Riemannian Time Warping: Multiple Sequence Alignment in Curved Spaces

Christian Scheurer; Christopher A. Erd\"os; Jochen J. Steil; Julian Richter; Niels Dehio

REVIEW 5 major objections 5 minor 33 references

Riemannian Time Warping: Multiple Sequence Alignment in Curved Spaces

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Riemannian Time Warping aligns many signals on curved spaces in linear time by running interpolation, averaging, and distance computation inside local tangent spaces.

desk verdict New and sensible Riemannian time-warping method with a real evaluation flaw: the NTW baseline is handicapped by zero initialization, and the 'consistently outperforms' claim outruns the paper's own statistics. read the letter →

arxiv 2506.01635 v3 pith:474KIBEH submitted 2025-06-02 cs.RO cs.LG

Julian Richter , Christopher A. Erdös , Christian Scheurer , Jochen J. Steil , Niels Dehio This is my paper

classification cs.ROcs.LG

keywords timewarpingmultiplesequencealignmentRiemannianmanifoldstangentspacedynamicrobotlearningfromdemonstrationsincinterpolationsymmetricpositivedefinitematrices

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Temporal alignment of multiple signals—matching the same events across recordings of different lengths—is standard in speech and robotics, but almost all existing methods assume the data lives in flat Euclidean space. This paper introduces Riemannian Time Warping (RTW), a method that performs the same alignment when each data point lies on a curved Riemannian manifold, such as orientations on the unit sphere or stiffness matrices on the manifold of symmetric positive definite matrices. The central claim is that RTW is the first such method that handles generic manifolds with many signals ($N \gg 2$) while keeping the computational cost linear in the number of signals and sample points, $O(NZ)$. The authors argue and demonstrate experimentally that aligning in the manifold's own geometry rather than forcing curved data into Euclidean coordinates gives lower alignment error and better learned robot motions.

What carries the argument

The load-bearing device is the local tangent space $T_{p_n[z]}\mathcal{M}$ chosen separately for each signal $n$ and time index $z$. The logarithmic map $\mathrm{Log}_{p_n[z]}$ pushes the $2\nu+1$ points of the sinc-interpolation window into this flat space, the warping is applied there, and the exponential map $\mathrm{Exp}_{p_n[z]}$ pulls the result back to the manifold. The origin $p_n[z]$ starts at the center of the interpolation window and is updated by a few Gauss–Newton iterations, so the flat approximation tracks the data. The same tangent-space pattern is reused for the Fréchet-mean update and for computing geodesic distances, which is what makes the whole pipeline manifold-respecting while retaining $O(NZ)$ complexity.

What would settle it

A decisive check is to run the paper's inverted-warping benchmark on a high-curvature manifold with known ground-truth warps—for example, $\mathbb{S}^2$ trajectories whose windows pass near the antipodal point of the tangent origin, or symmetric positive definite matrices with near-singular values. If RTW's restoration accuracy and barycenter loss degrade sharply relative to its $\mathbb{S}^1$ results as curvature increases or sampling becomes sparser, the local-tangent approximation is the limiting component.

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Extended reading notes

Core claim

The paper's discovery is a recipe for transplanting continuous-time trainable time warping from Euclidean space to Riemannian manifolds. Three operations that are undefined on a curved manifold—sinc interpolation, signal averaging, and distance evaluation—are replaced by tangent-space versions: data in each interpolation window is mapped into the flat tangent space at a locally chosen origin with the logarithmic map, the usual Euclidean computation is performed there, and the result is mapped back with the exponential map. The tangent origin is initialized at the window's center and refined by a few Gauss–Newton iterations, which keeps the approximation close to the manifold's geometry. RTW parameterizes the warping functions with a neural network as in Neural Time Warping and optimizes a Gaussian-weighted window loss; the resulting complexity stays $O(NZ)$. On synthetic inverted-warping benchmarks on $\mathbb{S}^1$ and $\mathbb{S}^3$, SPD-matrix experiments, the UCR time-series archive, and a real-robot teaching task in $\mathbb{R}^3 \times \mathbb{S}^3$, the paper reports that RTW consistently beats the relevant baselines in both averaging and classification.

Load-bearing premise

The load-bearing premise is that warping computed inside a locally flat approximation of the curved space—with the flat point re-centered a few times—stays faithful to the true geometry; on sharply curved regions or with sparse sampling, the Euclidean alignment found in that tangent space may no longer be a valid temporal alignment on the manifold.

