Perturbed Dynamic Time Warping: A Probabilistic Framework and Generalized Variants
Authors: Xiangqian Sun
Chaoqun Wang
Wei Zhang
Venue: ICLR 2026 Poster (2026)
Keywords: dynamic time warping
perturbation
dynamic programming
Primary area: learning on time series and dynamical systems

Abstract: Dynamic Time Warping (DTW) is a classical method for measuring similarity between time series, but its non-differentiability hinders integration into end-to-end learning frameworks. To address this, soft-DTW replaces the minimum operator with a smooth soft-min, enabling differentiability and efficient computation. Motivated by soft-DTW, we propose perturbed-DTW, a differentiable framework of DTW obtained by adding random perturbations to warping costs and taking the expected minimum. Under Gumbel noise, perturbed-DTW exactly recovers soft-DTW, providing a natural probabilistic interpretation of soft-DTW. We further generalize this framework by extending the Gumbel noise to the broader family of generalized extreme value (GEV) distributions, leading to a new class of soft-DTW variants. Building on this insight, we introduce nested-soft-DTW (ns-DTW), which integrates GEV perturbations into the dynamic programming formulation of perturbed-DTW. This extension induces alignments with tunable skewness, offering greater flexibility in modeling diverse alignment structures. We validate ns-DTW on barycenter computation, clustering, and classification, demonstrating its effectiveness over existing approaches.

Published as a conference paper at ICLR 2026
PERTURBED DYNAMIC TIME WARPING: A PROBA- BILISTIC FRAMEWORK AND GENERALIZED VARIANTS
Xiangqian Sun" Chaoqun Wang?, Wei Zhang?
School of Mathematics and Physics, Xi an Jiaotong- Liverpool University, China Schooo of Al and Advanced Computing. Xi'an Jiaotong-Liverpon University, China School of Management, Zhejiang University, China (Xiangqian. Sun, Chaoqun. Wang) @xjt lu. edu. cn, wei wei.zhang@zil eedaauuuuu.eeuu.....mmm edu. cn
ABSTRACT
Dynamic Time Warping (DTW) is classical method for measuring similarity be- tween time series, but its non-diiferentiabiliyy hinders integration into end-to-end learning frameworks. To address this, soft-DTW replaces the minimum opera- tor with smooth soft-min, enabling differentiability and efficient computation. Motivated by soft-DTTW, we propose perturbed-DT'W a differentiable framework of DTW obtained by adding random perturbations warping costs and taking the expected minimum. Under Gumbel noise, perturbed-DTW exactly recov- ers soft-DTW, providing a natural probabilistic interpretation of soft-DTW. We further generalize this framework by extending the Gumbel noise to the broader family of generalized extreme value (GEV) distributions, leading new class of soft-DTW variants. Building on this insight, we introduce nested-soft-DDy (ns-DTW), which integrates GEV perturbations into the dynamic programming formulation of perturbed-DTW. This extension induces alignments with tunable skewness, offering greater flexibility in modeling diverse alignment structures. We validate ns-DTW barycenter computation, clustering, and classification, demonstrating its effectiveness over existing approaches.
INTRODUCTION
Dynamic Time Warping (DTW) a classical measure of similarity between time series that com- putes the minimum-cost alignment between two sequences (Berndt & Clifford [994] Sakoe & Chiba 2003). Unlike Euclidean distance, DTW accommodates temporal distortions and uncqual sequence lengths, making it broadly applicable across domains such object recognition (Belongie (et al. 22002), time-series forecasting (Le Guen & Thome, 2019), and irregular sequence modeling (Zhang et 2023). Despite its effectiveness, DTW is limited by the non-differentiable nature of its minimum operator, rendering it incompatible with gradient-bar based optimization methods.
To address non-differentiability, Cuturi & Blondel (201)) introduced soft-DTW, which replaces the hard minimum with smooth soft-min operator. DoWt admits efficient dynamic programming and yields gradients that can be computed recursively. Thereby, it enables gradient-based optimiza- tion in applications such as music score alignment (Mensch & Blondel, 2018), video segmentation (Chang et 2019) and trajectory clustering (Brankovic al. 2020).
