The Future Is Finer: Multivariate Time Series Super-Resolution Forecasting

# Multivariate Time Series Super-Resolution Forecasting
Source: [https://arxiv.org/html/2610.33470](https://arxiv.org/html/2610.33470)
## The Future Is Finer: Multivariate Time Series Super\-Resolution Forecasting

DOI:[XXXXXXX\.XXXXXXX](https://doi.org/XXXXXXX.XXXXXXX)Conference:The ACM Web Conference 2027; May 10 to 14, 2027; Dublin, IrelandISBN:978\-1\-4503\-XXXX\-X/2018/06Shuhao LiAffiliation:Fudan University,Shanghai,ChinaAffiliation:Nanyang Technological University,Singaporeemail:[shli23@m\.fudan\.edu\.cn](mailto:[email protected])Weidong YangAffiliation:Fudan University,Shanghai,ChinaAffiliation:Zhuhai Fudan Innovation Research Institute,Zhuhai,China,Yue CuiAffiliation:Tongyi Laboratory Alibaba Group,Hangzhou,China,Fan ZhangAffiliation:Guangzhou University,Guangzhou,ChinaandLu QinAffiliation:University of Technology Sydney,Sydney,Australia

2027

###### Abstract\.

Web platforms run on multivariate time series, from service telemetry and Web traffic logs to sensor streams served through Web APIs\. Yet, retention policies often archive them at coarse granularity to save storage and bandwidth\. Decisions such as autoscaling, alerting, and billing operate at finer granularity than existing forecasters reach, since these models predict at the resolution of their input\. We therefore introduceMultivariateTimeSeriesSuper\-ResolutionForecasting \(MTS\-SRF\), which forecasts fine\-grained futures from coarse histories, and proposeTimeMagnifierto address its three challenges\. Fourier Anchor counters data sparsity by fitting each input window to a trend\-Fourier structure with sparse cross\-variable gating; De\-noise Only constrains fine\-grained ambiguity by refining along the resolution axis rather than the noise axis, lengthening the grid level by level; and banded DTW Alignment corrects cross\-resolution misalignment against the coarse forecast\. On the first MTS\-SRF benchmark, with ten datasets from five domains, real archived cross\-resolution pairs, strengthened baseline adaptations, and five\-seed significance tests, TimeMagnifier is best on every dataset, resolution coefficient, and metric against 15 baselines and cuts MSE by 12\.3% on average\. Its De\-noise Only and DTW stages also transfer as a plug\-in that lowers the MSE of existing forecasters by 5\.1% to 9\.0%\. Code will be made publicly available\.

###### Keywords:

Multivariate Time Series Forecasting, Super\-Resolution Forecasting, Coarse\-to\-Fine Temporal Modeling, Web Telemetry

## 1\.Introduction

Figure 1\.Same\-resolution forecasting versus MTS\-SRF from the same hourly history\. The hourly forecast returns only on\-hour values, while the 10\-minute forecast recovers the wind crest between them that crosses the alert level\.Multivariate time series underpin many Web services, from server and network telemetry and Web traffic logs to environmental and energy sensors published through Web platforms\([Runge et al\., 2019](https://arxiv.org/html/2610.33470#bib.bib1);[Avila and Mezić, 2020](https://arxiv.org/html/2610.33470#bib.bib2);[March et al\., 2021](https://arxiv.org/html/2610.33470#bib.bib3)\)\. They are collected at fixed intervals such as every minute, hour, or day\([Zhou et al\., 2021](https://arxiv.org/html/2610.33470#bib.bib4);[Lai et al\., 2018](https://arxiv.org/html/2610.33470#bib.bib5)\), and to bound storage and transmission cost they are commonly rolled up to coarser intervals as they age, as in the downsampling retention of Web monitoring stacks\([Adams et al\., 2020](https://arxiv.org/html/2610.33470#bib.bib55)\)\. Existing forecasters, however, predict at the resolution of their input\([Oreshkin et al\., 2020](https://arxiv.org/html/2610.33470#bib.bib6);[Cirstea et al\., 2022](https://arxiv.org/html/2610.33470#bib.bib7)\), so a coarse archive cannot directly support decisions made at finer granularity\.

For instance, when a Web\-based weather service trains its forecasting model on hourly sampled data, the model cannot capture short\-term variations, such as when rainfall will stop within the hour, which often matter more for real\-world decisions\.Figure[1](https://arxiv.org/html/2610.33470#S1.F1)illustrates this on a Weather test window\. From the same hourly history of saturated vapor pressure \(VPmax\) and maximum wind speed \(max\.wv\), an hourly forecast returns only on\-hour values that stay below the 5 m/s wind alert, whereas a 10\-minute forecast recovers the crest between two stamps that crosses it, a slow swing below the coarse Nyquist frequency\. The same gap arises on other Web platforms, where energy platforms schedule load in 15\-minute intervals while archiving hourly records, and navigation services need hourly congestion forecasts from coarser traffic archives\. Simply raising the sampling frequency is costly: \(1\)higher collection cost, as sensors and agents must report more often; \(2\)storage and transmission that grow linearly with the sampling rateacross every monitored series; and \(3\)greater computational overheadfor retraining\([Whang and Lee, 2020](https://arxiv.org/html/2610.33470#bib.bib8);[Wang et al\., 2020a](https://arxiv.org/html/2610.33470#bib.bib9);[Ekambaram et al\., 2023](https://arxiv.org/html/2610.33470#bib.bib10);[Lin et al\., 2024](https://arxiv.org/html/2610.33470#bib.bib11)\)\.

Instead, we formalize multivariate time series super\-resolution forecasting \(MTS\-SRF\), which forecasts a finer future grid from coarse history alone and uses fine data only as training supervision, a combination that mixed\-frequency, multi\-scale, and super\-resolution methods each relax\. MTS\-SRF is also not forecasting followed by interpolation, since interpolating a coarse forecast cannot by itself restore the between\-stamp variation shown inFigure[1](https://arxiv.org/html/2610.33470#S1.F1)\. It poses three challenges\. First,data sparsitylimits the model’s ability to capture critical trends and patterns\([Vandal et al\., 2017](https://arxiv.org/html/2610.33470#bib.bib50);[Kuleshov et al\., 2017](https://arxiv.org/html/2610.33470#bib.bib51)\)\. Coarse data often lack essential details, and their noise can be magnified in fine\-grained forecasts, leading to biased trend estimations\. Second,fine\-grained ambiguitycomplicates prediction, since many fine\-grained futures are compatible with the same coarse history\. Coarse\-grained patterns reflect broad time scales and fail to capture short\-term dynamics\([Ghysels et al\., 2007](https://arxiv.org/html/2610.33470#bib.bib45);[Koutsoyiannis and Montanari, 2007](https://arxiv.org/html/2610.33470#bib.bib12)\), which matter most in time\-critical tasks like real\-time weather or traffic forecasting\. Finally,cross\-resolution misalignmentdisrupts modeling\. A forecast refined on a finer grid can drift in phase against the coarse evidence, and real archives add timestamp jitter\. Effective forecasting requires alignment across resolutions to prevent error propagation\.

To address these challenges, we proposeTimeMagnifier, built on a single idea, treating the coarse forecast as a resolution\-degraded version of the fine future and refining it along the resolution axis, much as diffusion models refine along the noise axis\. TheFourier Anchormodule counters data sparsity by fitting each input window to a trend\-Fourier structure with sparse cross\-variable gating, yielding a structured coarse forecast and per\-window periodic priors\. TheDe\-noise Onlymodule constrains fine\-grained ambiguity by lengthening the grid level by level, smoothing expansion artifacts and re\-injecting these priors under fine supervision resampled to every level\. TheDTW Alignmentmodule corrects cross\-resolution misalignment by realigning each level to the coarse forecast, thereby keeping refinement anchored\. A multi\-granularity loss trains all three stages jointly\. Our main contributions are as follows:

- •We formulate the MTS\-SRF problem, where models forecast high\-resolution sequences from long\-interval observations, addressing limitations of conventional fixed\-resolution forecasting\.
- •We proposeTimeMagnifier, whose De\-noise Only stage refines along the resolution axis rather than the noise axis, lengthening the grid at each level under resampled fine supervision and per\-window periodic priors, and is coupled with banded DTW that realigns every level to the coarse forecast\.
- •We show that the De\-noise Only and DTW stages transfer to other forecasters, consistently improving DLinear and PatchTST when appended to their unchanged backbones\.
- •We build the first MTS\-SRF benchmark for deep multivariate forecasting on ten datasets from five domains, including real archived cross\-resolution pairs, and show significant gains over strong and strengthened baselines under five seeds\.

## 2\.Problem Formulation

As illustrated inFigure[2](https://arxiv.org/html/2610.33470#S2.F2), given a sampling time pointtt, letxit,Δ​τGx\_\{i\}^\{t,\\Delta\\tau\_\{G\}\}denote the observed value at sampling timetton theii\-th time series with granularityΔ​τG\\Delta\\tau\_\{G\}, whereΔ​τG=Γ​Δ​τmin\\Delta\\tau\_\{G\}=\\Gamma\\Delta\\tau\_\{\\text\{min\}\},Δ​τmin\\Delta\\tau\_\{\\text\{min\}\}is the smallest sampling interval, i\.e\., the highest resolution, andΓ\\Gammais the low\-resolution coefficient, indicating the factor by which the sampling interval of the coarse\-grained time series exceeds the highest resolutionΔ​τmin\\Delta\\tau\_\{\\text\{min\}\}\. Specifically, a time series with granularityΔ​τG\\Delta\\tau\_\{G\}has a longer sampling period thanΔ​τmin\\Delta\\tau\_\{\\text\{min\}\}, andΓ\\Gammais a positive integer representing the magnification of the lower resolution relative to the highest resolution\.

Thus, the setXt,Δ​τG=\{x1t,Δ​τG,x2t,Δ​τG,…,xNt,Δ​τG\}X^\{t,\\Delta\\tau\_\{G\}\}=\\\{x\_\{1\}^\{t,\\Delta\\tau\_\{G\}\},x\_\{2\}^\{t,\\Delta\\tau\_\{G\}\},\\dots,x\_\{N\}^\{t,\\Delta\\tau\_\{G\}\}\\\}represents the observed values ofNNvariables at granularityΔ​τG\\Delta\\tau\_\{G\}and time pointtt\.

