Time-Gated Multi-Scale Flow Matching for Time-Series Imputation
Authors: Hangtian Wang
Mahito Sugiyama
Venue: ICLR 2026 Poster (2026)
Keywords: Time-series imputation
Flow matching
ODE-based generative models
Transformers
Multi-scale modeling
Primary area: learning on time series and dynamical systems

Abstract: We address multivariate time–series imputation by learning the velocity field of a
data-conditioned ordinary differential equation (ODE) via flow matching. Our
method, Time-Gated Multi-Scale Flow Matching (TG-MSFM), conditions the
flow on a structured endpoint comprising observed values, a per-time visibility
mask, and short left/right context, processed by a time-aware Transformer whose
self-attention is masked to aggregate only from observed timestamps. To recon-
cile global trends with local details along the trajectory, we introduce time-gated
multi-scale velocity heads on a fixed 1D pyramid and blend them through a time-
dependent gate; a mild anti-aliasing filter stabilizes the finest branch. At inference,
we use a second-order Heun integrator with a per-step data-consistency projection
that keeps observed coordinates exactly on the straight path from the initial noise
to the endpoint, reducing boundary artifacts and drift. Training adopts gap-only
supervision of the velocity on missing data coordinates, with small optional regu-
larizers for numerical stability. Across standard benchmarks, Time-Gated Multi-
Scale Flow Matching attains competitive or improved MSE/MAE with favorable
speed–quality trade-offs, and ablations isolate the contributions of the time-gated
multi-scale heads, masked attention, and the data-consistent ODE integration

Published as a conference paper at ICLR 2026
TIME-GATED MULTI-SCALE FLOW MATCHING FOR TIME-SERIES IMPUTATION
Hangtian Wang!?, Mahito Sugiyama 1.2
National Institute of Informatics
The Graduate University for Advanced Studies, SOKENDAL (bangtinnwang,, mahito)@nii ip
ABSTRACT
We propose to perform multivariate time-series imputation by learning the ve- locity field of a data-conditioned ordinary differential equation (ODE) via flow matching (FM). Our method, called Time-Gated Multi-Scale Flow Matching (TG-MSFM), conditions the flow on a structured endpoint comprising observed values, per-time visibility mask, and short left/right context, processed by time-aware Transformer whose self-attention is masked to aggregate only from observed timestamps. To reconcile global trends with local details along the jectory, we introduce time-gated multi-scale velocity heads on a fixed 1D pyramid and blend them through time-dependent gate; mild anti-aliasing filter stabi- lizes the finest branch. At inference, we use second-order Heun integrator with a per-step data-consistency (DC) projection that keeps observed coordinates ex- actly on the straight path from the initial noise to the endpoint, reducing bound- ary artifacts and drift. Training adopts gap-only supervision of the velocity on missing data coordinates, with small optional regularizers for numerical stability. Across standard benchmarks, TG-MSFM attains competitive or improved per- formance with favorable speed-quality trade-offs,, and ablations demonstrate the isolated contributions of the time-gated multi-scale heads, masked attention, and the data-consistent ODE integration.
INTRODUCTION
Missing values are pervasive in multivariate time series of various domains such as sensors, health records, transportation, and finance. Practical deployments face three coupled challenges: (i) ir- regular sampling and blockwise gaps, which break short-range continuity; (ii) the coexistence of slow trends and sharp transients, which stresses model's 's spectral bias; and (iii) the need for reli- able, reproducible inference at moderate computational cost. While sequence architectures, notably Transformers /Vaswani al. (017], provide expressive context aggregation, naively applying them to imputation risks leakage from unobserved timestamps and leaves unspecified how to reconcile global structure with local detail.
We approach imputation through data-conditioned ordinary differential equation (ODE) whose velocity field is learned by flow matching (Lipman et al.) 2023 Liu et al.) 2023b). The evolves from Gaussian noise to a structured endpoint that encodes the partially observed sequence. Con- ditioning is implemented with a time-aware Transformer whose self-attention visibility-masked to aggregate only across observed timestamps, preventing information leakage. To balance fre- quency content along the trajectory, we decompose the velocity into multi-scale heads on a fixed 1D pyramid and blend them via a time-depentn -eee-deeendent gate. Coarse scales dominate early to establish long-range trends; finer scales are emphasized later to refine local variations. mild anti-aliasing filter stabilizes the finest branch following standard signal-processin
practice (Lin al.) 2017 Ron- neberger et al. 2015; Oppenheim et al.] 1996). At inference we integrate the learned velocity with the second-order Heun method (Hairer al. 1993) and apply per-step data-consistency projec- tion that keeps all observed coordinates exactly on the straight path from noise to data, yielding deterministic, teassurement-pressevviie trajectories.

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Published as a conference paper at ICLR 2026
In this paper present TG-MSFM, minimal and effective instantiation of the above design. Training uses gap-only supervision: the flow-matching objective is evaluated on missing entries, dedicating modeling capacity to what must be inferred while letting the data-consistency projec- tion enforce observed coordinates during inference. The backbone, gating rule, and integrator are deliberately simple; hyperparameters are fixed across datasets (see Appendix [ for details).
Our contributions are summarized as follows:
Formulation. We cast multivariate Lime-series imputation as learning a data-conditioned ODE via flow matching (Lipman al.) [023 Liu et al.] (023b) with visibility maskedd self-attention (Vaswani et al. 2017) and gap-only supervision.
Architecture. We introduce a time-gated multi-scale velocity decomposition that schedules coarse-to-fing refinement along the ODE, coupled with a light anti-aliasing filter suppress high-frequency ringing (Lin et al. 2017 Ronneberger et al. 2015 Oppenheim et al. 1996).
Inference. We pair second-order Heun integration (Hairer al.
1993) with per-step data- consistency projection that preserves all observed measurements exactly while evolving unknown entries under the learned dynamics.
Evidence. On ten public benchmarks, TG-MSFM attains competitive improved imputa- tion accuracies with favorable speed-quality trade-offs (Fig. (2)). degrades gracefully as the central gap length increases (Fig. Bap, and passes targeted ablations of its key components (Table
Scope and positioning. Our focus deterministic imputation: given the same input, TG-MSFM returns a unique reconstruction. We do claim determinism to be universally preferable; rather, it is particularly attractive in domains such as clinical records, industrial monitoring, and au- diting pipelines, where reproducibility, single-trajectory inspection, and pointwise error metrics (MSE/MAE) are the primary concern. In this context, a data-conditioned ODE with gap-only su- pervision and per-step data-consistency (DC) projection aligns the training objective on missing entries with the inference constraint that preserves observed coordinates, while offering a simple aceuracy-cost knob through the number of Heun steps. Diffusion-based models can also be run deterministically via probability-flow ODEs or DDIM-style samplers; goal is not replace them, but to provide a lighter- task-aligned alternative for long-gap imputation that comple- ments fully probabilistic approaches when uncertainty estimates required. From probabilistic viewpoint, sampling the initial state Zn ~ (0, I) together with the learned ODE induces a condi- tional distribution over imputations given (ã, .). In this work, however, adopt a deterministic single-trajectory regime: for each window we draw one 2n fixed by the random seed and then run fully deterministic Heun+DC solver, without multi-sample aggregation calibrated uncertainty estimation.
2 RELATED WORK
Time-series imputation with deep models. Early neural approaches address missingness by de- signing recurrent architectures and decay mechanisms to handle irregular observations (e.g.. GRU- D and BRITS). More recent encoder-decoder designs rely on self-attention to aggregate long-range temporal context. Representative ants and strong forecasting models that are often adapted to imputation, such as DLinear, TemesNet, PSt I'Tansformer, SAITS, SCINet and FreTS (Vaswani et al.] 2017: Zeng et al.] [023] Wu ctt al. 2023 Nie et al.] 2023 Liu et al. 2024 Du et al. 2023] Liu et al. 2022, Yue et al. 2025). These methods differ in how they encode temporal structure (channel-wise linearity, 2D tempo- ral kernels, patching, inverted attention, self-atteeniinn-baeer imputation, hierarchical or frequency- aware blocks), but they typically learn a point estimator that does not explicitly model the evolution from noise to data.
ge-geeneeraiee baselines in our study include Transformer vari-
Diffusion-based probabilistic imputation. Score-basee diffusion models have been adapted to fill gaps by conditioning on observed entries and denoising missing coordinates along reverse stochastic process. CSDI is a canonical representative that parameterizes the conditional score and samples imputations by iterative refinement (Tashiro et al. 2021). PriSTI extends this conditional
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diffusion paradigm to spatiotemporal sensor networks, SSSD couples diffusion with structured state- space models for long sequences, and frequency-ave generative models explicitly control spectral content during denoising for multivariate time-series imputation (Liu et al. 2023a Lopez Alcaraz & Strodthofft 2023; Yang et al.) 2024). More recent work further refines conditional diffusion for time-series data, including Mtsci for measureement-cooniiteen imputation, Diffusion-TSS for inter- pretable time-series generation, and Diffputer for efficient missing- imputation (Zhou et al.) 2024 Yuaa & Qiao 2024 Zhang et al. 2025). These approaches naturally provide uncertainty through sampling, but inference requires many reverse steps or ODE evaluations and can exhibit sampling variance in deterministic evaluation protocols. More recently, consistency models (Song eet al. 2023) train a network to approximate the solution operator of the probability-flow ODE of a diffusion model, enabling single- or few-step conditional sampling while retaining uncertainty modeling. CoSTI (Solis-García et al.) (2255) applies this idea to spatio-temporal imputation, pro- viding fast probabilistic imputations for multivariate time series; our work complementary in that focus on simpler, flo--matching ODE aimed at reproducible point estimates rather than full posterior sampling.