Motivated by soft-DTW, we extend the differentiable DTW framework through the lens of per- turbed optimizer (Berthet et al.) 2020). We propose perturbed-DTW. new scheme that introduces randomness into the alignment costs, computes the minimum perturbed cost, and then averages over the noise distribution. This formulation naturally yields differentiable relaxation: the optimal alignment matrix becomes distribution over paths rather than single deterministic solution. Inter- estingly, when the perturbations are Gumbel distributed, perturbed- provides probabilistic interpretation of soft-DTW as the expectation of DTW under Gumbel perturbations.
recovers soft-DTW. This
"corresponding author

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Building perturbed-DTW. we introduce nested-soft-DTW (ns-DTW), novel variant obtained by employing the generalized extreme value (GEV) distribution as the perturbation. By modeling correlations across groups of variables, ns-DTW captures richer aligmment structures. Remarkably, the resulting alignments ns-DTW can exhibit tunable skewness beyond what soft-DTW allows.
Our contributions are threefold:
We introduce DTtur general citturbbttion-basee framework for differentiable DTW. Within this framework, soft-DTW emerges naturally as the expectation of DTW when alignment costs are perturbed by Gumbel noise, thereby providing probabilistic interpretation of its smoothing behavior.
By adopting GEV perturbations, we derive ns-DTW, which offers greater modeling flexi- bility through correlated perturbations and skewed alignment distributions.
We demonstrate the effectiveness ns-DTW on diverse time-series tasks, showing that captures meaningful alignment structures while remaining computationally tractable.
The remainder of the paper is organized as follows. Section 2 reviews DTW and soft-DTW. Section presents perturbed-DDi and its connection to soft-DTW, and introduces ns- Ds- and its properties. Section 4 provides experimental results across benchmark datasets.
2
PRELIMINARIES
2.1 DYNAMIC TIME WARPING
Consider two p-dimensional time series x [ 3 TRxx and y 9y......11 yn) e RPx= Denote [m] {i..... m. and [n]] {1..... n.. Dynamic time warping (DTW) aims to find the optimal alignment between two sequences by allowing a point one sequence to be matched with one more points in the other. To formulate the optimal alignment problem, first define the alignment cost. A local cost matrix C E mR measures element-wise dissimilarities: [C(x, k)kk, c(æ;, yy), where c(-,-) IR R + R, is differentiable cost function. A commonly used cost function adopts the squared Euclidean distance, that is,
(C(x, clai, yj)
| v, Vi [m), j [m]]
Then, an alignment matrix A E {0, 11mxx encodes an alignment between data points æ; and yy: As, = if x; is aligned with y,, and ( otherwise. In addition, we call a aligmment matrix is valid it satisfies:
the nonzero entries of form a path starting from (1, 1) and ending (m, n); (ii) the moves in the path can only be one of the directions: {-, Y) Denote the set of all valid alignment matrices Am,m. Given the local cost matrix C and the alignment matrix E Am,us DTW is defined as
DCW = min (A,C),
(1)
AEAA...
where (A.C) = Traco(C A) is the sum of elementwise products (Frobenius inner product). Optimal alignment matrix A is defined as the one that achieves the minimum cost
A arg min C),
AEA.m..
However, directly computing DTW via T intractable due to the exponential size of Am... In- stead, DTW can be computed efficiently via dynamic programming (DP). To formulate the DP, define Dagj as the minimal cost of alignment from time series [II...., æz] and [yI....,9) y,]. Then, Da.; satisfies the Bellman equation (Bellman 1952).
D min Dig-1 Di-1.- D1-1J-1 } C.g. V<iss m. 1 <ssnn n A1 A A
(3)
with boundary conditions D = Eiz CKi e [m]. and D D1== E Cl. Vj E [n]. Then DTW distance is then given by D Dm,ns i.e. DTW(C) D Therefore, the optimal alignment
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path can be recovered by backtracking from Dm. D1.1 from 33. This approach guarantees that the optimal solution is found by considering all possible aligmment paths while avoiding the exponential complexity of exhaustive enumeration.
Equation (3) enables efficient computation both the DTW distance and its optimal alignment via dynamic programming. Yet, the inherent non-differentiabiliyy of DTW, stemming from the mini- mum operator, precludes its use in end-to-end learning pipelines. Introducing differentiability with- sacrificing computational efficiency is central challenge in extending DTW modern learning frameworks (Cuturi & Blondel 2017).