Figure 2\.An example of the multivariate time series super\-resolution forecasting problem, where the low\-resolution coefficient isΓ=4\\Gamma=4, i\.e\.,Δ​τG=4​Δ​τmin\\Delta\\tau\_\{G\}=4\\Delta\\tau\_\{\\text\{min\}\}, and the high\-resolution coefficient isγ=2\\gamma=2, i\.e\.,Δ​τg=2​Δ​τmin\\Delta\\tau\_\{g\}=2\\Delta\\tau\_\{\\text\{min\}\}\.Multivariate Time Series Super\-Resolution Forecasting \(MTS\-SRF\)\. Given a historical look\-back time windowTT, letX=\{X1,Δ​τG,X2,Δ​τG,…,XT,Δ​τG\}X=\\\{X^\{1,\\Delta\\tau\_\{G\}\},\\allowbreak\\,X^\{2,\\Delta\\tau\_\{G\}\},\\dots,\\,X^\{T,\\Delta\\tau\_\{G\}\}\\\}represent the observed values of the low resolution time seriesXXover the pastTTtime steps, where eachXt,Δ​τGX^\{t,\\Delta\\tau\_\{G\}\}is theNN\-variate observation with granularityΔ​τG\\Delta\\tau\_\{G\}at timett\. Given a requested target\-grid horizonhreqh\_\{\\mathrm\{req\}\}, letΓ,γ∈ℕ\+\\Gamma,\\gamma\\in\\mathbb\{N\}^\{\+\}withγ≤Γ\\gamma\\leq\\Gamma,d=gcd⁡\(Γ,γ\)d=\\gcd\(\\Gamma,\\gamma\),p=Γ/dp=\\Gamma/d, andq=γ/dq=\\gamma/d\. To keep both grids integral and temporally compatible for any rational ratio, we retain complete resolution blocks:

\(1\)m=⌈hreqp⌉,h=m​p,hG=m​q,r=hhG=Γγ\.m=\\left\\lceil\\frac\{h\_\{\\mathrm\{req\}\}\}\{p\}\\right\\rceil,\\quad h=mp,\\quad h\_\{G\}=mq,\\quad r=\\frac\{h\}\{h\_\{G\}\}=\\frac\{\\Gamma\}\{\\gamma\}\.HereΔ​τg=γ​Δ​τmin\\Delta\\tau\_\{g\}=\\gamma\\Delta\\tau\_\{\\text\{min\}\}withγ\\gammathe high\-resolution coefficient; the at mostp−1p\-1extra steps use real targets and are not scored\. The primary MTS\-SRF setting isγ<Γ\\gamma<\\Gamma, whileγ=Γ\\gamma=\\Gammareduces to same\-resolution forecasting\.

![Refer to caption](https://arxiv.org/html/2610.33470v1/TimeMagnifier1.png)Figure 3\.Illustration of TimeMagnifier’s architectural framework and inference data flow withΓ=4,γ=2\\Gamma=4,\\gamma=2as an example\.Objective:The goal is to generate future fine\-grained data from historical low\-resolution data, similar to super\-resolution restoration in computer vision\([Lei et al\., 2024](https://arxiv.org/html/2610.33470#bib.bib40);[Wang et al\., 2020b](https://arxiv.org/html/2610.33470#bib.bib42)\)\.

Remark on supervision\.MTS\-SRF requires fine\-resolution data only as training supervision; at deployment the model consumes coarse history alone, matching settings where fine\-grained collection is limited by cost, storage, or sensors\. The benchmark therefore constructs coarse inputs by point\-sampling, as image super\-resolution does with synthetic degradation\. ETT also offers real archived pairs, since its official hourly ETTh files are exactly the on\-hour records of the 15\-minute ETTm files, and theΓ=4\\Gamma=4ETTm columns ofTable[1](https://arxiv.org/html/2610.33470#S4.T1)use these pairs\.

Formal Expression\.We seek a mappingfffrom the low\-resolution historyXXto the future high\-resolution seriesY^\\hat\{Y\}, that is:

\(2\)Y^=f⁡\(X,θ\)\\hat\{Y\}=f\(X;\\theta\)wheref⁡\(⋅,θ\)f\(\\cdot;\\theta\)represents the forecasting function,θ\\thetaare the model parameters, andY^\\hat\{Y\}collects the firsthreqh\_\{\\mathrm\{req\}\}of thehhinternal steps, estimating the ground\-truth high\-resolution seriesY=\{Y1,Δ​τg,…,Yhreq,Δ​τg\}∈ℝhreq×NY=\\\{Y^\{1,\\Delta\\tau\_\{g\}\},\\dots,\\allowbreak Y^\{h\_\{\\mathrm\{req\}\},\\Delta\\tau\_\{g\}\}\\\}\\in\\mathbb\{R\}^\{h\_\{\\mathrm\{req\}\}\\times N\}, with coarse counterpartYΔ​τGY^\{\\Delta\\tau\_\{G\}\}\.

## 3\.TimeMagnifier Framework

To tackle MTS\-SRF, we propose TimeMagnifier, a lightweight coarse\-to\-fine framework illustrated inFigure[3](https://arxiv.org/html/2610.33470#S2.F3)\. TimeMagnifier decomposes super\-resolution forecasting into three coupled stages that mirror the three challenges identified in Section[1](https://arxiv.org/html/2610.33470#S1)\. TheFourier Anchorstage counteracts downsampling\-induced information loss by estimating the trend and periodic structure that survive coarse sampling and borrowing complementary information from correlated variables, producing a structured coarse forecastY¯\\bar\{Y\}together with Fourier coefficients\{Ai\(k\),Bi\(k\)\}\\\{A\_\{i\}^\{\(k\)\},B\_\{i\}^\{\(k\)\}\\\}\. TheDe\-noise Onlystage constrains fine\-grained ambiguity\. Instead of predicting the fine sequence in one shot, it decomposes theΓ/γ\\Gamma/\\gammaresolution gap into successive refinement levels, reusing the Fourier coefficients of the first stage as periodic priors to constrain the ill\-posed upsampling\. TheDTW Alignmentstage corrects cross\-resolution misalignment by warping the refined sequence back onto the coarse priorY¯\\bar\{Y\}, so that every refinement step remains temporally consistent with the structured coarse forecast\. The three stages are output\-coupled rather than concatenated, since the second reuses the periodic coefficients of the first, the third aligns against the output of the first, and a multi\-granularity loss optimizes all of them jointly\.

### 3\.1\.Fourier Anchor Module

Coarse sampling removes high\-frequency detail, but the components that survive it, namely long\-term trends, dominant periodicities, and cross\-variable correlations, are exactly what a super\-resolution forecaster can still rely on\([Shao et al\., 2022](https://arxiv.org/html/2610.33470#bib.bib13);[Zeng et al\., 2023](https://arxiv.org/html/2610.33470#bib.bib14);[Deng et al\., 2021](https://arxiv.org/html/2610.33470#bib.bib15)\)\. TheFourier Anchorstage therefore makes this surviving structure explicit by decomposing each variable into a linear trend, a Fourier\-based periodic term, and a learnable cross\-series interaction term, yielding both a stable coarse\-grained forecast at granularityΔ​τG\\Delta\\tau\_\{G\}and periodic coefficients that the subsequent stage reuses at finer resolutions\.

Input\-Conditioned Trend and Periodic Components\.Coarse sampling preserves the linear trend and the dominant periodicities below the coarse Nyquist frequency, which Fourier series theory lets us represent by sines and cosines\([Jenkins, 2022](https://arxiv.org/html/2610.33470#bib.bib16)\)\. We collect both into the basis

\(3\)ϕ⁡\(τ\)=\[1,τ,cos⁡\(ω1​τ\),sin⁡\(ω1​τ\),…,cos⁡\(ωK​τ\),sin⁡\(ωK​τ\)\]⊤\\phi\(\\tau\)=\\left\[1,\\,\\tau,\\,\\cos\(\\omega\_\{1\}\\tau\),\\,\\sin\(\\omega\_\{1\}\\tau\),\\dots,\\cos\(\\omega\_\{K\}\\tau\),\\,\\sin\(\\omega\_\{K\}\\tau\)\\right\]^\{\\top\}withωk=2​π​Γ​Δ​τmin/Πk\\omega\_\{k\}=2\\pi\\Gamma\\Delta\\tau\_\{\\text\{min\}\}/\\Pi\_\{k\}forKKdominant periodsΠk\\Pi\_\{k\}fixed on the coarse training split, and writeΦH∈ℝT×D\\Phi\_\{H\}\\in\\mathbb\{R\}^\{T\\times D\}andΦF∈ℝhG×D\\Phi\_\{F\}\\in\\mathbb\{R\}^\{h\_\{G\}\\times D\},D=2\+2​KD=2\+2K, for the basis matrices evaluated on the historical gridτtH=t−T\\tau\_\{t\}^\{H\}=t\-Tand the future coarse gridτnF=n\\tau\_\{n\}^\{F\}=n, both expressed in units ofΔ​τG\\Delta\\tau\_\{G\}, so that the last observed step sits at time zero and the basis is dimensionless\. For each inputXX, Fourier Anchor fits the coefficients by ridge regression and extrapolates them forward:

\(4\)C⁡\(X\)=\(ΦH⊤​ΦH\+λridge​R\)−1​ΦH⊤​X,Y~​\(X\)=ΦF​C​\(X\),C\(X\)=\\left\(\\Phi\_\{H\}^\{\\top\}\\Phi\_\{H\}\+\\lambda\_\{\\mathrm\{ridge\}\}R\\right\)^\{\-1\}\\Phi\_\{H\}^\{\\top\}X,\\qquad\\widetilde\{Y\}\(X\)=\\Phi\_\{F\}\\,C\(X\),whereR=diag⁡\(0,1,…,1\)R=\\operatorname\{diag\}\(0,1,\\dots,1\)leaves the intercept unpenalized\. For fixed frequencies this is an explicit linear mapY~​\(X\)=PFA​X\\widetilde\{Y\}\(X\)=P\_\{\\mathrm\{FA\}\}XwithPFA=ΦF​\(ΦH⊤​ΦH\+λridge​R\)−1​ΦH⊤P\_\{\\mathrm\{FA\}\}=\\Phi\_\{F\}\(\\Phi\_\{H\}^\{\\top\}\\Phi\_\{H\}\+\\lambda\_\{\\mathrm\{ridge\}\}R\)^\{\-1\}\\Phi\_\{H\}^\{\\top\}, shared across variables and independent ofNN\. The columnci​\(X\)c\_\{i\}\(X\)ofC⁡\(X\)C\(X\)holds the intercept, trend, and Fourier coefficients\{bi​\(X\),ai​\(X\),Ai\(k\)​\(X\),Bi\(k\)​\(X\)\}\\\{b\_\{i\}\(X\),a\_\{i\}\(X\),A\_\{i\}^\{\(k\)\}\(X\),B\_\{i\}^\{\(k\)\}\(X\)\\\}of variableii, ordered as inϕ⁡\(τ\)\\phi\(\\tau\), which are reused as input\-conditioned periodic priors during refinement\.

Cross Component\.Channel\-dependent strategies can outperform channel\-independent ones but incur higher computational cost\([Qiu et al\., 2025](https://arxiv.org/html/2610.33470#bib.bib39);[Liu et al\., 2024](https://arxiv.org/html/2610.33470#bib.bib26)\)\. We therefore adopt a sparse shared\-gating design that retains effectiveness on a compact parameter budget:

\(5\)ai​jn=exp⁡\(gi​jn\)∑ℓ∈𝒩iexp⁡\(gi​ℓn\),Y¯in​\(X\)=Y~in​\(X\)\+κ​∑j∈𝒩iai​jn⋅Y~jn​\(X\)a\_\{ij\}^\{n\}=\\frac\{\\exp\(g\_\{ij\}^\{n\}\)\}\{\\sum\_\{\\ell\\in\\mathcal\{N\}\_\{i\}\}\\exp\(g\_\{i\\ell\}^\{n\}\)\},\\qquad\\bar\{Y\}\_\{i\}^\{n\}\(X\)=\\widetilde\{Y\}\_\{i\}^\{n\}\(X\)\+\\kappa\\sum\_\{j\\in\\mathcal\{N\}\_\{i\}\}a\_\{ij\}^\{n\}\\cdot\\widetilde\{Y\}\_\{j\}^\{n\}\(X\)wheregi​jn=MLPθg​\(\[Y~in​\(X\);Y~jn​\(X\)\]\)g\_\{ij\}^\{n\}=\\text\{MLP\}\_\{\\theta\_\{g\}\}\(\[\\widetilde\{Y\}\_\{i\}^\{n\}\(X\);\\widetilde\{Y\}\_\{j\}^\{n\}\(X\)\]\)is a shared lightweight network learning the interaction strength between variablesiiandjjat coarse stepnn,𝒩i\\mathcal\{N\}\_\{i\}collects the\|𝒩i\|=min⁡\(16,N−1\)\|\\mathcal\{N\}\_\{i\}\|=\\min\(16,N\-1\)channels most correlated withiiunder absolute Pearson correlation on the coarse training split, andκ\\kappais a learnable scalar controlling the strength of the cross\-variable residual\. Since bothY~\\widetilde\{Y\}and the gates are functions ofXX, the coarse forecastY¯\\bar\{Y\}is fully input\-conditioned\. We abbreviateY¯in​\(X\)\\bar\{Y\}\_\{i\}^\{n\}\(X\)asY¯in\\bar\{Y\}\_\{i\}^\{n\}in what follows\.