Flow and path matching. Flow matching and rectified flows provide an alternative to diffusion by training a velocity field along prescribed bridge between simple noise and data (Lipman al. 2023 Liu et al.) 2023b). Compared with stochastic samplers, integrating a learned ODE at test time offers deterministic alternative and, in many empirical settings, can achieve competitive speed-quality trade-off; probability-flow ODEs for diffusion models provide a closely related for- mulation of deterministic sampling (Song et al., 202T). Our work follows this line but adapts it to multivariate time-series imputation with three task spec-fi elements: (i) subbiiity-asked attention that restricts aggregation observed timestamps, (ii) time-gated multi-scale parameterization of the velocity that allocates spectral emphasis along the trajectory, and (iii) Heun integrator coupled with a per-step cata-consistency projection to preserve measurements exactly.
Optimal transport and alignment for imputation. complementary thread formulates impu- tation as alignment/rransport between partially observed sequences and learned priors, including Sinkhorn-style objectives and transport-guided matching (Wang et al. (202aa). These methods are deterministic and can be compute-efficient; however, they typically rely on a hand-crafted matching cost and do not expose an explicit continuous-ting generative trajectory. Our method instead learns conditional velocity field and uses OT-style ideas only implicitly (through the linear bridge and data-consistency projection), resulting in a transparent ODE evolution.
Multi-scale and frequency-aware designs. Multi-resolution backbones and frequency-aware blocks have proved effective in forecasting and representation learning (Lin et al. 2017 Ron- neberger et al.] 2015) Oppenheim et al., [996; Wu et al.] 2023] Yi et al. 2023). Recent works have also explored frequency-
generative formulations for imputation (Yang et al.) 2024). We differ in where multi-scale modeling is injected: rather than fusing features only in the encoder, we parameterize the velocity field with scale-specific heads and time gate. This lets the solver traverse a coarse-to-fine trajectory that first stabilizes global trends and then refines high-frequency details as 1-
Continuous-time models and ODE solvers. Neural ODEs and continuous- time latent dynamics offer natural tools for irregularly sampled series, from variational latent ODEs to neural controlled differential equations. Our design is distinct in that the ODE not a latent dynamics model but the deterministic bridge of a conditional flow; its integration uses a second-order explicit method (Heun) with a hard projection that enforces data consistency at each step (Hairer et al. 1993). Empirically (Sec. 4.. this coupling reduces boundary artifacts and improves stability under long gaps.
Positioning against recent generative and graph methods. Our deterministic, DC-clamped flow differs from (i) trajectory flow matching for Neural SDEs, which targets stochastic and irregular clinical series with uncertainty-aware outputs (Zhang al.) (004): (ii)
rep- (Le Naour et al.) 2024); (iii) tiffusion-based imputers such as SSSD that combine conditional diffusion and structured state-space models (Lopez Alcaraz & Strodthoff 2023) and medical DA- TASWDM with density-aware temporal attention (Xu et al. 2023); and (iv) graphdspatioteeppraa imputers captur-
resentations that learn sample-conditinn
continuous fields for imputation/forecastin
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ing relational structure Ye et al. 2021 Suo al. 2020). A recent survey (Wang et al. 2025b frames uncertainty and architecture axes; our contribution sits in the deterministic/FM corner with gang-aap
explicit DC projection and time-gates multi-scale velocity, providing strong long gap behavior un- der fixed hyperparameters (Appendix E). In parallel, consistency-mode approaches (Song et al., 2023 SSofi-GGariia et al., (025) approximate probability-flow ODEs obtain few-step determin- istic samplers with calibrated uncertainty, and discriminative graph-based imputers such as GRIN or ImputeFormer (Cini et al. 2022 Nie et al., 2024) provide strong deterministic baselines when reliable graph is available; TG-MSFM instead targets graph-agnostic, long long-gap imputation with simple flow-matching ODE and per-step data consistency.
3 TIME-GATED MULTI-SCALE FLOW MATCHING
3.1 PROBLEM SETTING, STRUCTURED INPUT, AND MASKED TRANSFORMER BACKBONE
Let r E RTXD be a multivariate time series and M E [0,1]Tx a binary observation mask, where Tt,d is observed if Mr, 1 and z is missing otherwise. The objective is to impute r at missing positions using a deterministic flow defined on E [0,
We form structured imput
= x m, x4, F E RT((D44))
per-time visibility flag such that m. 1 if Me, 1 for some d and my 0 otherwise, and ,4, E RT are moving averages of observed points with the length w (we set w = 10) summarizing local left/right temporal context. These three auxiliary channels are treated as known and inform both attention and data-consistency at inference. We denote data channels by 'D (1...., D} and conditioning channels by C, so the full channel set is composed of both D and with |C] = 3.
where m E {0,
A time-aware Transformer Sa Vaswani et al. 2017) serves the backbone. It consumes (#x, L, T) and produces shared representation h fø t,ā) E (Tx()+3), Self-attention is masked by time visibility: query at index attends only to keys with m, 1. Let 4r, 4r,ke ki Rd be query and key vectors, respectively. The attention logits and weights are
2. ke.
if m 1, m4 O,
exp(a7t)
a,
dil
and art = E, exp(ars)
00,
Equivalently, with bias matrix B RXX where Br,, 0 m,=1 and By -o otherwise, scores are given as softmax((QK (OKT///KK//)) /vdx))++++ B). The scalar is encoded via a sinusoidal/timestep embedding and added to token features (Vaswani al. 2017). The conditioning channels C' prop- agate through the backbone and heads identically to data channels but are (i) excluded from the supervised set defined below and (ii) clamped by data consistency during inference maintain alignment with the conditioning state.
3.2 FLOW MATCHING WITH GAP-ONLY SUPERVISION
We adopt deterministic flow matching (ipmman et al. between Gaussian noise and data-anchored endpoint. For each sample, draw 2g ~N(00, I) and
2023 Liu et al. 2023b) on a linear bridge
21
21 = t) 20 + 21, t Uniform[0, 1).
The teacher velocity constant along the path:
n(2 t)
21 21 20- dt
velocity field vo is trained to match v conditioned on ã, with supervision restricted to the missing entries data coordinates. Define the index set of gaps
Ω {(1, d) Mid d D).
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Structured input (ohservedonly) (viaibility zt, 2²
Multi-scale velocity
Data Consistency
lawer leider to du 2.
Downl Angthoot
Downz Angghwell
Down (Aughool
7g
Head (Conv'
Headl (Comv
Head (Cope)
ODE Solver
Ups Equanplk
Up specto
Upr
Time-aware Transformer
Jts Uar
act))
visiilliy-masked self-atteution
multi-scale velocities
Imputed Output
Figure I: Overview Tate-Gated Multi-Scale Flow Matching, The partially observed sequence I and its visibility mask are encoded by time-aware masked Transformer produce shared representation h. A fixed ID pyramid with strides S {1, 4} builds scale-specific features h(*) = (h), which are mapped by lightweight ivelccty heads" Head, and upsampled to can- didate velocities ((*) = (Head, (h()))). time-dependent gate a(t) = softmax(MLP(()) E 1 11 blends these branches into the final velocity field (; = E.es at a(*), which parametrizes the ODE é oz(2; 1; i; During inference, a Heun solver together with per-step data-consistency projection keeps the observed coordinates on the linear bridge between 2p and 21, yielding deterministic, prasurement-preservvin imputation trajectory.
The objective is given
LPM
Σ
[vo(ze, (z)) 20]e,d
(I)
122
(t,))R
This restriction aligns the training signal with the imputation target: observed coordinates are already constrained inference by the data-consistency mechanism described in Sec. 3.4 Applying addi- tional loss on observed entries would be redundant with that constraint and can introduce conflicting gradients without improving reconstruction of the unknown portions. Optional stability regularizers, first or second temporal differences of og and light high-frequene suppression, are included with small fixed weights; exact forms are deferred to the Appendix Ajto keep the main objective concise.
3.3 TIME-GATED MULTI-SCALE VELOCITY HEADS
To capture both fow-freqeency trends and high- figh-frequency details, the shared representation h fans out multi-scale velocity heads (see Fig. [ for an overview).