2.2 SOFT-DTW
Cuturi & Blondel 2 proposed soft-DTW, differentiable relaxation of DTW that replaces the minimum operator in Equation (T) with a smooth approximation (Cuturi et al. 2007 Saago et al., 2006).
Technically, for vector (31.---; p.), the soft minimum operator defined
min, x - log
exp(
where y > O is the temperature parameter to control the smoothness and bias. In particular, min, a approaches min æ 7 >> O while approaches -y log y + cc. Formally, soft-DTW is defined as
exp (A,C) ?
soft-DTW,(C) = min, min,(,,), = -7
--llg log
Σ
(4)
AEAm.n
where (A, = ((A,C) AEA.... Rllmm. denotes the vector whose entries the alignment
costs (A, concatenated over all alignments in Am,n- The soft-minimum operator here maps vec- of all possible alignment costs into an "aggregated" cost. Noted that soft-DTW is differentiable discrepancy compared to DTW, and converges to DTW when y + 0. Conversely, as - soft-DTW approaches the mean of all possible alignment costs.
The differentiability of soft-DTW with respect to both the time series x, and the cost matrix C enables use gradient- maddedt--bsed learning algorithms. This property makes it particularly valuable for applications in machine learning and optimization. Notably, the optimal alignment matrix in soft-DTW no longer deterministic; instead, follows Gibbs distribution over Am,
exp ( (A, ()/,
P(A:C) = Ex'EAL exp (A1,C)//
(5)
The expected alignment matrix is
E E P(A:C) A.
AEAm,
Here, Egg E (0, 1) represents the marginal probability that is aligned with y₁, Although soft- DTW has succinct form, direct evaluating (4) is computationally challenging, as it requires sum- ming over all possible alignment matrices. However, MMnsch & Blondel (2018) showed that soft- DTW can reformulated as regula--eggllrizzed dynamic programming. Specifically, one can ob- tain the soft accumulated cost matrix S by replacing the hard minimum in the DTW recursion with the soft minimum:
S. = min, (S--15-1, S. S1z-1) + Ci..
This recursion yields soft-DTW. (C) Sm. Cuturi Blondel 2017). Additionally, soft-DTW admits alternative variational formulation, expressed as the solution to entropy-reguulrizzed linear program (Blondel et al. 2021). Let
pCC) = (P(A: AEAm.. E Ammml and s(C) (A, C) E R ([A(m.))

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denote the vector of alignment matrix probabilities and its corresponding alignment costs, respec- tively Thus, soft-DTW can be expressed in the variational form:
soft-DTW,
min PEA Ammi
pp. s) (p))
(6)
where Hp)) = (p, log p) denotes the negative Shannon entropy. This variational perspective high-
lights soft-DTW as an catropy-regularizer alignment cost (Blondel al. 2020 Sun et al. 2023). as illustrated in Figure
DTW alignment
soft- DTW alignment
ragularized
Figure 1: An illustration of soft-DTW. The right one is the expected alignment of soft-DTWW.
3 PERTURBED DYNAMIC TIME WARPING
3.1 PERTURBED-DE -DWW DEFINITION
In this section, we introduce perturbed-DTW. new scheme that provides an alternative means of rendering the original DTW differentiable. Let a denote perturbation vector drawn from distribution P. For temperature parameter > 0, define the perturbed minimum as
R" and let e (E1..... ......... En,
E minnfx - ] = Ren> min(fr - -78,, " - 7
(7)
Next, we define the perturbed-DTW by replacing the standard minimum operator in the DTW with this perturbed version.
Definition 1. The perturbed-DTW is defined as
perturbed-DTTW,(CC) = Eg-P min { ( - -
(8)
where E is perturbation vector of dimension Amm,ul distributed according to P. and y >> the temperature parameter.
Intuitively, method first perturbs the cost of each valid alignment with random noise term ye, and then takes the minimum over all possible alignment matrices. The final result the expecta- tion of this minimum value with respect the probability distribution of the noise. The expected alignment under perturbed-DTW can be treated as an aggregated version of DTW alignments under different realizations of E, as shown in Figure 2 This formulation closely related to random util- ity theory (Train 2009) where choices are made based on utilities perturbed by random shocks. Unlike DTW. which relies heuristic soft-minimum operator, perturbed-DTW achieves dif- ferentiability through randomization. This approach establishes deep connection between time series alignment and random utility models, where an alignment matrix A represents a choice with utility of - -AA, C",, and nE acts random utility shock.
