023 across test\-time sparsityvv, trained with uniform weights and PCA initialization,τ2=0\.25\\tau^\{2\}=0\.25andT=10T=10; weights are passed at inference time only\. The color distinguishes the three weight schedules; line style distinguishes*disabled*and*enabled*weight\-scaling\. \(A\) The test imputation error as a function of test\-time sparsity\. \(B\) The boundary fraction corresponds to the share of test users within 10% of the grid boundary\. \(C\) The distortion of the trained latent space is computed by re\-embedding synthetic users with uniform weights\.Table 10:Confidence\-weights summary on Smartvote 2023\. The model is trained once on candidates with uniform weights \(PCA initialization,τ2=0\.25\\tau^\{2\}=0\.25,T=10T=10iterations, grid resolutionR=100R=100\); each row is an inference\-time configuration of weight scaling and weight schedule\. Cells are averaged across sparsity levels and seeds\. The first four metric columns report test reconstruction and imputation error; the last three are latent\-geometry diagnostics: the spread as the total variance of the embedding, the boundary fraction \(BF\), and the distortion \(DIS\)\. Best in columnbold, runner\-up initalics\.![Refer to caption](https://arxiv.org/html/2609.06018v1/weights_distortion_smartvote_2023.png)Figure 17:Distortion of the trained latent space onSmartvote2023 across test\-time sparsityv∈\{0,0\.3,0\.6,0\.9\}v\\in\\\{0,0\.3,0\.6,0\.9\\\}\(columns\) with weight scaling disabled \(top row\) and enabled \(bottom row\)\. Synthetic users sampled on a30×3030\\times 30grid have a fraction1−v1\-vof their predicted answers retained and are re\-embedded\. Each arrow points from a synthetic user’s grid position to its re\-embedded position; the inset reports the mean and standard deviation of the displacement∥x^−x∥\\lVert\\hat\{x\}\-x\\rVertover the grid\.
### D\.2Discussion

As a novel extension to ideal point estimation,IXPLOREallows confidence weights for users to emphasize certain responses\. However, we could not demonstrate the effectiveness of this extension, because the weights provided in theSmartvote2023 data were not expressive enough: just 11% of answered items carry a non\-default confidence, and the provided values take only two levels \(0\.5 and 2\)\. Still,IXPLORE’s*weight\-scaling*—the setting that equalizes the total weight of sparse and dense responders—substantially reduces the distortion of sparse users at the cost of a higher boundary fraction\. This could be valuable when displaying users in the political map throughout a survey or deliberation process\([Small et al\., 2021](https://arxiv.org/html/2609.06018#bib.bib45);[Bachmann et al\., 2026](https://arxiv.org/html/2609.06018#bib.bib2)\), as it would allow a larger spread of users’ positions with few responses\.