Let S {1, 2, 4} denote stride factors of a fixed ID pyramid implemented with average-pooling downsampling and linear upsampling (Lin et al. 2017 Ronneberger al. 2015). For each 8 E S,
(+) Downs,(h),
(+) (h(o))
(()) Ups,(u(*)), va (;
Σ at) a(t)
SES
Each Head, lightweight local module (e.g., Conv-GELU-Conv) that proposes scale-specific velocity. Scales are combined through a time-dependent gate
(t softmax (MMMmaxxMMP(()) RIS (,(,,t; = Σ at,(t) ((.).
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This design allows the spectral emphasis to evolve deterministically with close to t 0, the gate can emphasize coarse components to stabilize the global trajectory; near t=1, can shift weight to the finest branch to resolve sharp transitions. To reduce spurious oscillations the finest path, apply a fixed ID anti-aliasing filter (3-5 taps, unit DC gain). An elementwise squashing (e.g., tanh) bounds velocity magnitudes without affecting the ODE fixed point.
3.4 DETERMINISTIC ODE INFERENCE WITH HEUN, DATA CONSISTENCY, AND COMPUTATIONAL ASPECTS
At test time integrate the learned velocity as am ODE from We use the second-order Heun method (predictor-correctoor;; explicit trapezoidal rule) (Hairer et al., 1993 with an optional monotone time warp left [0, 1]
1, starting at 2g 60~00 ~N~ NO0
corrector;
(e.g. leff (t) 1*t k 22:
At (vo(2n, t. u0 t, + At:
= 2n + At Uo (2n: Em: B),
ode #1+1 = 2n 2
After each step, data-consistency (DC) projection is applied. Let K be indices of observed data
coordinates (i.e., K = {(t, d) M together with conditioning channels C, which are treated as known for all t. With E and the same 20 used to initialize the trajectory.
2 [K] < (1 - loft) zol K] + lett 21 [K]
Zn+1 + ode K
Hence, known entries follow the linear bridge exactly at every step, while unknown entries evolve under the ODE.
Property (consistency under perfect velocity). If un va(2;;;; Heun+DC procedure recovers the exact linear bridge for coordinates, i.e., Zn = -1-1,) ) + 21 for every step n. Heun is exact for constant velocities (Hairer et al. 1993), and DC clamps K to the same linear path.
v(2,, = 21-20- then the
Computational considerations. Let L be the number of Transformer layers, H the number of heads, and d the head dimension. The backbone has O(L time and O(T²) attention-memory costs per batch; masked attention preserves these asymptotics. Multi-scale heads add O(SITD) per forward pass. Heun requires two velocity evaluations per step. so inference with N steps uses roughly 2N forward passes; the DC projection linear in KK. In practice, we use S= [, 4), w= 10=10, a 3-5 tap low-pass for AntiAlias ID with unit DC gain (Oppenheim et al. 1996). l E [1, for the time warp, and N E [200, 400] for robust accuracy cost trade-off. Optional smalh- temporal regularizers can be enabled for numerical stability; they are not critical to the main results and are detailed in the appendix.
EXPERIMENTS
We evaluate TG-MSFM on widely used multivariate time-series imputation benchmarks. The sec- tion proceeds as follows: Sec. S.c. states datasets, metrics, baselines, and compute; Sec. 4.2 presents the main accuracy table and analyzes dataset trends; Sec. 4.3 studies the speed-quality trade-off and
step efficiency; Sec. 4.4 ablates key design choices: Sec. 4.5 reports robustness to gap length with qualitative reconstructions.
4.1 EXPERIMENTAL SETUP
Datasets.
We use WTTT///TTT///TTT I/ETTm2 (electricity transformer temperature), Electricity, Traffic, Weather, Illness, Exchange, and PEMS03. We follow the standard preprocessing and official splits used in prior work to ensure comparability.
Metries. We use the mean squared error (MSE) and the mean absolute error (MAE), which are computed exclusively on missing entries. Unless otherwise noted, we we report the average over missing ratios {0.1, 0.3, 0.5, 0.7) and random seeds.
Baselines. Transformer, DLinear, TimesNet, FreTS, PatchTST, SCINet, iTransformer, SAITS, and CSDI; additionally, we include alignment-style methods Sinkhorn OT and TDM (Zhao et al. 2023 Muzellec al. 2020) contextualize non-generative approaches.

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Table 1: Imputation accuracy (MSE/MAE) on ten datasets, averaged over missing ratios
{0.1, 0.3, 0.73. Bold indicates our method; underline marks the best baseline per column.
ETTh1 MSF
ETTh2
ETTml
ETTm2
Electricity MAE
Method
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
Transformer
0.222
0.322
0.221
0.312
0.060
0.160
0.041
0.134
0,120
0.228
DLinear
0.144
0.267
0.108
0.231
0.103
0.221
0.084
0.201
0.137
0.271
TimesNet
0.253
0.353
0.133
0.263
0.061
0.173
0.068
0.186
0.129
0.254
PreTS
0.184
0.312
0.147
0.259
0.055
0.159
0.039
0.135
0.155
0.285
PatchTST
0.171
0.126
0.258
0.050
0.149
0.030
0.118
0.138
0.262
SCINet
0,149
0,275
0.128
0.248
0.067
0.176
0.064
0.179
0.125
0.239
Tanssormrr
0.163
0.281
0.101
0.211
0.056
0.156
0.034
0.125
0.128
0.251
SAITS
0,216
0.305
0.183
0.256
0.056
0.154
0.042
0.129
0.114
0.216
CSDI
0.151
0.269
0.098
0.263
0.101
0.177
0.158
0.113
0.533
0.269
PriSTI
0,143
0.270
0.110
0,220
0.053
0.153
0.033
0.116
0.118
0.220
Misci
0,141
0,268
0.093
0.226
0.086
0.164
0.085
0.109
0.420
0.259
Diffusion-TS
0,165
0.292
0.129
0.257
0.094
0.183
0.092
0.121
0.485
0.268
PGTI
0,176
0.301
0.141
0,264
0.098
0.191
0.099
0.128
0.512
0.276
TG-MSFM (Ours)
0.120
0.219
0.044
0.213
0.045
0.124
0.020
0.089
0.101
0.198
Traffic
Weather
Illesss
Exchange
PEMS03
Method
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
Transformer
0.216
0.214
0.195
0.132
0.240
0.300
0.224
0.186
0.081
0.184
DLinear
0.251
0,272
0.274
0.185
0.210
0.273
0.261
0.216
0.112
0.261
TimesNot
0,201
0.243
0.280
0.189
0.231
0.277
0,319
0.264
0.076
0.190
FreTS
0.120
0.278
0.325
0.228
0.189
0.109
PatchTST
0,235
0,238
0.247
0.167
0.605
0.505
0.237
0.197
0.065
0.179
SCTNet
0.290
0.314
0.198
0.136
0.617
0.473
0,298
0.247
0.106
0.249
Transformer
0.252
0.269
0
0.127
0.316
0.310
0,068
0.056
0,080
0,200
SAITS
0.224
0.207
0.132
0.089
0.167
0.216
1.005
0.835
0.083
0.189
CSDI
0.306
0.324
0.158
0.107
0.356
0.384
0.100
0.103
0.115
0,170
PRISTI
0,200
0,208
0.135
0.088
0.170
0.220
0.067
0.057
0.068
0.172
0.188
0.245
0.083
0.070
0.091
0.182
Diffusion TS
0,228
0,255
0.173
0.1
0.295
0.318
0.096
0.095
0.102
0.198
FGTI
0,241
0.268
0.186
0.124
0.530
0.345
0.108
100
0.110
0.214
TG-MSFM (Ours)
0.187
0.209
0.102
0.096
0.064
0.116
0.029
0.025
0.047
0.142
Implementation and environment. Our default model uses the time-aware Transformer with visibility-masked self-attention, time-gated multi-scale velocity heads, and Heun+data consistency inference (Sec. (3). Hyperparameters are fixed across datasets. All experiments run on server with 88 CPU threads (Xeon E7 8788880 v4 @ 2.20GHz) and 3 TB RAM. Deterministic inference uses N 300 ODE steps unless stated.
4.2
MAIN RESULTS
Analysis. Across ten datasets, TG-MSFM delivers the strongest average performance in both MSE and MAE, and it does so without -ataset-specific tuning. On periodic familics (ETTh/m), gains are steady but moderate. Visibility-masked attention already transports seasonal information from observed timestamps, so most of the improvement arises near gap boundaries: Heun+data consistency (DC) prevents drift of observed coordinates and curbs error propagation into the gap in- terior. On burst-plus-trnnd families TTraffic/Exchange),, improvements are larger: early flow phases emphasize coarse scales that stabilize the global trajectory, while the fine head, lightly anti-aliased, introduces localized corrections only when the state is close to the endpoint. This mitigates over- shooting and reduces the absolute error footprint in regions with rapid transients. On heteroge- neous/higher--vrinnce families (IInnes//EEMS03), TG-MSFM reduces both tail errors and median deviations, suggesting the time- -atne- heads act as an implicit spectral scheduler that avoids injecting high-frequency detail prematurely. Finally, compared with stochastic diffusion (CSDI), the deter- ministic ODE yields consistently lower MAE under the standard deterministic-impuattio protocol, eliminating sampling variance source of evaluation noise.