Example: Gumbel Perturbation. We consider the i.i.d. Gumbel distribution for the perturbation noise and this choice recovers the soft-DTW, Our result is based on the following lemma.
For notational simplicity, we omit the explicit dependence C and write p and

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perturber realization
DTW alignment
perturbed alignment
perturbed
aggregate
Figure 2: An illustration perturbed-DTWV The left one is the alignment under the DTW, the middle one is the alignments under different random noise realizations, and the right one can be treated as the aggregation of middle ones and is also the alignment under soft-DTWW.
Lemma 1. e are ii.d Gumbell 1) distributed, where e. 0.5772 is the Fuler-Mascheront constans, then
E
min - 7E1, , -n5# = -y log
exp(
2
This lemma leverages the property of Gumbel distribution, since the minimum of Gumbel variables corresponds to the negative maximum of their negations. Based on this lemma, we can obtain the following result which induces soft-DTW.
Proposition 1. The perturbed-DIT under i.i.d. Gumbel(- 1) perturbation is
perturbed-DTW, (C) = E min { (A, (8,,))-ee)
- y log
Σ
(A,
C)
exp
AEAm.
In addition, the optimal alignment matrix in perturbed-DTWV is
exp ( (A,C)/))
P(A:C) arg min { (A, -78 EA'EA exp (- (A'CC)/))
Example: : Generalized Extreme Value Perturbation. Beyond the Gumbel case, other perturba- tion families yield new differentiable relaxations. In partieular, extend the perturbation frame- work by modeling the random noise with the generalized extreme value (GEV) distribution. Before proceeding, we briefly recall the definition of the GEV distribution.
To formalize this setting, we partition Am, into J groups, with the j-th group containing Ký ele- ments, such that Σ K, Am, We index groups by j and denote by i the alignment matrix associated with the k-th element of group j. Then, the cumulative distribution function (CDF) of GEV joint distribution is
F(EIL...,EJRR exp
Σ
exp
Eyk
(10)
f=1
where 0 1 is the similarity parameter. Intuitively, the GEV distribution can be regarded as a correlated multivariate generalization of the Gumbel distribution. By adopting the GEV per- turbation, obtain the following result and denote the variant nested-soft-DTTW dubbed ns-DTW,,CC
Theorem 1. the GEV errors centered have mean zero (i.e. Est. = Ezk c), the perturbed- DTW under E has the following expression:
ns-DTW, (C) min ((A.C)
- =
= log
E
Σ
exp
-(A.0)))
(1I)
TTp
AEe
Moreover, if A is the kih one group j. the corresponding probability is
(ENEE exp
(-1
(A''0))
T--
(A,C)
P(A:C) Σ/= (ENEr exp
77.
eep
(A'.C)
TT)
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The idea of GEV can also be found in the nest logit model from random utiliity theory (Tain 2009). Note that perturbed-DTT under GEV perturbation presents groupwise correlation between different A when 0 ти and it reduces to soft-DTW when тę -1, Additionally, can also write into a variational form.
Proposition 2. The perturbed- ---b under GEV perturbation Equation variational form:
11 can be written into
ns- (C) =
min (p. s) + HH)( PEA (((.m))
(12)
where
K
K2 pim
H(p)
Σ Ł=1
TePim log Płm.
Σ
(Te-1)
(Te
Σ mm=1
pim
log
(13)
We call the regularization term 13 as nested Shannon entropy (Fosgerau et al. 2020 and it reduce
to Shannon entropy when = vl. The following proposition characterizes some properties of perturbed-DTWW
Proposition 3 (Properties of perturbed-DTW) The following properties hold for perturbed DTW:
1. (Scaling) perturbed-DTW, (C') "eettubbedDDTT,(CC/)))
2. (Optimal Alignment matrix distribution) The distribution of the alignment matrix given by P(A: C) = E arg min (A. -37C)
-yE}
3. (Gradient) The gradient of Berturbed-DTWW with respect to C is expected alignment matrix = AAEA,, A. P(A:C).