Editorial extensions

If this is right

RTW brings multiple-sequence alignment to data where Euclidean arithmetic gives invalid results, such as unit quaternions and symmetric positive definite matrices, at the same $O(NZ)$ cost as the best Euclidean methods.
Because cost scales linearly with the number of signals, RTW can align datasets with $N \gg 2$ signals, where DTW is quadratic and MMDDTW is exponential in $N$.
The Gaussian-weighted window loss helps match repetitive sub-patterns, which pointwise methods miss; the circle-drawing robot experiment is the paper's demonstration.
Using RTW as preprocessing for imitation learning lowers reproduction error for three different learners (DMP, GMM, Gaussian processes), so the benefit propagates downstream.
On the UCR archive, RTW significantly beats TTW and NTW on the majority of classification datasets, and roughly matches NTW on averaging.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the local-tangent construction makes RTW a first-order method: on strongly curved manifolds or thinly sampled signals, a fully intrinsic or higher-order retraction scheme would be the natural next step, and the paper's own limitation note about closed-form maps points there.
A testable extension would run the inverted-warping benchmark on $\mathbb{S}^2$ with signals passing near the antipodal point of the tangent origin; if restoration error rises sharply, the per-window tangent approximation is the limiting factor.
The same framework should transfer to other manifold-valued time series beyond robotics, such as orientation streams in graphics or covariance matrices in finance, because the method only assumes access to exponential and logarithmic maps.
Since the authors note RTW was not tested on multi-modal imitation learning systems, a natural open question is whether the alignment gains persist when demonstrations contain multiple motion modes with large spatial variance.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core idea here is genuinely new and the mathematics is sound. RTW is, as far as I can tell, the first generic multiple-sequence time-warping method for Riemannian manifolds that scales linearly with N and Z. The tangent-space machinery is natural: project windows to the tangent plane, do windowed sinc interpolation there, refine the tangent point with Gauss-Newton iterations, and use a Gaussian-weighted loss. That is a clean and sensible construction, and the paper does a good job of laying it out. The real-robot experiments on the LBR iiwa are a nice bonus and make the practical motivation concrete. I also credit the authors for explicitly disclosing that they did not initialize NTW the same way as RTW.

But that disclosure is exactly where the trouble starts. The NTW baseline uses zero weight initialization, which the authors say significantly degrades performance, while RTW uses Xavier. So every RTW-versus-NTW comparison in Tables II through VI varies two things at once: the Riemannian machinery and the initialization scheme. You cannot attribute the improvement to either one. The stress-test note is correct, and it is load-bearing because the paper's headline claim is empirical superiority. The paper's own Table II undercuts the abstract: for UCR averaging, RTW is significantly better than NTW on only 8.6% of datasets and significantly worse on 7.8%, with no significant difference on 83.6%. That is not 'consistently outperforms.' Classification is more favorable (35% vs 6%), but still a large no-difference zone. The overclaim should be fixed, not defended.

Other soft spots are minor in comparison. The synthetic inversion task generates warpings partly from the same parameterized family the model uses, which mildly favors RTW, but they also include spline-based warpings, so it is not airtight circularity. The absence of code and the missing error bars on several tables are annoying but fixable. The central algorithm, as far as I can tell, remains defensible; the weakness is in the evaluation protocol and the way the results are described.

This paper deserves a serious referee. The method is a real contribution to robot learning and time-series alignment on manifolds, and the flaws are correctable. I would send it to review with a clear request: rerun the baselines with matched initialization (e.g., Xavier for NTW, or at least report both), add error bars, soften the abstract to match what the statistics actually show, and ideally release code. If the authors do that, the paper would be solid.

For the reading group: maybe worth a slot to discuss baseline hygiene in manifold learning papers, but it is not a must-read. I would cite it if I worked on manifold time warping, and I take the authors as serious thinkers who made an honest but consequential experimental misstep.

Referee Report

5 major / 5 minor

Summary. The paper introduces Riemannian Time Warping (RTW), a method for temporally aligning multiple signals whose data lie on a Riemannian manifold. The approach extends existing trainable time-warping methods (TTW, NTW) by performing windowed sinc interpolation, mean computation, and distance evaluation in local tangent spaces, with Gauss-Newton refinement of the tangent point. The authors claim the first generic time-warping method for N>2 signals on manifolds (beyond quaternion-specific QDTW) with O(NZ) complexity. They evaluate RTW on the UCR archive in R^1, on synthetic data in S^1, on quaternion data in S^3, on SPD matrices, and on real robot teaching demonstrations in R^3 × S^3, reporting gains over baselines in averaging, classification, and imitation-learning preprocessing.