4.3 SPEED-QUALITY AND STEP EFFICIENCY
Figure Souree2ssuma summarizes the compute-accurace trade-off on ETThl under identical hardware and batch size. Panel 2a plots wall-clock time (ms per sample) against MSE; solid curves means over five
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7G-M5FM (AUPC1 -0
0.2150
7G-MPFM
0.2125
CSDI (AUPC) -0.390)
0.2123
CSDC
0,2100
0.2100
0,2075
0.2050
0.2050
0.2025
0.2000
0.2000
0.1975
0.1950
50
100
150
200
250
350
400
100
150
200
250
300
350
400
(maplemmpl))
(network torward ovaluations)
(a) Speed-quality on ETTh1: wall-clock time vs.
(b) ETTh1: MSF vs. number of function evaluations (NFE). For Heun, NFE=2N; for CSDI, NFE equals reverse steps.
MSE. Shaded bands show central 90% CI: legend re- ports AUPC (higher is better).
Figure 2:
Speed-quality analysis on ETThI. Means over five seeds with 90% CIs (shaded). Curves the same hardware and batch size.
Table 2: Ablation on Electricity and ETThl. Columns "MS Gate / Heun" denote the presence (V) or absence (X) of multi-scall scale heads, time gate, and Heun integrator. Bold is the full model.
MS
Gate
Heun
Electricity MAE
ETTh1 MSE
Variant
Single-scale (a=1) Static mising (no gute)
/
0.116
0.227
0.158
U.276
x
v
0.212
0.223
0.147
0.261
Ealer (no Heun)
/
x
0.115
0.218
0.143
0.257
TG-MSFM (full)
/
/
0.101
0.198
0.126
0.231
seeds, with shaded bands denoting the central 90% bootstrap interval. The legend reports AUPC (area under the Pareto curve; higher is better). TG-MSFM dominates the upper-left region: for fixed budget of milliseconds it attains lower error, and for target error it requires less time. Panel 2b complements this view by plotting MSE versus the number of function evaluations (NFE). For Heun, denoising steps.
NFE 2NV where N the number of ODE steps; for CSDI, NFE equals
TG-MSFM exhibits clear diminishing returns around N = 250 and graceful degradation below N S 100 due to the coarse-to-fine gate. contrast, CSDI's slope is flatter: extra reverse steps predominantly damp sampling noise rather than correcting structural bias. In practice this translates into simple recipe: set N [200, 300] for near-optimal accuracy, or N [80, 120] for fast validation runs with small accuracy loss.
4.4 ABLATION
Table Dioleeisslatess the effect of the main components ETTh1 and Electricity. Regarding multi-scale heads, collapsing to a single scale increases both MSE and MAE, indicating that a single receptive field cannot reconcile global trends with short transients. When we consider the time gate, replacing the gate by static mixing degrades accuracy consistently, supporting the view that spectral emphasis must evolve with the flow phase. Finally, about the integrator, using Fuler instead of Heun in- corrector average reduces local truncation eгтor at precisely
creases boundary errors: the the points where the DC projection constrains observed coordinates. yielding smaller leakage into neighboring missing timestamps. Overall, the components are complementary: the gate schedules what emphasized, while Heun+DC controls how updates propagate through the gap.
4.5 ROBUSTNESS TO GAP LENGTH AND QUALITATIVE ANALYSIS
(a) ETTh1: MSE versus gap length. We vary the length of a central missing block while keeping the overall observation ratio fixed. As shown in Figure 3a error increases with longer gaps for all methods, but TG-MSFM exhibits a slower growth and remains the most accurate across all lengths. Two design choices contribute this behavior: (i) the Heun+DC integrator keeps observed coordinates exactly on the deterministic bridge at each step, mitigating boundary drift and error accumulation gap edges; and (ii) the time-gated multi-scale heads delay fine-scale refinement
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TG-MSFM
TimesNet FreTS
iTranfformer
DLinear
PatchTST SAITS
CSDI
PriSTI
SCINet
Transformer
0.225
0.220
MSE
0.215
0.210
0.205
24
36
48
60
72
Central gap (hours)
(a) MSE versus central gap length (mean over seeds; shaded 90% CT).
HUFF
HULL
WW MAM
WA
Mm-
Vilue
m
40
72.
Row Index
(b) Imputation examples ETTh1: observations (dots), ground truth (orange), and TG-MSFM predictions (red) across four channels (2x2).
Figure 3: Robustness and qualitative analysis. (a) Longer gaps amplify uncertainty; TG-MSFM maintains the lowest error as the gap grows. (b) Predictions align with observations by construction and transition smoothly across gap edges while recovering seasonal structure.
until the state approaches the endpoint, avoiding premature overfitting local fluctuations when long spans are missing.
(b) ETThl imputation examples. Figure 3b presents four representative ETThl channels in 2x2 grid. Black dots denote observed entries; the orange is the TG-MSFM prediction. Predictions coincide with observations by construction (per-step data consistency) and follow the slow thermal trend through the interior of the gap. Transitions near the gap boundaries are smooth, while seasonal variations and localized deviations are progressively recovered along the flow, consistent with the intended coarse-to-fin evolution.
is the ground truth; the blue curve
5 CONCLUSION
This paper introduced Eite-Gated Multi-Scale Flow Matching, deterministic framework for mul- tivariate time-series imputation that learns the velocity field of a data-conditioned ODE via flow matching. The method uses structured endpoint encode partial observations, time-aware

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Published as a conference paper at ICLR 2026
Transformer with visibility- -aakkdd self-attention to aggregate context, time-gated multi-scale locity heads to balance global trends and local details along the trajectory. and a Heun integrator with per-step data consistency to exactly preserve measurements. Training focuses supervision missing entries through a gap-only objective, aligning the learning signal with the imputation target (Sec.
Empirically, TG-MSFM attains state-of-the-an art or competitive accuracy on ten widely used bench- marks while offering favorable speed-quality trade-offs (Table []] Fig. 2 On periodic series, de- terministic integration and data consistency reduce boundary drift; on burst-plus-trent series, the time- gated multi-scale parameterization mitigates overshoot and improves absolute error. Ablations confirm that each design element, multi-scale heads, the time gate, and the Heun+DC coupling, con- tributes additively (Table 2 and robustness tests show graceful degradation as gap length increases (Fig. 3).
Limitations and future work. First, the bridge is linear and the timestep schedule is global; learn- ing data-adaptive bridges and time warps may further reduce step counts and boundary artifacts. Second, the backbone uses quadratic self-attention; extending to long-sequence or streaming vari- ants (e.g., sparse/linear attention) would broaden applicability. Third, the method deterministic by design; in settings where calibrated uncertainty is required, combining Time-Gated Multi-Scale Flow Matching with lightweight posteriorization (e.g., ensembles around the velocity field, con- formal bands over ODE trajectories) is promising direction. Finally, while results span diverse public benchmarks, domain-specific evaluations (e.g., clinical telemetry with censoring or device dropouts) would test robustness to real-world missingness patterns.
Overall, TG-MSFM provides a transparent, measurement-preseeeeeee preeservig alternative to stochastic sam- pling for time-series imputation, isolating when and how spectrum-specifii structure is introduced along the generative trajectory. We hope the formulation and analysis here encourage further work on deterministic, phase-aware flows for irregular and partially observed sequences.
LLM usage: We have used LLMs to polish our texts and correct grammars.
ACKNOWLEDGEMENTS
This work was supported by JSPS. KAKENHI Grant Number JP25H01112, JP25H01124, Japan and JST, CREST Grant Number JPMICR22D3, Japan.
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APPENDIX
A TRAINING, OBJECTIVE IDENTITIES, AND DETERMINISTIC PROPERTIES
Notation alignment. We keep the same symbols as in the main text: time series E TX observation mask M E {0, 1 and the structured endpoint = [ O M, m, 24, I²| (T++++3) where m, = 1[Ed : Me,d = 1} and b ft are length-w w moving averages (default w=10). The three conditioning channels are always treated known during training and inference.
A.1 OBJECTIVE IDENTITIES AND UNBIASED GRADIENTS
Bridge construction and oracle velocity. define
For each sample, draw ~ ~N I), set 21 = T, and
d
24 (1 - t) + 21,
t²(, dt 21 21 2g.
Let D denote the data coordinates and {(t, Me.d 0, D} the missing-indes set.
Gap-only flow matching loss. With a velocity field ug, the training objective used in the paper is
LPM = Eo
Σ
(; - (=,, 1)
(3)
(i,d)ER
Mini-batch training with (i) sampling I, zo and (ii) uniformly sampling a subset Qr C Ω gives an unbiased estimator of LFM-
Gradient form. zeros others. Then
Let ro(2),t) Ir( o² (2e, t)), where Il keeps entries in and
a
VOLFM= = 2 (0)=,,1z)] ro 70(=,1))
Jø = a0 ( (;
(4)
In practice we use standard autograd; the identity clarifies that restricting supervision to Ω yiclds a variance reduction observed coordinates, focusing learning capacity unknown entries.