4. (Asymptotic) perturbed-DDW,, - (C) and E + A' as - 0.
3.2 PERTURBED-DTYY COMPUTATION
Analogously, perturbed-DTT DTW faces the computational challenge, as evaluating (8) still requires enu- merating over the exponentially large space Am... To address this, we adapt the dynamic program- ming formulation DTW by replacing the minimum operator in (3) with the perturbed minimum operator. Specifically, we define the perturbed accumulated cost matrix V recursively
Vr = E min (V-15-1 -E--1,1-1 Vi-1,5 - 7E:-1.j V15-1 -E1,-1) + C.j.
This recursive form will end at Vm, In this way, we just need to compute the expectation over three variables in recursion and set perturbed-DTW. (C) Vn,.. which makes the computation tractable. We now examine two specific perturbation distributions: the Gumbel and the GEV.
Example: Gumbel Perturbation. The dynamic programming formula of perturbed-DTWW for Gumbel noise is
Vi,i = [min (1.jvV - 76ij-1, Vi-1.j 7Ei-1js Vi-1.--1 - 78 18i-11j-131 + Ci,
kuy + exp (-V )دد + exp (- Vi 1. 1 )) + C..-
= - log [ex
7
(14)
which is consistent to soft-DTW. This connection casts perturbed-DTVV to differentiable dynamic programming (Mensch & Blondel 2018) and reveals that soft-DTW is special case of Gumbel perturbations.
Proposition 4. Define the (random) perturbed cost matrix Cj, where [C,les Cas YEa, and Eij is Gumbell -c, 1) distributed. Then
sofL-DT,,(C) =[(TW((.))
(15)
This shows that soft-DTW equals the expectation of DTW discrepancy for perturbed local cost matrix C under Gumbel perturbation.

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Example: Generalized Extreme Value Perturbation. Different from the linear programming formulation of perturbed-DTW in Section here we no longer enumerate alignments in Am,n- Instead, we just need to consider three admissible transition directions { ' at each stage (i, When the perturbation vector & (8:-1,j-1, 6;- 3;11j Eij-1) follows a GEV distribution, three distinct schemes of grouping need to be considered:
91 = { } =2= W (4) 93 = W {+}}-
92
To ease the illustration, we just present and a comprehensive discussion of grouping is deferred Appendix A.4 Formally, we divide the directions {-,,,, > of warping paths into groups:
J1 = {+ and J2 = I ) The perturbation (E;iiioon(E::)) (E1-1.5-1, 1. follows GEV distribution, then the dynamic programming formula is
Vi. (imin(V.,, 7, Vi-1,J YEi-1,1, Vi-1--1 115-1.1-111 + Ca.
--ylog (exp(( Viysi) expl= k=r)) + exp
------ O...
(16)
is important to elarify that the value Vm,, obtained via the dynamic programming recursion (16) is tractable algorithmic realization, rather than exact evaluation, of the theoretical ns-DTW defieed in (TT.. This distinction can be understood from two perspectives. First, the global definition in (II) implies single GEV perturbation of dimension [mm,al over the entire alignment space, whereas the DP formulation applies independent, low-dimensional GEV perturbations locally to the three transition directions at each step. Second, regarding the recursive structure: unlike the standard Log-Sum-Exy operator, which satisfies the stability property (i.e., the sum of Gumbel variables follows Gumbel distribution), the nested application of the generalized operators in (16) does not strictly preserve the form of the global GEV distribution. Despite this theoretical distinction, we refer to the efficient DP output Vm,, as ns- DTW throughout this work.
Effect of Grouping. The ns- DS-DTW offers greater flexibility in modeling diverse alignment structures by allowing different groups of directions. This enables the algorithm to produce alignments that skewed towards either the vertical (4) or horizontal (+) directions, as illustrated in Figure 3
+-OT ig 33
m-DTWis ip 2
OT ig
(a) ns-DTW (gl)
(b) ns-DTW (g2)
(c) ns-DTW (g3)
Figure 3: Comparison of warping paths with under different groupings of ns-DTWW The yellow path
depicted as the optimal warping path under DTW.