Optimization\.The trend and Fourier coefficients are computed from each input through Eq\. \([4](https://arxiv.org/html/2610.33470#S3.E4)\) and are not global model parameters; the trainable Fourier Anchor parameters are the shared interaction MLPθg\\theta\_\{g\}and the residual scalarκ\\kappa, whileλridge=10−3\\lambda\_\{\\mathrm\{ridge\}\}=10^\{\-3\}is fixed across all datasets and resolution settings\. They are trained by minimizing a weighted mean squared error \(WMSE\):

\(6\)ℒFA=∑i=1Nϖi​\(1hG​∑n=1hG\(Y¯in​\(X\)−Yin,Δ​τG\)2\)\\mathcal\{L\}\_\{\\text\{FA\}\}=\\sum\_\{i=1\}^\{N\}\\varpi\_\{i\}\\left\(\\frac\{1\}\{h\_\{G\}\}\\sum\_\{n=1\}^\{h\_\{G\}\}\\left\(\\bar\{Y\}\_\{i\}^\{n\}\(X\)\-Y\_\{i\}^\{n,\\Delta\\tau\_\{G\}\}\\right\)^\{2\}\\right\)where the series weightsϖi\\varpi\_\{i\}default to uniform andYin,Δ​τGY\_\{i\}^\{n,\\Delta\\tau\_\{G\}\}is the ground\-truth observation at granularityΔ​τG\\Delta\\tau\_\{G\}\. BecauseY~​\(X\)=ΦF​C​\(X\)\\widetilde\{Y\}\(X\)=\\Phi\_\{F\}C\(X\), the coarse forecast lies in the trend\-Fourier subspacecol⁡\(ΦF\)\\operatorname\{col\}\(\\Phi\_\{F\}\)before cross\-variable mixing\. At both training and inference,Y¯\\bar\{Y\}and\{Ai\(k\)​\(X\),Bi\(k\)​\(X\)\}\\\{A\_\{i\}^\{\(k\)\}\(X\),B\_\{i\}^\{\(k\)\}\(X\)\\\}are passed to the De\-noise Only Module as its initialization and periodic prior\.

### 3\.2\.De\-noise Only Module

Even with the surviving structure recovered, upsampling a coarse forecast onto a finer grid remains ill\-posed, since many fine\-grained futures are compatible with the same coarse estimate\. Inspired by the coarse\-to\-fine refinement principle of diffusion\-based generative models\([Rasul et al\., 2021](https://arxiv.org/html/2610.33470#bib.bib30);[Ho et al\., 2020](https://arxiv.org/html/2610.33470#bib.bib41)\), we treat the Fourier Anchor output as a resolution\-degraded version of the fine\-grained target and refine it deterministically, bypassing stochastic noise injection\. Starting fromY^\(0\)=Y¯\\hat\{Y\}^\{\(0\)\}=\\bar\{Y\}, theDe\-noise Onlystage appliesJJrefinement steps, each of which refines the sampling interval, smooths the expanded sequence, and re\-injects the periodic prior learned by Fourier Anchor, transformingY^\(0\)\\hat\{Y\}^\{\(0\)\}into a fine\-grained sequenceY^\(J\)\\hat\{Y\}^\{\(J\)\}\.

Granularity Expansion\.For the boundary caser=1r=1we setJ=1J=1with identity expansion, so the output still passes one level of smoothing, periodic correction, and alignment\. The refinement ladder is defined on sequence lengths: letL\(0\)=hGL^\{\(0\)\}=h\_\{G\}and

\(7\)L\(j\+1\)=min⁡\(2​L\(j\),h\),J=max⁡\(1,⌈log2⁡r⌉\),L^\{\(j\+1\)\}=\\min\\left\(2L^\{\(j\)\},\\,h\\right\),\\qquad J=\\max\\big\(1,\\left\\lceil\\log\_\{2\}r\\right\\rceil\\big\),so the discrete expansion factorq~j=L\(j\+1\)/L\(j\)∈\[1,2\]\\tilde\{q\}\_\{j\}=L^\{\(j\+1\)\}/L^\{\(j\)\}\\in\[1,2\]equals two except possibly at the final level, and one only whenr=1r=1\. Becauseh=r​hGh=rh\_\{G\}by Eq\. \([1](https://arxiv.org/html/2610.33470#S2.E1)\), the normalized sampling interval at leveljjis

\(8\)Δ​τ\(j\)=hGL\(j\),\\Delta\\tau^\{\(j\)\}=\\frac\{h\_\{G\}\}\{L^\{\(j\)\}\},so thatΔ​τ\(0\)=1\\Delta\\tau^\{\(0\)\}=1andΔ​τ\(J\)=1/r\\Delta\\tau^\{\(J\)\}=1/rhold exactly in coarse\-step units\. We index the leveljjgrid𝒢\(j\)\\mathcal\{G\}^\{\(j\)\}byn=1,…,L\(j\)n=1,\\dots,L^\{\(j\)\}, with interval\-end timestampsτn\(j\)=n​Δ​τ\(j\)\\tau\_\{n\}^\{\(j\)\}=n\\Delta\\tau^\{\(j\)\}, matching the future coarse gridτnF\\tau\_\{n\}^\{F\}; expansion is a resampling operator onto the next grid:

\(9\)Y^i\(j,1\)=ℐ𝒢\(j\)→𝒢\(j\+1\)​\(Y^i\(j\)\)\\hat\{Y\}\_\{i\}^\{\(j,1\)\}=\\mathcal\{I\}\_\{\\mathcal\{G\}^\{\(j\)\}\\to\\mathcal\{G\}^\{\(j\+1\)\}\}\\big\(\\hat\{Y\}\_\{i\}^\{\(j\)\}\\big\)which reduces to value replication whenq~j=2\\tilde\{q\}\_\{j\}=2and to nearest\-neighbor resampling when1<q~j<21<\\tilde\{q\}\_\{j\}<2\. The grid lengths double fromhGh\_\{G\}, with a possibly shorter final expansion that reacheshh, so the total refined length isO⁡\(h\)O\(h\)\.

Denoising via Local Smoothing\.Following granularity expansion, the interpolated sequence may introduce artificial noise or abrupt transitions\. To address this, we apply a smoothing operation that preserves local continuity and global trend consistency:

\(10\)Y^i\(j,2\)​\(n\)\\displaystyle\\hat\{Y\}\_\{i\}^\{\(j,2\)\}\(n\)=Y^i\(j,1\)​\(n\)−α⁡\(Y^i\(j,1\)​\(n\)CLOSE\\displaystyle=\\hat\{Y\}\_\{i\}^\{\(j,1\)\}\(n\)\-\\alpha\\Big\(\\hat\{Y\}\_\{i\}^\{\(j,1\)\}\(n\)OPEN−12​\(Y^i\(j,1\)​\(n−1\)\+Y^i\(j,1\)​\(n\+1\)\)\)\\displaystyle\-\\tfrac\{1\}\{2\}\\big\(\\hat\{Y\}\_\{i\}^\{\(j,1\)\}\(n\-1\)\+\\hat\{Y\}\_\{i\}^\{\(j,1\)\}\(n\+1\)\\big\)\\Big\)where the boundary neighbors are defined by replication,

\(11\)Y^i\(j,1\)​\(0\)=Y^i\(j,1\)​\(1\),Y^i\(j,1\)​\(L\(j\+1\)\+1\)=Y^i\(j,1\)​\(L\(j\+1\)\),\\hat\{Y\}\_\{i\}^\{\(j,1\)\}\(0\)=\\hat\{Y\}\_\{i\}^\{\(j,1\)\}\(1\),\\qquad\\hat\{Y\}\_\{i\}^\{\(j,1\)\}\\big\(L^\{\(j\+1\)\}\+1\\big\)=\\hat\{Y\}\_\{i\}^\{\(j,1\)\}\\big\(L^\{\(j\+1\)\}\\big\),which avoids artificial wrap\-around and writes the update asSα=I−α​𝐋S\_\{\\alpha\}=I\-\\alpha\\mathbf\{L\}with𝐋\\mathbf\{L\}symmetric positive semidefinite\. The learnable scalarα=σ⁡\(a\)∈\(0,1\)\\alpha=\\sigma\(a\)\\in\(0,1\)is shared across levels, withaainitialized to zero, soα\\alphastays in the range required by Remark[2](https://arxiv.org/html/2610.33470#Thmtmremark2)\.

Periodic Correction\.To restore the periodic structure learned by Fourier Anchor at the finer resolution, we reuse the Fourier coefficients\{Ai\(k\),Bi\(k\)\}\\\{A\_\{i\}^\{\(k\)\},B\_\{i\}^\{\(k\)\}\\\}and define:

\(12\)pi​\(τ\)\\displaystyle p\_\{i\}\(\\tau\)=∑k=1K\(Ai\(k\)​\(X\)​cos⁡\(ωk​τ\)\+Bi\(k\)​\(X\)​sin⁡\(ωk​τ\)\),\\displaystyle=\\sum\_\{k=1\}^\{K\}\\left\(A\_\{i\}^\{\(k\)\}\(X\)\\cos\(\\omega\_\{k\}\\tau\)\+B\_\{i\}^\{\(k\)\}\(X\)\\sin\(\\omega\_\{k\}\\tau\)\\right\),Pi\(j\)​\(n\)\\displaystyle P\_\{i\}^\{\(j\)\}\(n\)=pi​\(τn\(j\+1\)\)−pi​\(τρj​\(n\)\(j\)\),\\displaystyle=p\_\{i\}\\big\(\\tau\_\{n\}^\{\(j\+1\)\}\\big\)\-p\_\{i\}\\big\(\\tau\_\{\\rho\_\{j\}\(n\)\}^\{\(j\)\}\\big\),whereρj​\(n\)\\rho\_\{j\}\(n\)is the level\-jjindex that expansion copies tonn, andAi\(k\)​\(X\)A\_\{i\}^\{\(k\)\}\(X\)andBi\(k\)​\(X\)B\_\{i\}^\{\(k\)\}\(X\)are the periodic amplitudes at frequencyωk\\omega\_\{k\}estimated from the current window by Eq\. \([4](https://arxiv.org/html/2610.33470#S3.E4)\), so the prior injected at each level is input\-conditioned rather than a globally shared template\. Since expansion already replicates the coarse\-grid periodic values, we add only the attenuated in\-between residual:

\(13\)Y^i\(j,3\)​\(n\)=Y^i\(j,2\)​\(n\)\+β​Pi\(j\)​\(n\)\\hat\{Y\}\_\{i\}^\{\(j,3\)\}\(n\)=\\hat\{Y\}\_\{i\}^\{\(j,2\)\}\(n\)\+\\beta P\_\{i\}^\{\(j\)\}\(n\)whereβ=σ⁡\(b\)∈\(0,1\)\\beta=\\sigma\(b\)\\in\(0,1\)is a learnable scalar periodic attenuation coefficient shared across all refinement levels, withbbinitialized to zero\. This step injects periodic variations as a controlled prior, preventing over\-amplification while maintaining consistency with the coarse\-grained temporal context\.