Significance. If the empirical claims are substantiated, RTW is a timely and useful contribution: it attacks a real gap (time warping for manifold-valued signals with many sequences), the tangent-space formulation is principled, and the O(NZ) complexity is a genuine improvement over dynamic-programming baselines such as MMDDTW and QDTW. The paper also gives a clear treatment of the S^3 and SPD++ maps and a plausible real-robot validation, which is valuable for the robotics audience. However, the central claim of 'consistently outperforming' baselines is currently weakened by an uncontrolled comparison with NTW and by the absence of error bars and significance tests in several key tables.

major comments (5)

[Section V, implementation paragraph] The comparison between RTW and NTW is confounded by initialization: the paper states that RTW uses Xavier initialization while NTW uses zero initialization, and that zero initialization 'significantly degrades the performance.' Since RTW and NTW differ in both the warping machinery and the initialization, Tables II–VI cannot isolate the effect of the proposed Riemannian extension. To support the paper's claims, NTW should be run with the same (Xavier) initialization, or an ablation should be reported that shows the effect of initialization separately from the geometric extension.
[Table II, averaging task] The paper's own statistics contradict the abstract's claim of consistently outperforming state-of-the-art baselines in averaging: RTW is significantly better than NTW on only 8.59% of UCR datasets, significantly worse on 7.81%, and shows no significant difference on 83.60%. Even ignoring the initialization confound, this is a statistical tie, not an outperformance. The averaging claims in the abstract and conclusion should be tempered or supported with additional evidence.
[Section V-B, inverted time warping] The synthetic inversion test generates warping functions by sampling from (5), the exact parameterized family used by RTW's neural network, in addition to (15) and splines. Since the model can trivially represent generating warps drawn from (5), the Restoration Accuracy metric is biased in RTW's favor. The experiment should either report results separately for each generating family (including those outside the model class) or generate warps from a genuinely different family, such as random monotone functions not expressible by (5).
[Tables III, IV, V, and VI] These tables report point estimates without error bars or significance tests. For the S^1 experiments, the average over 100 runs is given, but no standard deviations or confidence intervals; for S^3 and SPD++, no measure of variability is provided at all. The claims that RTW 'achieves the best performance' and 'is significantly faster' need statistical support, particularly since some differences (e.g., NTW versus RTW for N=4 in Table III) appear small in absolute terms.
[Section IV-A] The paper states that exponential and logarithmic maps are 'distance-preserving,' but this is only locally true in a normal neighborhood; Fig. 1 itself acknowledges distortion for points far from the tangent origin. The method's reliance on local tangent spaces with a few Gauss-Newton iterations needs a correctness argument or an empirical check (e.g., comparing the tangent-space warping with a more faithful geodesic-based warping on a high-curvature example) to justify that the computed alignment is meaningful for data with strong curvature or sparse sampling.

minor comments (5)

[General] Typographical errors include 'tempral' in footnote 11, 'AVGERAGE' in the captions of Tables VII and VIII, and 'Modfied' in Figure 6.
[Section III-B, Eq. (8)] The definition of the Gaussian-weighted window loss is hard to parse; the normalization in \hat{g}(v,z) could be written more explicitly, and a brief justification for the chosen \kappa and \rho values would improve readability.
[Section V-A] The UCR evaluation is conducted only in R^1 and thus does not exercise the Riemannian extension; the paper should state this limitation explicitly in the experimental section so that the reader does not infer manifold evaluation from the UCR benchmark.
[Section V-C, Table V] The description of the modified QDTW baseline is unclear: QDTW is originally for N=2, but the experiment uses N=4; the text should clarify how the modification handles the multi-signal case and whether the comparison is fair.
[Section V-D] The real-robot experiments are qualitative and lack error bars or statistical tests on the reproduction errors in Tables VII and VIII; adding standard deviations or a paired test would strengthen the claim that RTW 'significantly improves' learning performance.

Circularity Check

1 steps flagged · score 2.0 of 10

No circular derivation; the manifold extension is self-contained, with only a mild self-favorable S1 benchmark and a confounded NTW comparison.

other [Section V-B, Inverted Time Warping in S1; Eqs. (5), (10)-(11), (19a)]
"This is achieved by generating N random warping functions ωn, obtained by uniformly sampling from both (5) and (15) with random parameters, as well as from random spline interpolations of various degrees."