Effect of r. The flexibility of ns-DTW is also governed by the parameter т. Setting T = 1 recovers the standard soft-DTW. Conversely, as - 0, the model becomes increasingly selective; transitions with higher accumulated costs receive vanishing weights, while those with lower costs are empha- sized. This mechanism introduces structural skew toward the optimal path. As illustrated in Figure 4 decreasing 7 progressively enlarges the expected warping path toward the direction of lower cost (e.g., the vertical direction ( allowing the model to adaptively prune high-cost deviations.
Proposition 3 reveals that the expected alignment pathes of perturbed-DTW. which is exactly the expected alignment matrix E under the distribution P(A; C.. However, this general alignment ma- trix distribution is hard to compute for general perturbation Specifically, let transition probability tensor E (0, 1] 1m x For any given (i.j), G.J. E R specifies a stochastic alignment policy
feasible actions {-,, Y) that P(A,y-1 1) G. P(A;-1.) = 1) Gi and
P(A;-15-1 = G The transition probability tensor can be computed as
G E aarg min [Vi--1 -Vi--1-855, 783.j-1, V 7E;-1.jo Vi- 781-15-111
(( (exp((-Mur -uy-") expl V--2) )
+ exp ((-y--1)
- log
(17)
Pt can be revealed that grouping all three directions together-or placing each direction in its group yields the classical soft-DDTW recursion.
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0.3
15.0W 1aw06.3)
(a) ns- DS-DTW (7 0.50)
(b) ns DS-DTW 0.3)
(c) ns- (r 0.1)
Figure 4: Comparison of warping paths with different temperature parameters 7 of ns-DDTW
where the gradient is taken with respect to {Vij-1, V1-1. Vi-15-1-- Then the expected alignment matrix is computed
Fa.j G.i+1,1 Pig+1 + G.1.,,2 + G411+11BBt11H
This recursive formula can be treated as the differentiable dynamic programming with nested Shan- non entropy. The ns-DTW allows the model control the intensity (skewness) toward directions associated with lower accumulated costs, by using and grouping schemes to introduce skewness. By promoting broader mixture over feasible paths, ns-DTW can approximate the true aligmment cost more effectively than soft-DTW. Algorithm| presents the pseudocode-for computing ns-DTT and transition probability tensor. more detailed discussion is provided in the appendix.
Algorithm ns-DTW and transition probability tensor computation
Require: Cost matrix E Rm y 2 0, < 5 1: Initialize: V1,0 < oo for all i; Vog
for all j; Vo,0 0 2: for E [I,..., m/] and j E [[.....n] do
3: Compute Vi, via (6
4: Compute Gi.j.: via
5: end for
6: Return:
7: Ds-DTW,(C)) Vn,n E G (0, 1] xxxx3
4 APPLICATIONS
In this section, conduct experiments using the UCR Time Series Classification Archive (Chen ctt al. 2015). We consider a subset of the archive containing 47 datasets for average, classification and clustering tasks. We report summary of our results in the manuscript, with full details provided in the appendix.
4.1 AVERAGING
investigate the problem of computing Fréchet mean of time series with respect to ns-DTW. Given collection of time series: , our goal is to compute a barycenter x that minimizes the total ns-DTW discrepancy:
M
min XERXX
ns- (CC(T)(((x.))))
We evaluate the quality of the barycenters in terms of DTW discrepancy and compare ns-DTW with several established baselines. including subgradient method (Schultz Jain 2018). DBA (Petitjean)
et al. 2011) and Soft-DTW (Cuturi & Blondel 2017) methods.
We evaluate the methods based on three direction grouping schemes: (91, 92. gs}. parameters 7 E {0.80, 0.85, 0.90, 0.95) and {0.1, 0.01, 0.001, 0.0001}. Table [able[[ssmmmrize the barycenter
As datasets contain only class, we datasets for the classification task.
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averaging results across varying Y. By selecting the optimal grouping scheme 9i and parameter 7, ns-DTW demonstrates robust performance improvements. Specifically, achieves a lower objec- tive value than the Subgradient method 100% of datasets and DBA on 97.87% %. Furthermore, when compared against the strongest baseline, soft-DTW (both methods tuned for optimal 7), ns- DTW yields superior results 74.47% of the datasets. Qualitative results the Beef dataset visualized in Figure Comprehensive dataset-wise results are provided in Appendix B.I.