Iterative Refinement\.Each level applies expansion, one smoothing step, one periodic correction, and the DTW correction ofSection[3\.3](https://arxiv.org/html/2610.33470#S3.SS3), mappingY^\(j\)\\hat\{Y\}^\{\(j\)\}toY^\(j\+1\)\\hat\{Y\}^\{\(j\+1\)\}\. AllJJlevels are executed without early stopping, so the output is always defined on the target gridΔ​τg\\Delta\\tau\_\{g\}\. A single smoothing pass attenuates the high\-frequency artifacts introduced by expansion and, by Remark[2](https://arxiv.org/html/2610.33470#Thmtmremark2), never amplifies the sequence\. After allJJlevels, the final sequenceY^\(J\)\\hat\{Y\}^\{\(J\)\}is obtained\.

### 3\.3\.DTW Alignment Module

Refinement alone can drift, since on finer grids small per\-step errors accumulate into phase shifts against the coarse evidence\. TheDTW Alignmentstage closes this loop by warpingY^\(j\)\\hat\{Y\}^\{\(j\)\}back onto the structured coarse priorY¯\\bar\{Y\}produced by Fourier Anchor\. We align toY¯\\bar\{Y\}rather than raw observations because future coarse values are unavailable at deployment;Y¯\\bar\{Y\}is supervised byℒFA\\mathcal\{L\}\_\{\\text\{FA\}\}, and since, by Remark[1](https://arxiv.org/html/2610.33470#Thmtmremark1), the update interpolates between refinement and anchor errors, the stepη\\eta, trained through the fine\-grid loss, can learn a small value when the anchor is unreliable\. It integrates classical DTW\([Berndt and Clifford, 1994](https://arxiv.org/html/2610.33470#bib.bib17)\)with learnable weights, following the global structure ofY¯\\bar\{Y\}while staying flexible locally\.

Temporal Alignment via Dynamic Time Warping\.Given the structured coarse\-grained sequenceY¯=\{Y¯1t,Δ​τG,…,Y¯Nt,Δ​τG\}\\bar\{Y\}=\\\{\\bar\{Y\}\_\{1\}^\{t,\\Delta\\tau\_\{G\}\},\\dots,\\bar\{Y\}\_\{N\}^\{t,\\Delta\\tau\_\{G\}\}\\\}and the pre\-alignment sequenceZ\(j\)=Y^\(j,3\)Z^\{\(j\)\}=\\hat\{Y\}^\{\(j,3\)\}on𝒢\(j\+1\)\\mathcal\{G\}^\{\(j\+1\)\}, the alignment objective is to minimize the cumulative distance between them:

\(14\)D⁡\(Y¯,Z\(j\)\)=∑i=1Nmin⁡∑\(tx,n\)∈πi\(j\)πi\(j\)⁡d⁡\(Y¯itx,Zi\(j\)​\(n\)\)D\(\\bar\{Y\},Z^\{\(j\)\}\)=\\sum\_\{i=1\}^\{N\}\\min\_\{\\pi\_\{i\}^\{\(j\)\}\}\\sum\_\{\(t\_\{x\},n\)\\in\\pi\_\{i\}^\{\(j\)\}\}d\\left\(\\bar\{Y\}\_\{i\}^\{t\_\{x\}\},Z\_\{i\}^\{\(j\)\}\(n\)\\right\)whereπi\(j\)\\pi\_\{i\}^\{\(j\)\}is the alignment path for variableiiat leveljjandd⁡\(⋅,⋅\)d\(\\cdot,\\cdot\)is the squared Euclidean distance on standardized values\. To restrict the search around the scaled diagonal, each coarse indextx∈\{1,…,hG\}t\_\{x\}\\in\\\{1,\\dots,h\_\{G\}\\\}is matched only against

\(15\)ℬj​\(tx\)=\{n:\|n−\(1\+\(tx−1\)​\(L\(j\+1\)−1\)max⁡\(hG−1,1\)\)\|≤B\},\\mathcal\{B\}\_\{j\}\(t\_\{x\}\)=\\left\\\{n:\\left\|n\-\\left\(1\+\\frac\{\(t\_\{x\}\-1\)\(L^\{\(j\+1\)\}\-1\)\}\{\\max\(h\_\{G\}\-1,1\)\}\\right\)\\right\|\\leq B\\right\\\},with half\-widthB=⌈r⌉B=\\lceil r\\rceil, which also contains the interval\-end matchn=⌈tx​L\(j\+1\)/hG⌉n=\\lceil t\_\{x\}L^\{\(j\+1\)\}/h\_\{G\}\\rceil, so each level costsO⁡\(N​h\)O\(Nh\)instead of theO⁡\(N​hG​h\)O\(Nh\_\{G\}h\)of unconstrained DTW\.

Learnable Alignment Weights for Adaptive Adjustment\.While traditional DTW weighs all cell pairs equally, it cannot emphasize the pairs that matter during refinement\. To overcome this limitation, we introduce a learnable weighting function that reweights the matched pairs in the alignment loss\. Given an input pair\(Y¯itx,Zi\(j\)​\(n\)\)\(\\bar\{Y\}\_\{i\}^\{t\_\{x\}\},\\allowbreak Z\_\{i\}^\{\(j\)\}\(n\)\), the alignment weight is computed as:

\(16\)ν~i\(j\)​\(tx,n\)\\displaystyle\\tilde\{\\nu\}\_\{i\}^\{\(j\)\}\(t\_\{x\},n\)=softplus⁡\(𝒜ψ​\(Y¯itx,Zi\(j\)​\(n\)\)\),\\displaystyle=\\operatorname\{softplus\}\\big\(\\mathcal\{A\}\_\{\\psi\}\(\\bar\{Y\}\_\{i\}^\{t\_\{x\}\},Z\_\{i\}^\{\(j\)\}\(n\)\)\\big\),νi\(j\)​\(tx,n\)\\displaystyle\\nu\_\{i\}^\{\(j\)\}\(t\_\{x\},n\)=ν~i\(j\)​\(tx,n\)/mean\(u,v\)∈πi\(j\)⁡ν~i\(j\)​\(u,v\)\.\\displaystyle=\\tilde\{\\nu\}\_\{i\}^\{\(j\)\}\(t\_\{x\},n\)\\Big/\\operatorname\{mean\}\_\{\(u,v\)\\in\\pi\_\{i\}^\{\(j\)\}\}\\tilde\{\\nu\}\_\{i\}^\{\(j\)\}\(u,v\)\.where𝒜ψ\\mathcal\{A\}\_\{\\psi\}is a learnable alignment network parameterized byψ\\psi\. The softplus keeps the weights positive and the mean over the cells of the path fixes their overall scale per variable, so the network expresses relative importance without being able to lower the loss by uniformly shrinking all weights\. These weights define the weighted alignment cost used in the loss:

\(17\)D′​\(Y¯,Z\(j\)\)=∑i=1N∑\(tx,n\)∈πi\(j\)νi\(j\)​\(tx,n\)​d​\(Y¯itx,Zi\(j\)​\(n\)\)D^\{\\prime\}\(\\bar\{Y\},Z^\{\(j\)\}\)=\\sum\_\{i=1\}^\{N\}\\sum\_\{\(t\_\{x\},n\)\\in\\pi\_\{i\}^\{\(j\)\}\}\\nu\_\{i\}^\{\(j\)\}\(t\_\{x\},n\)\\,d\\left\(\\bar\{Y\}\_\{i\}^\{t\_\{x\}\},Z\_\{i\}^\{\(j\)\}\(n\)\\right\)To rule out a cheap low\-weight path,πi\(j\)\\pi\_\{i\}^\{\(j\)\}is chosen under the unweighted cost of Eq\. \([14](https://arxiv.org/html/2610.33470#S3.E14)\) andν\\nuenters only the loss\. Since the path searchminπi\(j\)\\min\_\{\\pi\_\{i\}^\{\(j\)\}\}is non\-differentiable, we fix the DP\-computed path per forward pass with a stop\-gradient and back\-propagate only throughν\\nu, the sequence values, andη\\eta, avoiding a soft relaxation\.

Refinement\-Based Alignment Correction\.At the end of each level, DTW warps the corrected sequenceY^\(j,3\)\\hat\{Y\}^\{\(j,3\)\}towardY¯\\bar\{Y\}:

\(18\)Y^i\(j\+1\)​\(n\)=Y^i\(j,3\)​\(n\)\+η⁡\(Y¯itx​\(n\)−Y^i\(j,3\)​\(n\)\)\\hat\{Y\}\_\{i\}^\{\(j\+1\)\}\(n\)=\\hat\{Y\}\_\{i\}^\{\(j,3\)\}\(n\)\+\\eta\\left\(\\bar\{Y\}\_\{i\}^\{t\_\{x\}\(n\)\}\-\\hat\{Y\}\_\{i\}^\{\(j,3\)\}\(n\)\\right\)wheretx​\(n\)t\_\{x\}\(n\)is, among the coarse indices matched tonnon the DTW path, the one nearest in time \(the larger on ties\), initialized astx​\(n\)=min⁡\(hG,1\+⌊\(n−1\)​L\(0\)/L\(j\+1\)⌋\)t\_\{x\}\(n\)=\\min\\\!\\big\(h\_\{G\},\\,1\+\\lfloor\(n\-1\)L^\{\(0\)\}/L^\{\(j\+1\)\}\\rfloor\\big\)and recomputed each level, andη=σ⁡\(ξ\)∈\(0,1\)\\eta=\\sigma\(\\xi\)\\in\(0,1\)with a single trainableξ\\xi\.

Alignment Loss Function\.While the alignment correction operates at every refinement step, the alignment loss is imposed on the final outputY^\(J\)\\hat\{Y\}^\{\(J\)\}, with the band of Eq\. \([15](https://arxiv.org/html/2610.33470#S3.E15)\) taken at lengthhh:

\(19\)ℒDTW=∑i=1N∑\(tx,n\)∈πi\(J\)νi\(J\)​\(tx,n\)​\(Y¯itx−Y^i\(J\)​\(n\)\)2\\mathcal\{L\}\_\{\\text\{DTW\}\}=\\sum\_\{i=1\}^\{N\}\\sum\_\{\(t\_\{x\},n\)\\in\\pi\_\{i\}^\{\(J\)\}\}\\nu\_\{i\}^\{\(J\)\}\(t\_\{x\},n\)\\left\(\\bar\{Y\}\_\{i\}^\{t\_\{x\}\}\-\\hat\{Y\}\_\{i\}^\{\(J\)\}\(n\)\\right\)^\{2\}MinimizingℒDTW\\mathcal\{L\}\_\{\\text\{DTW\}\}encourages the refined outputs to stay close to the structured prior while adapting across granularities through learned alignment weights\.

### 3\.4\.Training Strategy

The three stages are optimized jointly rather than sequentially, and a multi\-granularity loss supervises the coarse estimate, the intermediate refinements, and the cross\-resolution alignment on the same forward pass, so that errors made at one resolution are corrected before they propagate to the next\.