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Regularizers (explicit forms). We optionally add small penalties (weights A2, AHP < 1):
RIvI =
Σ
[10]k+1, - [vola.d]22
(5)
[
(i,d)
RTV2
[[[v]] 2(voleat + [voli-
(6)
[
(K
RHP
Σ
11(h (d))l.))ll2:
S2
(U,))R
where is ID convolution over time, and fixed high-pass kernel (e.g., [1, -2, 1).. The total loss is LPM+ + Rrv + AuP (These terms are not critical to reproduce our results; we set them to very small constants in all experiments.)
2,
A.2 TIME-GATED MULTI-SCALE VELOCITY: BOUNDEDNESS AND CONVEXITY
Let the shared hidden h E RTx(D+3)) feed scale-specific heads u+)) = Head, (Down,hh), upsam- pled to (!*), The time gate (tt softmax(MLP(tt) E IRS defines
(o t; 8) =
(8)
.ES
where á is an elementwise squashing (we use tanh). Since a (1) 2 0 and Σ, a,(()) = 1, equation
convex combination of bounded proposals ((ū(*),, hence ||ug ll I 1. The fixed low-pass the finest branch is a linear time-invariant operator with unit DC gain, so it preserves the bridge's mean and reduces high-frequency energy: ||]/|||2 < ||<|| for all I orthogonal the DC component.
An empirical view of a(l) is provided in Appendix Fig. B2
and per-scale contributions in Fig. B3
A.3 DETERMINISTIC HEUN+DC: EXACTNESS AND STABILITY
Heun step.
With step size At, one step of Heun (explicit trapezoidal rule) is
2n + At up tni Ø),
(9)
ode 2++1
2n + At
2 (o(2nntn: + vo t, +At;
(10)
We then apply the data-consistency (DC) projection on known indices K (observed data coordinates and the three conditioning channels) with the linear bridge:
Zr++1 [K] < (1-4x+1) zo[K] [K] bn+1 z1 [K],
Zn (K] 6-
2
ode
K
(11)
Proposition A.1 (Exactness under oracle velocity). vo t: = 02 t) 2 - 2g (constant in (z,, ()), then for any step size At, the Heun update satisfies 2x+1 (1 11-mm1) =0 + 1n+1121 Con-
sequently, the DC projection in equation [T] leaves the state unchanged and coordinates (known and unknown) follow the exact linear bridge every step.
Proof. For constant velocity, explicit trapezoidal equals exact integration of linear function (local =ode
truncation error 0), hence an++1 2n + A/ (o)= By induction with ln+1 In At, 2n+1 (1-tn+1)80 + fnt121. The DC step matches the same value on K, leaving 2x+1 unchanged.
Proposition A.2 (Projection preserves measurements; non-expansiveness) Let PK be the affine projection defined by equation 1I at step n+1. Then for any u, -D-3() [[ (1( < ||u v/12. with equality u-v has support only on K. Moreover, Px overwrites K with the lincar-bridge value, thus preserving all observed measurements exactly at every step.
Proof. PK is an orthogonal projection on the affine subspace {2 2|K] = bridge(tn+1)); orthogo- nal projections are I-Lipschitz. The second claim follows by construction.
Heun accuracy and stability (classical). If Ug is C² and Lipschitz in 2, Heun has local truncation ( and global error (At) (Hairer al. 1993). The DC step does not increase the error (Prop. A.2) and prevents drift on K, which reduces boundary artifacts observed with pure Euler. A controlled toy visualization provided in Appendix Fig. B7
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A.4 EXISTENCE/UNIQUENESS AND DETERMINISTIC TRAJECTORIES
Well-posedness. Assume ( [) is globally Lipschitz in uniformly over / E [0. 1 (true heads and backbone are Lipschitz and @ is bounded). Then the IVP & (=,211;3) z(0) 20
admits unique solution =(() on [0, 1]. Given a fixed =o (and fixed i), the numerical path produced by Heun+DC deterministic function of step size, and time grid.
Fixed point at t=1. If uz(z, 6; σ) is such that 21 is an equilibrium of the autonomous system at t in neighborhood (e.g., 0g 1, 1;3)= 20), then z(t) approaches 21 as Independently of this, the DC step enforces 2 [ 21 [ exactly.
A.5 COMPLEXITY ACCOUNTING
Let L be Transformer depth, H heads, and head dim dg. One forward pass costs O T dy)) in time and ( in attention memory per batch. Multi-scale heads add O(SS) D). Heun uses two velocity evaluations per step, so N steps require 2N passes; the DC projection is O([K]).
A.6 OPTIMIZATION, DATA, AND DEFAULTS
Optimization. AdamW (weight decay x 10 10_4), cosine LR with 5k warmup, peak LR 2 10 batch size E [16, 64]. Training for 200-400K steps with early stopping on validation MSE.
Preprocessing and masking. Per-channel standardization the training split; missing ratios (0.1, 0.3, 0.5, 0.73 for random-missing; central-gap masking for length sweeps.
lists shared defaults; unless otherwise stated Default hyperparameters. Table AT
(1,,2 4), anti-alias taps = 5, time warp k=1.5, and 3
Table Al: Default hyperparameters (shared across datasets unless specified).
Component
Symbol
Value
Component
Symbol
Valuee
Transfomner layers
L
6
Atlention heads
H
Head dimension
d,
64
MI.P maioo
Pyramid strides
S
4)
Context window
Auti-alias taps
5
Time warp
k
1.5
ODE steps
N
Batch
Optimizer
Adam
LR (peak)
2x
Weight decay
1 x 10
Warmup steps
5k
B ALGORITHMS AND LIGHTWEIGHT DIAGNOSTICS
B.I TRAINING: GAP-ONLY FLOW MATCHING
This subsection provides concrete training pseudocode for the gap-only flow matching objective described in Sec. 3 Algorithm B1 highlights (i) how construct the structured endpoint E from partially observed input and auxiliary conditioning channels, and (ii) how the supervision restricted to missing coordinates only (the index set Ω optionally subsampled P2),, that model capacity is focused on imputing unknown values while known entries are handled by the inference- time data-consistency mechanism.
B.2 INFERENCE: HEUN + DATA CONSISTENCY (DC)
Algorithms B1 BI-22
B2
match the main text (Sec. and the identities in Appendix
B.3
LIGHTWEIGHT DIAGNOSTIC FIGURES
This subsection collects small, dataset-agnostic diagnostics that help interpret the proposed solver and the time-gated multi-scale parameterization. Figure B1 visualizes how the per-step DC pro-
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Algorithm B1 Training loop (gap-only flow matching)
Require: dataset {(x, M)); window length T; channels D; context window w; steps per epoch S: optimizer (AdamW) for epoch 1, do
1:
2:
for step = 1, 1......3 do
Sample mini-batch {(2, M))
4:
Build structured endpoint 5 l M, m, Iz, z"!
5:
Sample ~ Uniform [0, 1] (per-sample or per-batch)
6:
Sample 2g (O, set 21 + z; compute 24 + (1 - t) + t2
7:
Forward backbone h 4- fo(2,,
8:
Multi-scale heads and gate: ng E.es al( (Head, (Down,
9:
Oracle velocity t* <
10:
Define missing index set f = {(i,d) Mid = 0, DJ: optionally subsample
24
C Ω
11:
Gap-only loss: LFM
1
Σ [uolicd [0 ledll2
(i.d)e
12:
Optional regularization (small weights): add equation 5 equation 7 to loss Backprop and optimizer step on 0,
13:
14:
end for
15:
end for
Algorithm B2 Deterministic inference (Heun + DC)
Require: (x, M), structured ã, steps N, time warp lett (t), initial zo (.
Set t 2g: define 21 ž; define known-index
f(i.d) M,,d 1) uc
2: for 0,...,N 1 do
3:
tu f Rt. At + N.
T tof (tn)
4:
k +- uo(2, In; .))
5:
++ z + At
6:
k2 (2 tAt 2)
7:
ode + + +f (k: + k2)
Heun (explicit trapezoid)
8:
Data consistency (bridge on K):
9:
=[K] (1 - Lesl (4n+1)) 20 [K] len (ln+1) [K]];
=[K] + odee K
10: end for
11: return
jection mitigates boundary drift under ODE integration; Figure B2 illustrates representative gate schedule at) that shifts from coarse to fine scales over time; and Figure B3 shows how per-scale proposals are combined by the gate under early- versus late-phase weights.
Ground truth
40
Euler (no DC)
x Heun + DC
value
20
x
0-
40
timestep
Figure B1: Heun+DC toy gap. ID signal with central gap: Euler (no DC) drifts at bound- aries; Heun+DC stays on the linear bridge on known indices and reduces ringing inside the gap. Synthetic data; training required.