The synthetic ground-truth warpings in the S1 experiment are sampled from Eq. (5), the same parameterized family that RTW optimizes via its neural-network warping functions. For the subset generated from (5), the inverse warping is therefore a member of RTW's hypothesis class by construction, so the Restoration Accuracy metric (19a) partly tests the optimizer's ability to recover a known member of its own family rather than providing independent evidence about discovering arbitrary manifold warpings. This is a mild benchmark self-favoritism, not a load-bearing derivation.

full rationale

The core RTW derivation is self-contained: tangent-space sinc interpolation (Eqs. 10-11), iterated Fréchet mean (Eqs. 12-13), and geodesic distance (Eq. 14) are standard Riemannian operations, and no parameter is fitted to a target and then renamed as a prediction. The Euclidean UCR comparison and the robot experiments use held-out or independent evaluation metrics that do not coincide with the training objective. The only mild self-referential element is the S1 inverted-warping experiment, where the generative model and the learned warpings share the parameterization of Eq. (5), making the restoration task partly in-family; this affects only that synthetic benchmark. Separately, the RTW-vs-NTW comparison is confounded by initialization (Xavier for RTW, zero for NTW, as the paper acknowledges), and Table II shows RTW is not significantly better than NTW on 83.6% of UCR averaging datasets; this weakens the abstract's 'consistently outperforms' claim and is a correctness/statistical concern, not a circularity. There is no load-bearing self-citation or imported uniqueness theorem.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The method's reported wins depend on several hand-selected hyperparameters (κ, ρ, λ, ν, Z) and on standard Riemannian geometry assumptions. No new physical entities are introduced.

free parameters (5)

Gaussian window size κ = 5
Chosen by hand for the windowed loss (8); larger values add context but more compute.
Window step ρ = 5
Distance between sampled indices in the Gaussian window loss (8).
Monotonicity penalty weight λ = 100
Weight for the non-monotonicity penalty (6)-(7).
Sinc window size ν = 10
Truncation window for sinc interpolation (4),(10); covers >99% of sinc power as in TTW [10].
Output length Z = N * Tmax
Resampled signal length chosen to satisfy the continuity constraint; a user-specified experimental choice.

assumptions (5)

standard math Exponential and logarithmic maps exist and are computable for the manifolds considered (S^D, S^D_++, R^3×S^3).
Used in Sections IV-A to IV-D and in experimental maps (16)-(17),(20)-(21).
domain assumption Signals are bandlimited and sampled at or above the Nyquist rate, so windowed sinc interpolation (4)/(10) reconstructs the continuous signal.
Assumed in Section III-B footnote 3; may be violated by robot recordings at 30 Hz.
ad hoc to paper The warping functions ω_n parameterized by the neural network (5) can represent the temporal shifts needed for alignment.
No expressiveness guarantee; relies on training.
domain assumption Gradient descent on the penalized loss (6) finds a feasible near-optimal warping.
No convergence proof; optimization is heuristic.
domain assumption The Fréchet mean via Gauss-Newton iterations converges after a few iterations to the true mean.
Assumed in Section IV-B; commonly used but no proof given.

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Cite this review

Pith. "Pith review of Riemannian Time Warping: Multiple Sequence Alignment in Curved Spaces." pith.science (2026). https://pith.science/paper/474KIBEH

@misc{pith2026250601635,
  author       = {Pith},
  title        = {Pith review of: Riemannian Time Warping: Multiple Sequence Alignment in Curved Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/474KIBEH}},
  note         = {Machine review of arXiv:2506.01635}
}

read the original abstract

Temporal alignment of multiple signals through time warping is crucial in many fields, such as classification within speech recognition or robot motion learning. Almost all related works are limited to data in Euclidean space. Although an attempt was made in 2011 to adapt this concept to unit quaternions, a general extension to Riemannian manifolds remains absent. Given its importance for numerous applications in robotics and beyond, we introduce Riemannian Time Warping (RTW). This novel approach efficiently aligns multiple signals by considering the geometric structure of the Riemannian manifold in which the data is embedded. Extensive experiments on synthetic and real-world data, including tests with an LBR iiwa robot, demonstrate that RTW consistently outperforms state-of-the-art baselines in both averaging and classification tasks.

Figures

Figures reproduced from arXiv: 2506.01635 by the authors.

**Figure 1.** Schematic overview of RTW: our approach enables the temporal alignment of multiple signals, represented through data on a Riemannian manifold [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** Visualization of the Gaussian-weighted window loss between two [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Classification performance of TTW (red), NTW (yellow), RTW (green) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: A planar robot (solid lines) follows endeffector trajectories (dashed [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: Comparison of the results for both robot teaching tasks with naive interpolation, modified TTW/NTW and the proposed RTW approach. The hand [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Reference graph

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Pith tools

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