Denoising and Refinement Loss\.The De\-noise Only Module iteratively smooths expansion artifacts and refines the time series\. Its training process involves two loss functions:

Denoising Loss:To enforce local smoothness, we define:

\(20\)ℒdenoise\(j\)=∑i=1N∑n=2L\(j\)−1\(Y^i\(j\)​\(n\)−12​\(Y^i\(j\)​\(n−1\)\+Y^i\(j\)​\(n\+1\)\)\)2\\mathcal\{L\}\_\{\\text\{denoise\}\}^\{\(j\)\}=\\sum\_\{i=1\}^\{N\}\\sum\_\{n=2\}^\{L^\{\(j\)\}\-1\}\\left\(\\hat\{Y\}\_\{i\}^\{\(j\)\}\(n\)\-\\frac\{1\}\{2\}\\left\(\\hat\{Y\}\_\{i\}^\{\(j\)\}\(n\-1\)\+\\hat\{Y\}\_\{i\}^\{\(j\)\}\(n\+1\)\\right\)\\right\)^\{2\}which penalizes deviations from the local two\-point average on the refined grid of stepjj, encouraging smoother transitions\.

Refinement Loss:To ensure the predicted sequence converges toward the fine\-grained target, we minimize:

\(21\)ℒrefine\(j\)=∑i=1N∑n=1L\(j\)\(Y^i\(j\)​\(n\)−Yi\(j\)​\(n\)\)2\\mathcal\{L\}\_\{\\text\{refine\}\}^\{\(j\)\}=\\sum\_\{i=1\}^\{N\}\\sum\_\{n=1\}^\{L^\{\(j\)\}\}\\left\(\\hat\{Y\}\_\{i\}^\{\(j\)\}\(n\)\-Y\_\{i\}^\{\(j\)\}\(n\)\\right\)^\{2\}whereY\(j\)Y^\{\(j\)\}is the fine\-resolution target resampled onto the leveljjgrid𝒢\(j\)\\mathcal\{G\}^\{\(j\)\}, so that supervision is applied at the resolution actually produced by that level; atj=Jj=Jit extendsYΔ​τgY^\{\\Delta\\tau\_\{g\}\}\.

Overall Optimization Objective\.The Fourier Anchor objectiveℒFA\\mathcal\{L\}\_\{\\text\{FA\}\}ofSection[3\.1](https://arxiv.org/html/2610.33470#S3.SS1)enters unchanged\. Writingℒ2=∑j=1J\(ℒdenoise\(j\)\+ℒrefine\(j\)\)\\mathcal\{L\}\_\{2\}=\\sum\_\{j=1\}^\{J\}\(\\mathcal\{L\}\_\{\\text\{denoise\}\}^\{\(j\)\}\+\\mathcal\{L\}\_\{\\text\{refine\}\}^\{\(j\)\}\)for the aggregated refinement objective andℒ3=ℒDTW\\mathcal\{L\}\_\{3\}=\\mathcal\{L\}\_\{\\text\{DTW\}\}for the alignment objective ofSection[3\.3](https://arxiv.org/html/2610.33470#S3.SS3), the total loss is

\(22\)ℒtotal=e−s1​ℒFA\+s1\+∑m=23\(e−sm​ℒm\+sm\)\\mathcal\{L\}\_\{\\text\{total\}\}=e^\{\-s\_\{1\}\}\\mathcal\{L\}\_\{\\text\{FA\}\}\+s\_\{1\}\+\\sum\_\{m=2\}^\{3\}\\left\(e^\{\-s\_\{m\}\}\\mathcal\{L\}\_\{m\}\+s\_\{m\}\\right\)where the effective weightsλm=e−sm\>0\\lambda\_\{m\}=e^\{\-s\_\{m\}\}\>0balance trend supervision, refinement, and alignment\. The additivesms\_\{m\}penalize shrinking any weight, so the trivial all\-zero solution is excluded, and thesms\_\{m\}are learned jointly withθ\\theta\. The refinement and alignment objectives stay active whenr=1r=1, so both modules remain trained in the same\-resolution case\. All objectives are optimized in the same forward and backward pass, and their costs are analyzed inAppendix[G](https://arxiv.org/html/2610.33470#A7)\.

## 4\.Experiments

We conduct extensive experiments on multivariate time series benchmarks to evaluate TimeMagnifier and establish performance baselines for the MTS\-SRF task\. FullΓ\\Gammaandγ\\gammaresults and case visualizations will be released with the code\.

Table 1\.Main results\. Thebrown backgroundmarks the best result and thegray backgroundthe second best\.Note:T\.Magnifier is short for TimeMagnifier\. Mean rows average the models of each category and are not ranked\.

### 4\.1\.Datasets and Evaluation Metrics

To assess TimeMagnifier’s generalization across diverse scenarios, we use ten benchmark datasets spanning environment, energy, economics, healthcare, and transportation\([Shao et al\., 2023](https://arxiv.org/html/2610.33470#bib.bib18);[Wang et al\., 2024b](https://arxiv.org/html/2610.33470#bib.bib19);[Qiu et al\., 2024](https://arxiv.org/html/2610.33470#bib.bib20)\)\. Several are themselves Web\-published rollups, as the hourly Electricity series aggregates 15\-minute UCI readings and Traffic uses hourly PeMS aggregates\([Lai et al\., 2018](https://arxiv.org/html/2610.33470#bib.bib5)\)\. The descriptions and statistical properties of these datasets are presented inAppendix[A](https://arxiv.org/html/2610.33470#A1)\. To ensure consistency with prior studies, we employ Mean Squared Error \(MSE\) and Mean Absolute Error \(MAE\) as the primary evaluation metrics, complemented by CRPS, interval coverage, and band\-wise spectral diagnostics inAppendix[D](https://arxiv.org/html/2610.33470#A4)\.

### 4\.2\.Baselines Description

To evaluate TimeMagnifier in super\-resolution forecasting, we compare it with 15 state\-of\-the\-art \(SOTA\) and widely used long\-term time series forecasting \(LTSF\) baselines\. These baselines cover diverse architectures, ensuring a fair and representative comparison\. We group them into four categories: 1\)Transformer\-based models:Transformer\([Vaswani et al\., 2017](https://arxiv.org/html/2610.33470#bib.bib21)\), Informer\([Zhou et al\., 2021](https://arxiv.org/html/2610.33470#bib.bib4)\), Autoformer\([Wu et al\., 2021](https://arxiv.org/html/2610.33470#bib.bib22)\), FEDformer\([Zhou et al\., 2022](https://arxiv.org/html/2610.33470#bib.bib23)\), Crossformer\([Zhang and Yan, 2023](https://arxiv.org/html/2610.33470#bib.bib24)\), PatchTST\([Nie et al\., 2023](https://arxiv.org/html/2610.33470#bib.bib25)\), iTransformer\([Liu et al\., 2024](https://arxiv.org/html/2610.33470#bib.bib26)\)\. 2\)CNN\-based model:TimesNet\([Wu et al\., 2022](https://arxiv.org/html/2610.33470#bib.bib29)\)\. 3\)Diffusion\-based models:TimeGrad\([Rasul et al\., 2021](https://arxiv.org/html/2610.33470#bib.bib30)\), TimeDiff\([Shen and Kwok, 2023](https://arxiv.org/html/2610.33470#bib.bib31)\)\. 4\)Linear\-based models:NLinear\([Zeng et al\., 2023](https://arxiv.org/html/2610.33470#bib.bib14)\), DLinear\([Zeng et al\., 2023](https://arxiv.org/html/2610.33470#bib.bib14)\), TiDE\([Das et al\., 2023](https://arxiv.org/html/2610.33470#bib.bib27)\), TimeMixer\([Wang et al\., 2024a](https://arxiv.org/html/2610.33470#bib.bib28)\), TQNet\([Lin et al\., 2025](https://arxiv.org/html/2610.33470#bib.bib43)\)\. Implementation details are provided inAppendix[B](https://arxiv.org/html/2610.33470#A2)\.

### 4\.3\.Experimental Setup

All experiments are conducted on a unified computing platform equipped with a 2\.70 GHz Intel® Xeon® CPU Max 9462 processor and two NVIDIA A100 80GB SXM GPUs\. We train with Adam at learning rate5×10−45\\times 10^\{\-4\}and batch size 64 for at most 100 epochs, with early stopping at patience 3 on the validation set and learning\-rate halving after any epoch without validation gain, and all methods, baselines included, share a 20\-trial search\. Splits are chronological, 12/4/4 months for ETT and 7:1:2 otherwise, and are made on the native grid before downsampling, so no future fine value leaks\. Main results report the mean of three runs; for every cell, TimeMagnifier and the runner\-up are additionally rerun with five seeds, and these five\-seed means, which differ slightly from the three\-run values, underlie the significance tests inAppendix[C](https://arxiv.org/html/2610.33470#A3)\.

### 4\.4\.Main Results

We setΓ=2\\Gamma=2andΓ=4\\Gamma=4withγ=1\\gamma=1, a look\-back window of 168 coarse steps, and a forecast horizon of 336, or a look\-back of 24 and horizon of 36 for Illness\.Table[1](https://arxiv.org/html/2610.33470#S4.T1)reports the super\-resolution forecasting results\. TimeMagnifier is best in every cell, andAppendix[C](https://arxiv.org/html/2610.33470#A3)shows that over five seeds its gain over the runner\-up passes a Diebold\-Mariano test atp<0\.01p<0\.01in 19 of 20 MSE settings, all but the short and noisy Illness atΓ=2\\Gamma=2\. AtΓ=2\\Gamma=2it reduces MSE over the strongest baseline by 0\.9% to 33\.3%, with the largest margins of 33\.3% on Weather and 19\.4% on ETTm1, while the MAE reductions exceed 25% on ETTh2, ETTm1, and Weather, reflecting smaller typical errors\. The margin also widens as inputs grow coarser on nine of ten datasets, since the relative MSE gain over the runner\-up rises withΓ\\Gammafrom 5\.9% to 12\.9% on ETTh1, from 19\.4% to 24\.0% on ETTm1, and from 3\.2% to 10\.8% on ExRate, so a dedicated coarse\-to\-fine design matters more as the gap grows\. No baseline is consistently competitive, suggesting that no standard LTSF architecture is consistently suited to MTS\-SRF\. AtΓ=2\\Gamma=2, iTransformer and TimeMixer come closest, together runner\-up on five datasets, while FEDformer is strong on ETTh1, whose periodic structure suits its frequency\-domain decomposition\. NLinear takes second place on ETTm2 and Illness at both coefficients, whereas Transformer and Informer perform worst, as they also do in standard same\-resolution forecasting\. AtΓ=4\\Gamma=4coarser inputs make forecasting harder, as 147 of 150 baseline MSE entries degrade and the other three improve by under 1%; even iTransformer and TQNet lose about 18% on average, while PatchTST and TimesNet occasionally reach second best, for instance on Traffic and on the MAE of ETTm2, suggesting that patching and multi\-scale designs partially mitigate the sparsity of coarse observations\. TimeMagnifier improves in MSE atΓ=4\\Gamma=4on Illness and ETTm1, possibly in part because the look\-back, counted in coarse steps, then covers at least one full annual or weekly cycle, 96 rather than 48 weeks and 7 rather than 3\.5 days\.

Table 2\.MSE of baseline adapters atΓ=4\\Gamma=4; P1 to P4 use the same splits and 20\-trial budget as TimeMagnifier\.
### 4\.5\.Strengthened Baseline Protocol

LTSF baselines are not built for MTS\-SRF, soTable[1](https://arxiv.org/html/2610.33470#S4.T1)equips each with a learned linear head that maps itshGh\_\{G\}coarse steps tohhfine steps and contains linear and nearest\-neighbor interpolation as special cases\. To test whether this head handicaps them, we give TQNet and iTransformer four alternatives atΓ=4\\Gamma=4with the same splits and 20\-trial budget: P1, a head that outputs the fine grid directly; P2, forecasting followed by cubic\-spline interpolation; P3, spline upsampling of the history followed by forecasting; and P4, a parameter\-matched sub\-pixel upsampler from image super\-resolution\([Shi et al\., 2016](https://arxiv.org/html/2610.33470#bib.bib44)\)\. We also evaluate mTAN\([Shukla and Marlin, 2021](https://arxiv.org/html/2610.33470#bib.bib46)\), a continuous\-time model queried at fine\-grid timestamps\. InTable[2](https://arxiv.org/html/2610.33470#S4.T2), the 16 new configurations range from 0\.2% better to 5\.5% worse than the linear head, and TimeMagnifier is 14\.4% and 34\.0% below the best adapters, P4 on ETTh1 and the original head P0 on Weather\.