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1.00
Coarse (s=1)
T1=0.29
T2=0.61
Mid (s=2)
(s=4
0.75
Gate weight a,(t)
0.50
0.25
a,tt) son eax (P.(8)) 20
0.00
0.00
0.25
0.50 Time along ODE
0.75
1.00
Figure B2: Time gate schedule. Softmax gate att) for three scales (s E {1, 2, 4)): higher coarse- scale weight early, higher fine-scale weight late. This conveys the intended coarse-to-fine evolution independent of any dataset.
Coarse
Gated
Vilne
Value
0.05
0.25
0.50
6.75
1.00
0.00
0.25
0.50
0.75
1.00
0.00
0.23
0.50
0.75
1.00
0.00
6.25
0.59
0.75
1.0
Timm (mvennaliant)
Time (ourmalloed)
Time (sormalized)
Time
(nncmaliznt)
a-10 25,02
Figure B3: Multi-scale contributions and gated sum. Columns show Coarse/Mid/Fine proposals and the Gated sum; top row uses Early gate a = [0.85, 0.13, 0.02], bottom row uses Late gate [0.10, 0.25, 0.65). Early emphasizes low-frequency trend, while Late restores fine details the same trend. (All panels share axis limits; GT in orange, proposals black dashed, gated sum in blue.)
C DATA AND MASKING PROTOCOL
C.1 PREPROCESSING AND INDEX SETS
Let RTxD be the multivariate series and M {0, T the observation mask. Per-channel standardization uses train-split statistics (Ma oa):
std Ftd
Ft.d
Pd
d. da aid + pd.
0
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and all metries computed after the inverse transform. We denote data-channel indices D {1, {I...., D} and the missing index set by
= { (l,d) 0,
C.2 STRUCTURED ENDPOINT AND VISIBILITY
Define the per-time visibility flag m, 1(3d D Mi,d 1). The structured endpoint concate- nates observed values and three conditioning channels:
I C m, I' =²] R (x(43))
The left/right context channels are finitewindow moving averages with window (default w=10);
(t) = {s: 15 w
Σ x[s].
[t)
SEIL())
IR(() = (s: t 1 TT, t w
1
Σ
x[s].
[z(((]
sEIR(()
These three conditioning channels are always treated as known during training and inference (i.e., they belong the known-index set together with observed data).
C.3 MASKING REGIMES
We evaluate two complementary protocols.
Random-missing (RM).
For a ratio p E
Mi. Bernoulli(1 p) i..d. over (t,d);
and results averaged over the four ratios and multiple seeds. All supervision and metrics only Q.
Central-gap (CG). block
For given gap length L, let [T- +1. Define single contiguous
0, g g 1, Pea =
Mi, 1, otherwise.
S 5 1, de de
We sweep L while keeping the global observation ratio fixed, and compute metrics on Reg.
C.4 VISIBILITY-MASKED ATTENTION (IMPLEMENTATION DETAIL)
Queries time attend only to visible timestamps with m,=1. With query/key matrices Q, K and an additive bias
0, me
Byd
-00,
m. = O,
the attention weights are
QK Vdh. + B).
softmax
Conditioning channels propagate through the backbone/heads like data channels but are excluded from supervision and clamped by data consistency at inference.
C.5
METRICS (ON MISSING INDICES ONLY)
After inverse standardization.
MSE
Σ
(Rt,d - 21,a)²
MAE
L - Fe,(
Ω
(1,))n
(t.))ER
Unless otherwise stated, we report deterministic imputations and average over RMM ratios E (0.1, 0.3, 0.5, 0.7) and multiple seeds.
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C.6 INFERENCE NOTE (CONSISTENCY OF CONDITIONING)
During deterministic integration (Heun + DC), the known-index set
K {(1, d) M2,a 1} U C with C {cchannels m, x , "
is overwritten at each step by the linear bridge (1 z) [K] + t [K], ensuring exact measurement preservation on K and preventing boundary drift.
D MINIMAL QUALITATIVE EXAMPLES
This appendix focuses solely single qualitative setting: single long central gap. We intention ally drop random- random-missing and all extended tables/ablations here; for quantitative metrics, see the main text.
WEICHTTED ILI
UNWEICHTED ILI
1.3
1.0
hand
0.5
0.0
-0.5
-1.0
-13
Volee
ILITOEAL
PROVIDERS
M
20
12
70
Done Toddr
Figure D1: Long-gap interpolation (single example). contiguous central gap of length I. shaded (grey). We plot GT (orange), our imputed centerline (red crosses), and the uncertainty band (green, shown only in the gap). The trajectory respects boundary measurements via DC and avoids ringing inside the gap.
E HYPERPARAMETER ROBUSTNESS AND WHY WE FIx THEM
Motivation. In the main text we adopt a single set of hyperparameters across datasets. Here justify why this is acceptable and how robust the method is to reasonable variations under the long- gap setting used in Appendix D
E. PARAMETER SENSITIVITY ANALYSIS
Our design attenuates hyperparameter sensitivity through four ingredients:
1. DC projection elamps measurements. During inference, known indices are overwritten by the linear bridge (App. A.3 Eq. equation [1). Errors observed coordinates are iden- tically zero and do accumulate even if the integrator step size or capacity suboptimal.
2. Bounded, convex, multi-scale velocity. The time-gated head (App. A.2 Eq. equation g)
forms a convex combination of bounded proposals; this curbs runaway dynamics and makes stability less dependent precise head weights the exact gate schedule.
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3. Second-order integration with non-expansive projection. Heun has global error O and the DC step is I-Lipschitz (App. A.33. increasing ODE steps shows diminishing re- turns, and small step-size changes rarely cause qualitative shifts.
4. Scale normalization and gap-only supervision. Per-channel standardization and evalu- ating the loss only on missing entries (App. (A.T) reduce gradient scale variability across datascts and masks, optimizer settings transfer.
E.2 SENSITIVITY PROTOCOL
We vary one hyperparameter at a time around the defaults in App. A.6 with all others fixed. Each setting uses the same window length T=48, central long gap with ratio L/7=0.355 and 3 random seeds. Metries are computed only on missing indices after inverse standardization, consistent with the main text and Appendix D
E.3 QUANTITATIVE SUMMARY ON ILLNESS
Table E1
reports RMSE/MAE for representative sweeps. Differences within reasonable ranges 2%,,, confirming robustness and supporting the use single fixed configuration.
small
Table E1: Long-gap sensitivity on Illness. Window 7=48, gap ratio L/7-0.35, 3 seeds. Metrics computed only on missing indices after inverse standardization. Default is marked with Dif- ferences are small, showing robustness.
Group
Setting
RMSEJ
MAET
ARMSE
Solver
Steps
0.3344:0.00
0.251#:0 005
+5.0%
300
0.318::0000 0.04
0.236:10000
200
0.321110.004
0.238.10.0 004
+0.9%
500
0.3171000004
0.236:100000
-0.3%
Scale
warp
0.321100.00
0.237 0.004
+1.0%
1.5"
0.318
0.236:10.009 003
0.320:t0. 000
0.237 003
+0.0%
Scales S
{2.4)
0.329=20.008 0006
0.215:: 005
+3.5%
(1,2,4)
0.318:10 004
0.236:+0 003
(1,2,488
0.317=0.004
0.236 003
0.3%
Capacity
Layers
0. 0.323:10.00 000
0.230 000
6'
0.318 004
0.236:0.00 003
0.319::0.00
0.23744 003
+0.3%
Head dim
0.32110.000
0.238110. 004
+0.9%
64
0.318:10.00 004
0.236:10 003
0.319=
0.237 003
+0.3%
Heads F7
0.320=10.004
0.237-10.00
+ +0.6%
0.318:10 004
0.236:0.
Optim.
Peak L.R
1 10
0.321110.000
0.238.10.00
+ +00.9%
2x
4
0.318:10.00 0.00
0.236:10. 003
10
0.322:10.cm
0.240 005
+1.3%
Warmup
0.319=20.000
0.237: 003
+0.3%
5k"
0.318:10 004
0.236:40 003
0.319=10.00
0.237 003
+0.3%
Batch B
16
0.319:10.005
0.237:10.000
+0.3%
0.318
0.236.10.008
0.319=20.004
0.23740.003
+0.3%
Weight docay
x 10
0.319::0.0 004
0.23740 003
+0.3%
x 10
0.318.10.000
0.236.10.00
0.32110.005
0.230-0.004
+0.9%
E.4 RECOMMENDED DEFAULTS AND SAFE RANGES
Table summarizes the recommended defaults and safe ranges; within these bands, we observe marginal metric changes and visually indistinguishable long-gap curves.
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Table E2: Recommended defaults and safe ranges (long-gap).