### 4\.6\.Study on Horizon and Look\-back Window

We fixΓ=2\\Gamma=2,γ=1\\gamma=1, and a look\-back of 168\. Forecast horizons are set to 96, 192, 336, and 729, where 729 is indivisible byΓ=2\\Gamma=2and thus exercises the block retention of Eq\. \([1](https://arxiv.org/html/2610.33470#S2.E1)\) withh=730h=730and one unscored step\. For the short Illness set, the look\-back is 24 and the four rows use horizons 12, 24, 36, and 60\.Table[3](https://arxiv.org/html/2610.33470#S4.T3)reports results across all ten datasets\. MSE and MAE rise monotonically with the horizon on every dataset, most steeply on the short Illness, whose MSE grows about 13 times from the shortest to the longest horizon, followed by ExRate at 3\.3 times and ETTm2 at 2\.8 times, while the other datasets grow by at most about two times\.

Look\-back window\.With the horizon fixed,Figure[4](https://arxiv.org/html/2610.33470#S4.F4)varies the look\-back over 48, 96, 168, 336, and 729, or 12 to 48 for Illness\. Error is lowest at an intermediate length on eight datasets, as short windows lack context and long ones add noise, while ETTm1 and Weather do best at 729\. The shared look\-back of 168 in main results is best for TimeMagnifier on only two datasets, so its gains do not rest on a tuned window\.

![Refer to caption](https://arxiv.org/html/2610.33470v1/MSE_heatmap.png)Figure 4\.Look\-back study atΓ=2\\Gamma=2and horizon 336\. MSE is shaded by its gap to each dataset’s best window \(bold\)\.∗Illness uses horizon 36 and windows 12, 18, 24, 36 and 48\.Table 3\.Results across different forecast horizons\.Table 4\.Comparison of baseline linear adaptation and De\-noise Only & DTW plug\-in adaptation for the MTS\-SRF task\.
### 4\.7\.Ablation Study

\(a\)BJAir Dataset\(b\)ETTh1 Dataset\(c\)ExRate Dataset\(d\)Traffic Dataset
Figure 5\.Ablation study results\. Longer lines indicate greater degradation in predictive accuracy\.Under the main settings atΓ=2\\Gamma=2andγ=1\\gamma=1, we remove one component at a time and retune the rest under the full model’s 20\-trial budget, since inherited hyperparameters would confound removal with poor tuning\. The variants arew/o CC, without cross\-variable interaction;w/o FA, a shared linear coarse forecaster without gates or periodic prior;w/o DTW, without alignment; andw/o D&D, replacing De\-noise Only and DTW by a linear output layer\. On all ten datasets and both metrics, the ordering is TimeMagnifier<<w/o CC<<w/o FA<<w/o DTW<<w/o D&D, with MSE increases of 5\.5% to 54\.6%, largest on Traffic, Electricity, and ETTm2;Figure[5](https://arxiv.org/html/2610.33470#S4.F5)shows four of the ten datasets\. Removing DTW alone already accounts for 77% to 97% of the degradation caused by removing both stages, so level\-wise refinement drifts without alignment\. Fourier Anchor with a bare linear head trails DLinear at 0\.6878 vs\. 0\.6426, as expected of the low\-capacity prior analyzed inAppendix[I](https://arxiv.org/html/2610.33470#A9), whose value emerges once refinement is added, asSection[4\.9](https://arxiv.org/html/2610.33470#S4.SS9)shows\.

### 4\.8\.Study on Resolution Coefficients

Figure[6](https://arxiv.org/html/2610.33470#S4.F6)variesΓ\\Gammafrom 1 to 6 withγ=1\\gamma=1, andγ\\gammafrom 1 to 6 withΓ=6\\Gamma=6over a fixed physical horizon, whereΓ=1\\Gamma=1andγ=6\\gamma=6are same\-resolution controls\. Error grows gently withΓ\\Gammaon both BJAir and ETTh1, by about 5% in MSE fromΓ=2\\Gamma=2toΓ=6\\Gamma=6, so accuracy decays gracefully as archives coarsen\. Acrossγ\\gamma, MSE varies by under 0\.3% on ExRate and Traffic, even at the controlγ=6\\gamma=6, so atΓ=6\\Gamma=6a finer grid adds little error on these sets\.

\(a\)Γ\\Gammaon BJAir\(b\)Γ\\Gammaon ETTh1\(c\)γ\\gammaon ExRate\(d\)γ\\gammaon Traffic
Figure 6\.Resolution study:Γ\\Gammawithγ=1\\gamma=1\(a, b\) andγ\\gammawithΓ=6\\Gamma=6\(c, d\);Γ=1\\Gamma=1andγ=6\\gamma=6are same\-resolution controls\.
### 4\.9\.De\-noise Only & DTW Plug\-in Analysis

We examine whether the De\-noise Only & DTW components \(D&D\) can serve as a plug\-in that gives existing LTSF baselines MTS\-SRF capability\. Under the main settings, we append D&D to DLinear and PatchTST, keeping their tuned backbones and hyperparameters; since these expose no Fourier coefficients, the plug\-in estimates them from each input window by the ridge regression of Eq\. \([4](https://arxiv.org/html/2610.33470#S3.E4)\), so it carries the full periodic prior\. D&D reduces MSE by 5\.1% to 9\.0% in all twelve cases ofTable[4](https://arxiv.org/html/2610.33470#S4.T4)\. Because the plug\-in and the full model share D&D and its prior, their gap isolates the coarse base, and the full model leads the best plug\-in in all six settings by 1\.7% to 20\.9% in MSE\. Base and refinement also interact: without D&D, Fourier Anchor trails DLinear on Traffic at 0\.8134 vs\. 0\.7835, but with D&D it leads at 0\.5262 vs\. 0\.7393, a crossover that additive effects cannot produce\.

### 4\.10\.Further Analyses

Appendices[D](https://arxiv.org/html/2610.33470#A4)to[F](https://arxiv.org/html/2610.33470#A6)extend the evaluation beyond point accuracy\. On ETTm1, TimeMagnifier stays ahead of TQNet when coarse inputs are mean\-aggregated, 30% missing, or jittered, and unlike TQNet it adds no excess power above the coarse Nyquist frequency\. On ETTh1, it reaches a CRPS of 0\.48 with 91% coverage, against 0\.54 to 0\.58 for the diffusion baselines\. Its learned alignment weights also outperform Euclidean, cosine, and Soft\-DTW alignment, with Soft\-DTW needing 2\.22 times our runtime\.

## 5\.Related Work

MTS\-SRF jointly assumes coarse\-only inputs at inference, a finer future grid, and fine data only as training supervision; the lines of work below each relax at least one of these assumptions\.

Mixed\-Frequency and Cross\-Resolution Modeling\.Mixed\-data sampling \(MIDAS\) regressions\([Ghysels et al\., 2007](https://arxiv.org/html/2610.33470#bib.bib45)\)predict low\-frequency targets from high\-frequency regressors, and reverse MIDAS\([Foroni et al\., 2018](https://arxiv.org/html/2610.33470#bib.bib47)\)predicts high\-frequency variables but still conditions on their high\-frequency lags, whereas MTS\-SRF observes no fine value at inference\. Temporal disaggregation\([Chow and Lin, 1971](https://arxiv.org/html/2610.33470#bib.bib48);[Denton, 1971](https://arxiv.org/html/2610.33470#bib.bib49)\)distributes observed past totals over finer periods and can extrapolate using fine\-grained indicators\. Temporal hierarchies\([Athanasopoulos et al\., 2017](https://arxiv.org/html/2610.33470#bib.bib54)\)reconcile forecasts across aggregation levels but never forecast below the observed resolution\. Imputation and continuous\-time models such as mTAN\([Shukla and Marlin, 2021](https://arxiv.org/html/2610.33470#bib.bib46)\)interpolate between observed points, and we evaluate mTAN under our protocol in Section[4\.5](https://arxiv.org/html/2610.33470#S4.SS5)\. Climate downscaling\([Vandal et al\., 2017](https://arxiv.org/html/2610.33470#bib.bib50)\)and audio super\-resolution\([Kuleshov et al\., 2017](https://arxiv.org/html/2610.33470#bib.bib51)\)share the coarse\-to\-fine goal but super\-resolve a given span rather than extrapolate a future one\.

Multi\-Scale Time Series Forecasting\.Multi\-scale models exploit dependencies across scales through graphs\([Chen et al\., 2023](https://arxiv.org/html/2610.33470#bib.bib32)\), iterative refinement\([Shabani et al\., 2022](https://arxiv.org/html/2610.33470#bib.bib34)\), adaptive pathways\([Chen et al\., 2024](https://arxiv.org/html/2610.33470#bib.bib33)\), warping of irregular series\([Zhang et al\., 2023](https://arxiv.org/html/2610.33470#bib.bib35)\), or MLP mixing\([Zhong et al\., 2023](https://arxiv.org/html/2610.33470#bib.bib36);[Hu et al\., 2024](https://arxiv.org/html/2610.33470#bib.bib37)\)\. They rescale only internally and observe at the native resolution, so the fine data that MTS\-SRF lacks at inference remain available to them\.

Diffusion for Time Series Forecasting\.Diffusion models corrupt data with noise and learn to reverse the process\. TimeGrad\([Rasul et al\., 2021](https://arxiv.org/html/2610.33470#bib.bib30)\)pioneered diffusion for forecasting, TimeDiff\([Shen and Kwok, 2023](https://arxiv.org/html/2610.33470#bib.bib31)\)stabilized denoising, and ScoreGrad\([Yan et al\., 2021](https://arxiv.org/html/2610.33470#bib.bib38)\)introduced score matching, all at a fixed resolution with costly multi\-step sampling\. Closer to our setting, Cold Diffusion\([Bansal et al\., 2023](https://arxiv.org/html/2610.33470#bib.bib52)\)replaces noise with deterministic degradations such as downsampling, and mr\-Diff\([Shen et al\., 2024](https://arxiv.org/html/2610.33470#bib.bib53)\)denoises from coarse to fine trends\. Both keep a fixed\-length grid and learn a denoising network, whereas De\-noise Only lengthens the grid at every level, supervises each level with resampled fine targets, injects per\-window periodic priors, and adds only two scalar parameters\.

## 6\.Conclusion and Future Work

In this study, we introduce Multivariate Time Series Super\-Resolution Forecasting \(MTS\-SRF\), a new task that predicts high\-resolution sequences from low\-resolution inputs, addressing fine\-grained forecasting from coarse archives in Web telemetry, energy, and environmental monitoring\. To overcome data sparsity, fine\-grained ambiguity, and cross\-resolution misalignment, we propose TimeMagnifier, which couples a Fourier Anchor stage, a progressive De\-noise Only stage, and a DTW alignment stage\. Extensive experiments confirm that TimeMagnifier outperforms SOTA LTSF models across diverse datasets, and its De\-noise Only and DTW components further transfer as a portable plug\-in that improves existing forecasters\. MTS\-SRF also provides a foundation for large\-scale pretraining and multi\-resolution time series integration\. Promising directions include self\-supervised pretraining that obtains cross\-resolution supervision without paired data, and adaptive strategies that adjust granularity to data characteristics\.

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- Zhouet al\.\(2022\)T\. Zhou, Z\. Ma, Q\. Wen, X\. Wang, L\. Sun, and R\. JinFedformer: frequency enhanced decomposed transformer for long\-term series forecasting\.InInternational conference on machine learning,pp\. 27268–27286\.Cited by:[§4\.2](https://arxiv.org/html/2610.33470#S4.SS2.p1.1)\.

Table 5\.Dataset statistics\.## Appendix ADataset Description

We utilize ten benchmark multivariate time series datasets across environment, energy, economics, healthcare, and transportation\.Table[5](https://arxiv.org/html/2610.33470#A0.T5)lists their statistics\.

## Appendix BBaseline Setup

All baseline models are obtained from publicly available benchmark implementations for time series forecasting111https://github\.com/decisionintelligence/TFB,222https://github\.com/GestaltCogTeam/BasicTSor from their official repositories\. To ensure fairness and reproducibility, we reproduce and evaluate all models in a unified experimental environment\.

None of the baselines has an upsampling path, so we reshape each baseline’shGh\_\{G\}coarse outputs intohhfine steps with the learned linear layer ofAlgorithm[1](https://arxiv.org/html/2610.33470#algorithm1), which preserves every backbone and contains linear and nearest\-neighbor interpolation as special cases\.Section[4\.5](https://arxiv.org/html/2610.33470#S4.SS5)compares this head with four alternative adapters and mTAN under the same 20\-trial budget\. For the plug\-in ofTable[4](https://arxiv.org/html/2610.33470#S4.T4), the Fourier coefficients needed by the periodic correction are estimated from each input window by the ridge regression of Eq\. \([4](https://arxiv.org/html/2610.33470#S3.E4)\), since DLinear and PatchTST do not produce them\.

Hyperparameter settings\.We setK=3K=3\. For each dataset, the three dominant physical periodsΠk\\Pi\_\{k\}are taken from the periodogram of the coarse training split, shared by all variables, and kept fixed for training and testing; at a givenΓ\\Gammaeach spansΠk/\(Γ​Δ​τmin\)\>2\\Pi\_\{k\}/\(\\Gamma\\Delta\\tau\_\{\\text\{min\}\}\)\>2coarse steps, below the coarse Nyquist frequency, e\.g\., 12 steps atΓ=2\\Gamma=2and 6 atΓ=4\\Gamma=4for the daily period of hourly data\. The smoothing coefficientα\\alphaand the periodic attenuationβ\\betaare initialized to 0\.5 and learned during training, and the alignment coefficientη=σ⁡\(ξ\)\\eta=\\sigma\(\\xi\)is initialized withξ=0\\xi=0\. We initializes1=s2=0s\_\{1\}=s\_\{2\}=0ands3=log⁡2s\_\{3\}=\\log 2, corresponding toλ1=λ2=1\.0\\lambda\_\{1\}=\\lambda\_\{2\}=1\.0andλ3=0\.5\\lambda\_\{3\}=0\.5, and learn them during training\. The DTW alignment uses the scaled\-diagonal band of Eq\. \([15](https://arxiv.org/html/2610.33470#S3.E15)\) withB=⌈r⌉B=\\lceil r\\rceil\. The ridge coefficient of Eq\. \([4](https://arxiv.org/html/2610.33470#S3.E4)\) is fixed toλridge=10−3\\lambda\_\{\\mathrm\{ridge\}\}=10^\{\-3\}for all datasets\. The cross\-variable residual strengthκ\\kappais initialized to 0 and learned during training\. Neighborhoods use\|𝒩i\|=min⁡\(16,N−1\)\|\\mathcal\{N\}\_\{i\}\|=\\min\(16,N\-1\)and are computed once from the training split\.

Algorithm 1Baseline Model AdaptationInput:Low\-resolution coefficient

Γ\\Gamma, High\-resolution coefficient

γ\\gamma, Look\-back window length

TT, Forecast horizon

hreqh\_\{\\mathrm\{req\}\}, Coarse\-grained time series data

X∈ℝT×NX\\in\\mathbb\{R\}^\{T\\times N\}
Output:Fine\-grained prediction

Y^∈ℝhreq×N\\hat\{Y\}\\in\\mathbb\{R\}^\{h\_\{\\mathrm\{req\}\}\\times N\}
1Step 1: Baseline Model Prediction

2

Y^coarse←BaselineModel​\(X\)\\hat\{Y\}\_\{\\mathrm\{coarse\}\}\\leftarrow\\text\{BaselineModel\}\(X\);//Baseline model predictshGh\_\{G\}coarse steps

3Step 2: Linear Upsampling for Super\-Resolution

4

Y^←LinearLayer​\(Y^coarse\)\\hat\{Y\}\\leftarrow\\text\{LinearLayer\}\(\\hat\{Y\}\_\{\\mathrm\{coarse\}\}\);//Linear transformation to adjust output length tohh

5returnthe first

hreqh\_\{\\mathrm\{req\}\}steps of

Y^\\hat\{Y\}

## Appendix CStatistical Reliability

For every cell ofTable[1](https://arxiv.org/html/2610.33470#S4.T1), we rerun TimeMagnifier and the runner\-up with five seeds on the fixed chronological splits and apply a two\-sided Diebold\-Mariano test with HAC variance to the seed\-averaged per\-window errors\.Table[6](https://arxiv.org/html/2610.33470#A3.T6)lists the fourΓ=2\\Gamma=2cells with the smallest relative gaps inTable[1](https://arxiv.org/html/2610.33470#S4.T1)\. The first three are significant \(p<0\.01p<0\.01\) with TimeMagnifier’s standard deviation at most 2\.0% of its mean \(at most 1\.7% for the runner\-ups\)\. Illness is not \(p=0\.255p=0\.255\), as its 966 weekly points leave few test windows\.

Table 6\.Smallest\-gap cells over five seeds \(Γ=2\\Gamma=2, MSE\)\.
## Appendix DSpectral, Probabilistic, and Anchor Checks

Band\-wise recovery\.The periodic prior injects only modes below the coarse Nyquist frequencyfNf\_\{N\}, and each level smooths once, so content belowfNf\_\{N\}is recoverable in principle up to aliasing from higher bands, while content abovefNf\_\{N\}is not identifiable from coarse samples alone\.Table[7](https://arxiv.org/html/2610.33470#A4.T7)tests this on ETTm1 \(Γ=4\\Gamma=4\) with the ratioρ\\rhoof predicted to true power and the coherenceCC\. AbovefNf\_\{N\}, TimeMagnifier attenuates power \(ρ=0\.81\\rho=0\.81\) but has low coherence \(C=0\.16C=0\.16\), while TQNet injects excess power with lower coherence\.

Table 7\.Band\-wise power ratioρ\\rhoand coherenceCCon ETTm1 \(Γ=4\\Gamma=4\)\.Probabilistic evaluation\.TimeMagnifier is a point forecaster, so an empirical residual construction forms its predictive distribution by adding the empirical validation residuals, per variable and horizon step, to the point forecast; CRPS is computed from this distribution and the 90% interval from its 5% and 95% quantiles\. On ETTh1 \(Γ=2\\Gamma=2\), TimeMagnifier attains CRPS 0\.48 and 91% empirical coverage, versus 0\.58 and 88% for TimeGrad and 0\.54 and 86% for TimeDiff, which under\-cover\.

Anchor bias\.On ETTh1 \(Γ=2\\Gamma=2\) the learned alignment step isη=0\.23\\eta=0\.23, so alignment corrects only partially\. ReplacingY¯\\bar\{Y\}by an oracle anchor built from the true future coarse values lowers MSE only from 0\.5865 to 0\.5541, and on the 25% of windows with the largest anchor error alignment still lowers MSE by 0\.03\.

## Appendix ERobustness to Acquisition Artifacts

We train on clean point\-sampled data and test on ETTm1 \(Γ=4\\Gamma=4\) with perturbed coarse inputs\. TimeMagnifier versus the best baseline, TQNet, obtains 0\.4923 versus 0\.5857 under mean aggregation, 0\.5296 versus 0\.6474 with 30% of coarse points missing and forward filled, and 0\.5142 versus 0\.6034 with timestamp jitter within the coarse interval \(MSE; clean inputs give 0\.4531 versus 0\.5962\)\.

## Appendix FAlignment Alternatives

Table[8](https://arxiv.org/html/2610.33470#A6.T8)replaces our alignment by simpler alternatives atΓ=4\\Gamma=4\. Banded DTW, Euclidean keeps our band, path search, and correction but weights all matched pairs equally, and shared path, cosine finds one banded path per sample and level for all channels from cosine distances of cross\-channel vectors\. All alternatives improve over no alignment but trail ours, and Soft\-DTW, the closest on Traffic, needs 2\.22 times the runtime\. Since the Euclidean variant and ours share the path search and the correction of Section[3\.3](https://arxiv.org/html/2610.33470#S3.SS3), their gap comes fromν\\nualone\. A uniformℒDTW\\mathcal\{L\}\_\{\\text\{DTW\}\}pulls every fine step toward the stepwise anchor, including in\-between steps where the anchor is a poor target, and so works againstℒrefine\\mathcal\{L\}\_\{\\text\{refine\}\};ν\\nulets training relax this pull where the two disagree and keep it where the anchor is reliable, consistent with Remark[1](https://arxiv.org/html/2610.33470#Thmtmremark1)\.

Table 8\.Alignment alternatives \(Γ=4\\Gamma=4; MSE, time ratio\)\.
## Appendix GComplexity Analysis

We describe the computational cost of one forward pass of TimeMagnifier in terms of the number of variablesNN, the coarse\-grid lengthTT, the internal horizonhh, the refinement ratior=Γ/γr=\\Gamma/\\gammabetween coarse and fine resolutions, and the number of Fourier modesKK\.

Fourier Anchor\.The ridge projection matrix depends only onλridge\\lambda\_\{\\mathrm\{ridge\}\}and the frequencies, so it is precomputed\. FittingC⁡\(X\)C\(X\)and extrapolating it tohGh\_\{G\}future coarse steps costO⁡\(N​K​\(T\+hG\)\)O\(NK\(T\+h\_\{G\}\)\), while the sparse shared gates on the future coarse grid costO⁡\(N​\|𝒩i\|​hG\)O\(N\|\\mathcal\{N\}\_\{i\}\|h\_\{G\}\)for a fixed hidden width, adding onlyθg\\theta\_\{g\}andκ\\kappa\.

De\-noise Only\.This stage uses local one\-dimensional stencils and the reused Fourier coefficients\. At leveljjthe smoothing update and the periodic correction act on the expanded grid of lengthL\(j\+1\)L^\{\(j\+1\)\}, and both are element\-wise \(up to nearest\-neighbor accesses andKKsinusoids\), so the cost isO⁡\(N​L\(j\+1\)​K\)O\(NL^\{\(j\+1\)\}K\)\. The expanded lengths follow a doubling schedule2​hG,4​hG,…2h\_\{G\},4h\_\{G\},\\dotscapped athh, so the total refined length satisfies∑j=0J−1L\(j\+1\)=O⁡\(h\)\\sum\_\{j=0\}^\{J\-1\}L^\{\(j\+1\)\}=O\(h\)and is dominated by the final grid\. The overall cost of De\-noise Only is thereforeO⁡\(N​K​h\)O\(NKh\), independent ofrr, which enters only through the number of levelsJ=O⁡\(log⁡r\)J=O\(\\log r\)\. The stage reuses the Fourier coefficients from Fourier Anchor and adds only two scalar parameters \(α\\alphaandβ\\beta\), so its parameter count does not grow withTT,hh, orrr\.