Hyperparameter
Default
Safe range
Layers L
4-8
Heads
4-8
Head dim d,
64
48-96
Scales
(12,44)
(1.2,4) or (1,2,888
Time warp
1.0-2.0
Ant-aalis lups
5
3-7
steps
Batch size B
32
16-64
Peak LR
2x10
(1-5)x10 10
Weight decay
1x10
10-8--5x 10
Warmup stops
5k
3k-10k
Table E3: MSE on ETTh1 and Electricity, under the same setting as Table 1
Method
ETThi (MSE)
Electricity (MSE)
TG-MSFM
0.123 0.001
0.112 + 0.003
CSDI
0.154 + 0.0301
0.568 + 0.028
E.5 SEED-WISE VARIABILITY ON ETTHl AND ELECTRICITY
We additionally evaluate the robustness to random seeds. We train TG-MSFM and CSDI with K different global random seeds and report the mean and standard deviation of the test MSE in Table B3
21


Reviews:
[{"review_id": "XPewIrGzSi", "reviewer": "Reviewer_migb", "summary": "This paper proposes the TG-MSFM model, which introduces a data-conditioned ODE framework trained via flow matching using gap-only supervision of the velocity on missing data coordinates. This approach adopts a data-conditioned ODE through flow matching, leading to a deterministic imputation mechanism. The model further incorporates time-gated multi-scale velocity heads on a fixed 1D pyramid, blended through a time-dependent gating mechanism to reconcile global trends with local details. In addition, a second-order Heun integrator combined with a per-step data-consistency projection ensures recovery of the exact linear bridge across all coordinates. The paper presents several innovations, and experiments conducted on diverse datasets across multiple domains demonstrate the effectiveness of the proposed model.", "questions": "1 Motivation: The authors state that the motivation for focusing on deterministic imputation is to improve reproducibility and ensure a unique reconstruction. Is this the main motivation for adopting a flow-matching framework and replacing the SDE formulation with a data-conditioned ODE as a discriminative model? Clarifying this reasoning would help readers better understand the core design choice.\n\n2 In the Positioning against recent FM/continuous/diffusion and graph methods section, the paper emphasizes how the proposed model differs from existing approaches. However, from the architecture shown in Figure 1, the model appears to combine several design elements from prior works. Moreover, some baseline models with related components are missing in the experimental comparison. Could the authors more clearly articulate the essential differences and innovations of the proposed method, beyond assembling components from existing architectures?\n\n3 In the Flow and Path Matching section, the paper states that “Compared with stochastic sampling, a learned ODE allows deterministic integration at test time and often reduces the number of function evaluations needed for a target quality.” This suggests that using an ODE instead of an SDE provides a favorable speed–quality trade-off. However, as shown in Song et al. (Section 4.3, Probability Flow and Connection to Neural ODEs), the reverse process of diffusion models can also use an ODE formulation, allowing an explicit trade-off between accuracy and efficiency. In this context, what are the specific advantages of the proposed deterministic ODE framework compared to the Probability Flow ODE derived from generative stochastic models?\n\n[1] Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., & Poole, B. (2020). Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456.\n\n4 Regarding the time-gated multi-scale velocity mechanism for suppressing high-frequency ringing and addressing the data-consistency issue, several recent diffusion-based models have proposed similar solutions, Could the authors further clarify how TG-MSFM fundamentally differs from or improves upon these approaches?\n\n[1] Zhou, J., Li, J., Zheng, G., Wang, X., & Zhou, C. (2024, October). Mtsci: A conditional diffusion model for multivariate time series consistent imputation. In Proceedings of the 33rd ACM International Conference on Information and Knowledge Management (pp. 3474-3483).\n\n[2] Yuan, X., & Qiao, Y. Diffusion-TS: Interpretable Diffusion for General Time Series Generation. In The Twelfth International Conference on Learning Representations.\n\n[3]  Yang, X., Sun, Y., & Chen, X. (2024). Frequency-aware generative models for multivariate time series imputation. Advances in Neural Information Processing Systems, 37, 52595-52623.\n\n5 Some datasets used in the experiments, such as Traffic and PEMS03, exhibit strong spatio-temporal dependencies. Including GNN-based baselines for comparison could enhance the comprehensiveness and fairness of the evaluation. For example, the discriminative models such as Grin, Spin and ImputeFormeras follows:\n\n[1] Cini, A., Marisca, I., & Alippi, C. Filling the G_ap_s: Multivariate Time Series Imputation by Graph Neural Networks. In International Conference on Learning Representations, 2022.\n\n[2] Marisca, I., Cini, A., & Alippi, C. (2022). Learning to reconstruct missing data from spatiotemporal graphs with sparse observations. Advances in neural information processing systems, 35, 32069-32082.\n\n[3] Nie, T., Qin, G., Ma, W., Mei, Y., & Sun, J. (2024, August). ImputeFormer: Low rankness-induced transformers for generalizable spatiotemporal imputation. In Proceedings of the 30th ACM SIGKDD conference on knowledge discovery and data mining (pp. 2260-2271).", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1 The proposed model employs a data-conditioned ODE trained via flow matching as a deterministic framework, which enhances the reproducibility and stability of the imputation process.\n\n2 The introduction of time-gated multi-scale velocity heads organized on a fixed 1D pyramid, blended through a time-dependent gating mechanism, effectively reconciles global trends with local details along the trajectory, achieving efficient fusion of global and local information.\n\n3 The time-gated multi-scale velocity decomposition schedules coarse-to-fine refinement along the ODE trajectory and incorporates a light anti-aliasing filter to suppress high-frequency ringing artifacts.\n\n4 The Heun+DC procedure successfully recovers the exact linear bridge across all coordinates, providing a robust accuracy–cost trade-off under appropriate hyperparameter settings.\n\n5 The proposed model is easy to deploy and achieves competitive performance across diverse datasets.", "weaknesses": "1 Some abbreviations should be clearly defined upon first appearance to improve readability—for example, Data Consistency (DC) and Flow Matching (FM). In addition, several sentences could be refined for clarity. For instance, the sentence “At inference we integrate the learned velocity with the second-order Heun method (line 48)” could be revised to “At inference, we integrate the learned velocity using the second-order Heun method.” The visualization in Figure 1 could also be improved—for example, the two arrows under v_theta in the second block could be revised to make the diagram clearer.\n\n2 The Diffusion-based probabilistic imputation section in Related Work requires improvement. Several relevant diffusion-based imputation models are missing, such as PriSTI, SSSD, and Frequency-aware Generative Models for Multivariate Time Series Imputation. In addition, recent works closely related to this paper should be discussed, including:\n\n\n[1] Zhou, J., Li, J., Zheng, G., Wang, X., & Zhou, C. (2024, October). Mtsci: A conditional diffusion model for multivariate time series consistent imputation. In Proceedings of the 33rd ACM International Conference on Information and Knowledge Management (pp. 3474-3483).\n\n\n[2] Yuan, X., & Qiao, Y. Diffusion-TS: Interpretable Diffusion for General Time Series Generation. In The Twelfth International Conference on Learning Representations.\n\n\n[3] Zhang, H., Fang, L., Wu, Q., & Yu, P. S. (2025). Diffputer: Empowering diffusion models for missing data imputation. In The Thirteenth International Conference on Learning Representations.\n\n3 The baseline models in the experimental section could be updated and compared against more recent and relevant approaches, such as the works listed below. Including stronger baselines would make the experimental validation more convincing.\n\n[1] Zhou, J., Li, J., Zheng, G., Wang, X., & Zhou, C. (2024, October). Mtsci: A conditional diffusion model for multivariate time series consistent imputation. In Proceedings of the 33rd ACM International Conference on Information and Knowledge Management (pp. 3474-3483).\n\n[2] Yuan, X., & Qiao, Y. Diffusion-TS: Interpretable Diffusion for General Time Series Generation. In The Twelfth International Conference on Learning Representations.\n\n[3] Yang, X., Sun, Y., & Chen, X. (2024). Frequency-aware generative models for multivariate time series imputation. Advances in Neural Information Processing Systems, 37, 52595-52623.\n\n4 One citation is not properly formatted: Freeformer: Frequency Enhanced Transformer for Multivariate Time Series Forecasting and should follow a formal citation style.\n\n5 The motivation of the proposed approach could be better articulated. The paper emphasizes the differences between the proposed model and prior methods, but it remains unclear why a deterministic flow-matching framework is preferable to an SDE-based or neural ODE-based stochastic model. Is the main goal to improve reproducibility or reduce computational efficiency? If reproducibility is the motivation, increasing the sampling steps or applying ODE in the reverse process of diffusion model as probability flow could achieve similar improvements. If computational efficiency is the target, DDIM-based comparisons should be included. Moreover, generative stochastic models inherently provide insights into the underlying data generation process, offering better interpretability and uncertainty estimation—this trade-off deserves further discussion.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "riNBfxdF6b", "reviewer": "Reviewer_QuMD", "summary": "This paper proposes Time-Gated Multi-Scale Flow Matching (TG-MSFM), an approach for multivariate time-series imputation. The method learns a velocity field of a data-conditioned ODE using flow matching, enabling the reconstruction of missing values in partially observed temporal data. The model incorporates (1) a time-aware Transformer with masked attention over observed timestamps, (2) time-gated multi-scale velocity heads operating over a fixed 1D pyramid to balance global trends and local detail, and (3) a data-consistent Heun integrator that projects observed dimensions back to their known values at each step to prevent drift. The approach is trained using gap-only supervision and demonstrates competitive or superior performance on standard benchmarks. However, the paper lacks a clear positioning with respect to system identification–based methods (concurrent learning, event-based learning) that deal with irregular data, and reservoir network–based systems that also learns a delay-embedding, deterministic high-dimensional vector fields, which could provide useful context regarding model interpretability and dynamical consistency.", "questions": "1.