DTW Alignment\.The band in Eq\. \([15](https://arxiv.org/html/2610.33470#S3.E15)\) assigns at most2​B\+12B\+1refined candidates to each of thehGh\_\{G\}coarse indices, so the dynamic programming table hasO⁡\(hG​B\)O\(h\_\{G\}B\)admissible cells per variable\. WithB=⌈r⌉B=\\lceil r\\rceilandh=r​hGh=rh\_\{G\}, the per\-level cost isO⁡\(N​hG​B\)=O⁡\(N​h\)O\(Nh\_\{G\}B\)=O\(Nh\), and since alignment runs at allJ=max⁡\(1,⌈log2⁡r⌉\)J=\\max\(1,\\lceil\\log\_\{2\}r\\rceil\)levels plus one final pass, the total isO⁡\(N​h​J\)O\(NhJ\)\.

Overall complexity\.Putting the three stages together, the cost of one forward pass of TimeMagnifier is

\(23\)costTM=O⁡\(N​K​\(T\+hG\)\+N​\|𝒩i\|​hG\)\+O⁡\(N​K​h\)\+O⁡\(N​h​J\)\.\\text\{cost\}\_\{\\text\{TM\}\}=O\\\!\\left\(NK\(T\+h\_\{G\}\)\+N\|\\mathcal\{N\}\_\{i\}\|h\_\{G\}\\right\)\+O\(NKh\)\+O\(NhJ\)\.HereJ=max⁡\(1,⌈log2⁡r⌉\)J=\\max\(1,\\lceil\\log\_\{2\}r\\rceil\)is the number of resolution levels: refinement is linear in the final horizonhh, while banded alignment is repeated over theJJlevels\. TimeMagnifier thus avoids the quadratic sequence\-length cost of self\-attention and the multi\-step sampling of diffusion models; its 0\.18M parameters do not grow withhh, and narrowing the bandBBtrades accuracy for speed\.

![Refer to caption](https://arxiv.org/html/2610.33470v1/efficiency_bubble.png)Figure 7\.Training time per iteration and MSE on Electricity atΓ=2\\Gamma=2, with bubble area proportional to trainable parameters; all models use the same GPU and batch size\.Measured cost\.Figure[7](https://arxiv.org/html/2610.33470#A7.F7)compares training cost, model size, and accuracy on Electricity atΓ=2\\Gamma=2\. TimeMagnifier needs 0\.07 s per iteration, no more than NLinear and DLinear, with 0\.18M parameters, behind only these two in size, while both diffusion models need over 4 s per iteration\.

## Appendix HRecoverability and Challenge Analysis

Signal Model and Downsampling\.LetX=\{X1,Δ​τG,X2,Δ​τG,…,XT,Δ​τG\}X=\\\{X^\{1,\\Delta\\tau\_\{G\}\},X^\{2,\\Delta\\tau\_\{G\}\},\\dots,\\\\ X^\{T,\\Delta\\tau\_\{G\}\}\\\}denote the coarse\-grained historical observations with sampling granularityΔ​τG=Γ​Δ​τmin\\Delta\\tau\_\{G\}=\\Gamma\\Delta\\tau\_\{\\text\{min\}\}\. In MTS\-SRF, we assume the low\-resolution input is obtained from the underlying high\-resolution sequence through a deterministic downsampling operator𝒟Γ​\(⋅\)\\mathcal\{D\}\_\{\\Gamma\}\(\\cdot\):

\(24\)Xt,Δ​τG=𝒟Γ​\(Xt,Δ​τmin\)\.X^\{t,\\Delta\\tau\_\{G\}\}=\\mathcal\{D\}\_\{\\Gamma\}\\big\(X^\{t,\\Delta\\tau\_\{\\text\{min\}\}\}\\big\)\.
Sources of Recoverable Structure\.MTS\-SRF is not exactly identifiable, since coarse sampling removes or aliases components above the coarse Nyquist limit, so TimeMagnifier does not claim to recover arbitrary missing detail\. Instead it exploits three complementary sources that constrain the plausible fine\-grained futures: dependencies among variables, low\-dimensional trend and periodic structure, and conditional patterns learned from paired training data\. Thew/o CCandw/o FAablations probe the first two directly\.

Challenge 1: Information Loss\.Downsampling lowers the Nyquist limit tofNyquistL​R=12​Γ​Δ​τminf^\{LR\}\_\{\\text\{Nyquist\}\}=\\tfrac\{1\}\{2\\Gamma\\Delta\\tau\_\{\\text\{min\}\}\}, so higher\-frequency components are lost or aliased into lower frequencies\. Fourier Anchor models the low\-frequency trend and periodic structure supported by the coarse observations and uses cross\-variable correlations to constrain plausible fine\-grained forecasts\.

Challenge 2: Fine\-Grained Ambiguity\.Many fine\-grained futures are compatible with the same coarse history, so MTS\-SRF estimates a plausible future rather than uniquely inverting𝒟Γ\\mathcal\{D\}\_\{\\Gamma\}\. De\-noise Only refines throughJ=max⁡\(1,⌈log2⁡\(Γ/γ\)⌉\)J=\\max\(1,\\lceil\\log\_\{2\}\(\\Gamma/\\gamma\)\\rceil\)levels, smoothing each intermediate grid and adding the coarse\-scale periodic prior\. This provides architectural regularization rather than a formal conditioning guarantee\.

Challenge 3: Temporal Misalignment\.Coarse\-to\-fine correspondence is one\-to\-many and can induce phase shifts\. DTW searches constrained warping paths under an unweighted cost and applies level\-wise corrections, with learned weights only in the loss, to stay consistent with the coarse forecast\.

Proof\.By definition,Y^\(j\+1\)−Y\(j\+1\)=\(1−η\)​\(Z\(j\)−Y\(j\+1\)\)\+η⁡\(Y¯∘tx\(j\)−Y\(j\+1\)\)\\hat\{Y\}^\{\(j\+1\)\}\-Y^\{\(j\+1\)\}=\(1\-\\eta\)\\big\(Z^\{\(j\)\}\-Y^\{\(j\+1\)\}\\big\)\+\\eta\\big\(\\bar\{Y\}\\circ t\_\{x\}^\{\(j\)\}\-Y^\{\(j\+1\)\}\\big\), and applying the triangle inequality yields the bound\.□\\square

The bound shows that alignment interpolates between the refinement error and the discrepancy of the warped coarse prior: it can never amplify the error beyond the worse of the two terms, and it strictly reduces the error whenever the coarse prior read along the warping path is closer to the target than the refined sequence, which is the refinement\-drift regime the module targets\.

## Appendix IStructured Representation of Fourier Anchor

For each inputXX, Eq\. \([4](https://arxiv.org/html/2610.33470#S3.E4)\) givesY~​\(X\)=ΦF​C​\(X\)\\widetilde\{Y\}\(X\)=\\Phi\_\{F\}C\(X\), so its columns lie in the trend\-Fourier subspacecol⁡\(ΦF\)\\operatorname\{col\}\(\\Phi\_\{F\}\)of dimension at mostD=2\+2​KD=2\+2K\. Ridge regularization withλridge\>0\\lambda\_\{\\mathrm\{ridge\}\}\>0keeps the solve forC⁡\(X\)C\(X\)unique, and, asPFAP\_\{\\mathrm\{FA\}\}is a fixed linear map, the base forecast is Lipschitz inXX\. The data\-adaptive cross\-variable gates can move the final mixed forecastY¯\\bar\{Y\}outside this subspace\.

## Appendix JAnalysis of De\-noise Only

Under the replicated boundary of Eq\. \([11](https://arxiv.org/html/2610.33470#S3.E11)\),𝐋\\mathbf\{L\}is symmetric positive semidefinite with eigenvalues0=ζ1≤⋯≤ζL\(j\+1\)≤20=\\zeta\_\{1\}\\leq\\dots\\leq\\zeta\_\{L^\{\(j\+1\)\}\}\\leq 2, so the eigenvalues ofSα=I−α​𝐋S\_\{\\alpha\}=I\-\\alpha\\mathbf\{L\}areμk=1−α​ζk\\mu\_\{k\}=1\-\\alpha\\zeta\_\{k\}\.

Proof\.Write𝐋=V​diag⁡\(ζk\)​V⊤\\mathbf\{L\}=V\\operatorname\{diag\}\(\\zeta\_\{k\}\)V^\{\\top\}, soSα=V​diag⁡\(μk\)​V⊤S\_\{\\alpha\}=V\\operatorname\{diag\}\(\\mu\_\{k\}\)V^\{\\top\}\. Since0≤ζk≤20\\leq\\zeta\_\{k\}\\leq 2andα∈\(0,1\)\\alpha\\in\(0,1\), we have\|μk\|≤1\|\\mu\_\{k\}\|\\leq 1for everykk, with strict inequality wheneverζk\>0\\zeta\_\{k\}\>0\. Hence‖Sα‖2≤1\\\|S\_\{\\alpha\}\\\|\_\{2\}\\leq 1\.□\\square

This characterizes the operation actually implemented, namely exactly one pass per level: smoothing never amplifies the sequence and strictly attenuates non\-constant modes\.

## Appendix KApplicability and Extensions

Supervision is a training\-time requirement, not a deployment constraint\.TimeMagnifier uses paired coarse\-fine sequences only during training, and at inference it consumes coarse history alone, matching settings where fine\-grained collection is limited by cost, storage, or sensor budget\. Such pairs are readily obtained, since any high\-resolution segment retained before rollup yields supervision by downsampling\. TheΓ=4\\Gamma=4ETTm results are real point\-sampled archives, but they do not cover sensors that aggregate\. Under the mean aggregation, 30% missing coarse points, and timestamp jitter ofAppendix[E](https://arxiv.org/html/2610.33470#A5), TimeMagnifier still leads the best baseline on ETTm1 by 15\.9%, 18\.2%, and 14\.8%, yet its error rises more \(8\.7%, 16\.9%, 13\.5%\), so acquisition mismatch remains a limitation\. The modular loss in Eq\. \([22](https://arxiv.org/html/2610.33470#S3.E22)\) also makes the framework compatible with self\-supervised extensions, because the refinement term is the only component that consults fine\-grained targets, and it can be replaced by a consistency objective that re\-downsamples the refined output and matches it against the future coarse targets\.

Graceful degradation under widening resolution gaps\.The Nyquist analysis inAppendix[H](https://arxiv.org/html/2610.33470#A8)shows that content above the coarse Nyquist limit is not identifiable from coarse samples alone, so the recoverable bandwidth generally decreases asΓ\\Gammagrows, and what matters is how gracefully it decays\. OurΓ\\Gammastudy inFigure[6](https://arxiv.org/html/2610.33470#S4.F6)shows that TimeMagnifier degrades smoothly up toΓ=6\\Gamma=6, andTable[1](https://arxiv.org/html/2610.33470#S4.T1)shows that its advantage over the strongest baseline widens as inputs become coarser on nine of ten datasets\. This is consistent with the design, because the trend\-Fourier subspace and the cross\-variable gates model the predictable structure that survives aggressive downsampling\. For still larger gaps, the architecture accommodates external evidence without modification, since domain priors or physical constraints can be injected through the periodic prior of Eq\. \([13](https://arxiv.org/html/2610.33470#S3.E13)\) alongside the learned Fourier coefficients\.