\tHow does TG-MSFM perform under irregular and sparse sampling conditions compared to continuous-time diffusion models?\n2.\tCould the data-consistency projection be viewed as a constrained ODE solver? If so, how does it affect the learned velocity field during training?\n3.\tWhat is the computational complexity relative to baseline flow-matching or diffusion-based methods, and what are the benefits of using the proposed method versus traditional systems theoretic methods?\n4.\tDid you experiment with adaptive time-stepping for the Heun integrator to further improve efficiency or stability?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The idea of learning the flow of an ODE for time-series imputation via flow matching is elegant and relatively underexplored compared to diffusion or autoregressive approaches.\n2. The introduction of multi-scale time-gated velocity heads is a clever mechanism to blend local and global temporal patterns.\n3. The time-aware Transformer with masked attention is well-motivated and aligns with the irregular sampling structure of real-world time-series data.\n4. The method is described with sufficient detail, and the components (velocity heads, time gates, anti-aliasing) each have clear motivations.", "weaknesses": "1. While results are strong, it is unclear how TG-MSFM compares against the most recent latent ODE or diffusion-based imputation models that use similar continuous-time formulations.\n2. The ablation study is mentioned but could be expanded to include the quantitative impact of each design choice (e.g., anti-aliasing, Heun vs. Euler integration, time-gate parametrization).\n3. Some parts of the model (e.g., the 1D pyramid structure and time-dependent gating) may be difficult to follow without schematic diagrams.\n4. The paper occasionally relies on terminology (“velocity heads,” “time-gated blending”) that could use more formal mathematical grounding.\n5. Learning deterministic dynamic models for irregular time series is studied in dynamic systems theory, event-triggered learning, and more explicitly with reservoir models. The paper could benefit from reviewing and contextualizing the flow-matching-based methods with these theoretically grounded methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lhWuZKAQS0", "reviewer": "Reviewer_3oz7", "summary": "The paper introduces TG-MSFM for deterministic multivariate time-series imputation using a data-conditioned ODE learned via flow matching. It employs a time-aware Transformer with visibility-masked self-attention to aggregate only observed timestamps, and a time-gated multi-scale velocity decomposition on a fixed 1D pyramid, blended by a softmax gate for coarse-to-fine refinement, stabilized by an anti-aliasing filter. Inference uses a second-order Heun integrator with per-step data-consistency projection to preserve observed values. Training focuses on gap-only supervision, with optional regularizers. Tested on 10 benchmark models, TG-MSFM achieves competitive or better MSE/MAE, excels in long-gap scenarios, and offers favorable speed-quality trade-offs, validated by ablations.", "questions": "See weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The method innovatively integrates flow matching with time-series-specific adaptations, including visibility-masked self-attention and time-gated multi-scale velocity, providing a transparent and deterministic alternative to stochastic diffusion models.\n2. The proposed model delivers superior or comparable performance to state-of-the-art methods across diverse benchmarks, particularly shining in long-gap cases, while maintaining efficient speed-accuracy trade-offs.\n3. The approach shows consistent performance as gap lengths increase, with component benefits confirmed through ablations and fixed hyperparameters across datasets.", "weaknesses": "1. The reliance on fixed pyramid scales and a simple MLP gate may not optimally adapt to all data spectra, potentially underperforming against more dynamic multi-resolution techniques.\n2. The paper focuses exclusively on block missing imputation. This narrow focus neglects point missing imputation, a common scenario in many time-series applications (e.g., sensor noise or sporadic data loss), rendering the method inadequate for datasets where isolated missing points predominate.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kISnbuAF4f", "reviewer": "Reviewer_9n5Q", "summary": "This paper proposes the TG-MSFM, a method for multivariate time-series imputation by flow matching (FM). To make the conditional generation wrok well in time series setting, they use time-aware transformer and time-gated multiscale velocity heads. At inferene stage, they use a Heun ODE integration with a per-step data-consistency projection, which keep observed coordinates exactly on the linear bridge. They then evaluate the TG-MSFM on 10 benchmars, and show it ttains competitive/ improved performance efficiently.", "questions": "How sensitive of the porposed methods to outliers?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is well written and idea is clear.\n2. The transformer is carefully designed to handle properties of time series.\n3. The experiements are extensive.", "weaknesses": "1. The work is mainly engineering (on transformer), and the insight it provided to the FM & TS community can be limited (don't take points off on this)\n2. Linear bridge and global time warp may be suboptimal for long/heterogeneous gaps.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6uNPwILjzG", "reviewer": "Reviewer_Jt1N", "summary": "The paper introduces Time-Gated Multi-Scale Flow Matching (TG-MSFM), a deterministic framework for multivariate time-series imputation based on flow matching. TG-MSFM learns the velocity field of a data-conditioned ODE using gap-only supervision, which focuses training on the missing entries while enforcing data consistency for observed points during inference. The model features a time-aware Transformer backbone with visibility-masked attention and a multi-scale velocity decomposition modulated by a time-dependent gating mechanism, allowing coarse-to-fine refinement along the generative trajectory. During inference, TG-MSFM integrates the learned flow using the Heun solver combined with a per-step data-consistency projection, ensuring measurement preservation and stable trajectories. Extensive experiments across standard benchmarks demonstrate competitive performance with favorable speed-accuracy trade-offs.", "questions": "- Deterministic vs. probabilistic behavior: The paper states that TG-MSFM produces deterministic imputations, yet I found this somewhat unclear. Based on Section 3.2, since the process samples $z_0 \\sim \\mathcal{N}(0,I)$, wouldn’t the final imputation depend on this random initialization, effectively yielding stochastic (probabilistic) outputs rather than fully deterministic ones? Could the authors clarify whether the inference is performed with a fixed $z_0$ or if randomness is averaged out in some way?\n\n- Model capacity: Could the authors provide the number of parameters in TG-MSFM (and optionally compare it with key baselines)?\nThis would help assess the method’s complexity and the fairness of computational comparisons\n\n- Addressing reviewer concerns: I would be glad to revise my evaluation if the authors can improve upon some of the points mentioned above.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- **Timely and relevant topic**: The paper addresses time-series imputation through flow matching, a rapidly emerging research direction that provides an efficient and deterministic alternative to diffusion-based models.\n\n- **Innovative training strategy**: The use of gap-only supervision, where the model is trained exclusively on missing entries, is conceptually elegant and departs from the conventional full-sample training paradigm dating back to GAIN [1]. This design choice aligns the learning signal directly with the imputation target.\n\n- **Strong empirical evidence**: The experimental section is comprehensive and well structured, covering ten widely used benchmarks and demonstrating consistently strong performance across varying missing ratios and datasets.\n\n- **Clear exposition and illustrations**: The method is well presented, with clear mathematical formulation and figures that effectively convey the architecture and flow process, making the paper easy to follow despite its technical depth.\n\n[1] Yoon, J., Jordon, J., & Schaar, M. (2018, July). Gain: Missing data imputation using generative adversarial nets. In International conference on machine learning (pp. 5689-5698). PMLR. https://arxiv.org/abs/1806.02920", "weaknesses": "- **Limited analysis of the gap-only supervision effect**: Although the empirical results are strong, the paper does not fully explain why training only on missing values leads to better performance. From a representation learning perspective, one might expect that also reconstructing observed entries, as in GAIN [1], helps maintain a richer compression of the overall time-series distribution. A deeper analysis or an ablation comparing gap-only versus full-sample supervision could strengthen the justification for this design choice.\n\n- **Missing discussion on consistency models**: Since the paper draws a clear connection between flow matching and diffusion models, it would be valuable to include a brief discussion on consistency models, which can be viewed as a discrete, distilled, and consistency-enforced formulation of flow matching aimed at improving inference efficiency [2]. In the context of time-series imputation, the recently proposed CoSTI model [3] follows this direction: it can produce probabilistic imputations in a single step, while requiring multiple runs to obtain deterministic estimates such as the median. Adding a short discussion, or even a small efficiency comparison, would help clarify how TG-MSFM relates to this broader family of consistency-based approaches and where it stands in terms of inference trade-offs.\n\n[2] Song, Y., Dhariwal, P., Chen, M., & Sutskever, I. (2023). Consistency models.https://arxiv.org/abs/2303.01469\n\n[3] Javier Solís-García, Belén Vega-Márquez, Juan A. Nepomuceno, and Isabel A. Nepomuceno-Chamorro. Costi: Consistency models for (a faster) spatio-temporal imputation. Knowledge-Based Systems, 327:114117, 2025. https://arxiv.org/abs/2501.19364", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)