Table I: Percentage of the datasets on which the proposed ns-DTW barycenter is achieving lower DTW loss than competing methods.
subgradient
DBA
soft-DTW
0.1
68.09%
46.81 %
36.17%
7 = 0.01
80.85%
72.34%
59.57%
7 0.001
95.74%
87.23%
80.85%
= 0.0001
100.00%
91.49%
91.49%
m--0"W
DSA
achent
Comparison of barycenters obtained ns-DTW, soft-DTW and DBA methods the UCR time series Beef dataset, using Euclidean averaging for initialization.
Figure 5:
4.2 CLASSIFICATION
Nearest Centroid Classifier. We evaluated time series classification performance using the Nearest Centroid Classifier (NCC), where class centroids were computed as barycenters using the respective averaging algorithms. We employed 50% train-validation-tes split, with the smoothing parameter y E {0.1, 0.01, 0.001, 0.0001} selected via cross-validation.
Overall, ns- Ds-DTW demonstrated superior performance, achieving equal or higher accuracy compared to Subgradient methods on 93.02 % of datasets, DBA on 88.37 %. and soft-DTW on 86.05 %. Figure 6 presents the pairwise performance comparison between ns- ns-DWW and baselines for NCC classifi- cation.
10
(a) Subgradient ns-DTW
(b) DBA v.s. ns-DTW
(c) soft-DTW v.s. ns-
Figure 6: Points above the diagonal indicate datasets where the ns-DTW outperforms (a) subgradi- ent; (b))BA; (c) soft-DTW in nearest centroid classifier.
INN Classifier. We also evaluated the I-Nearest Neighbor (INN) classifier, where test samples assigned to the class the nearest training example based on the minimized DTW discrepancy. The data splits and cros-vvaldattin for y remained consistent with the NCC experiments.
Overall, ns- ns-DTW achieved equal or higher accuracy compared to DBA on 88.37 datasets and Soft-DTWW on 86.05% %.

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4.3 CLUSTERING
We study the k-means clustering task: the distances between each time series are ns-DTW discrep- ancy and the centroids of each class their barycenters. Formally, the objective find clusters C CI...., Cx that minimize total within-cluster ns-DTW discrepancy:
ΣΣ nC-DTW(C(y..)))), k=1jJECz
min CL...CC
Overall, ns-DTW achieved equal or higher accuracy compared to DBA on 88.37% datasets and soft-DTW 86.05%. Our ns-DTW outperforms DBA on 100.00% and soft-DTW 76.60% % Figure presents the clustering results on the CBF dataset and the complete results are shown in Appendix B.4.
Certrand
106
300
100
330
samplest
(32 samples))
Centrod
156
300
330
sampleo
samples)
8/4
Figure 7: Comparison of clustering results obtained by ns-DTW (first row), soft-DTW (second row) and DBA (third row) methods on the UCR time series CBF dataset, initialized using Euclidean averaging.
CONCLUSION
We introduced perturbed-DTWW probabilistic framework that makes DTW differentiable by adding random perturbations to the warping costs. This perspective recovers soft-DTW under Gumbel perturbation, providing natural probabilistic interpretation. Extending the generalized extreme value family leads to nested Soft-DTW, which enables tunable skewness in alignments and greater modeling flexibility.
Experiments on barycenter computation and clustering demonstrate competitive performance over existing methods. Looking forward, extending Arturbbd-DDTW to divergence-ber based formulations and to broader classes of perturbed dynamic programming operators offers promising directions for future work.
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ACKNOWLEDGMENTS
Xiangqian Sun was supported by grant from XJTLU Research Development Fund [RDF-23-02- 052]. Chaoqun Wang was supported by grant from XJTLU Research Development Fund IRDF-24- 02-033J. Wei Zhang was partly supported by grants from the National Natural Science Foundation of China [Grants 72401257, 72431010, 1 71821002];
ETHICS STATEMENT
This research does not raise any potential violations of the ICLR Code of Ethics. In particular, does not involve human subjects, dataset release practices, or potentially harmful insights, method- ologies, or applications. It also raises no concerns regarding conflicts of interest or sponsorship, discrimination, bias, or fairness, nor does present issues related privacy, security, legal compli- ance, or research integrity (e.g., IRB approval, documentation, or research ethics).
REPRODUCIBILITT STATEMENT
The datasets used in this research are publicly available at https:// www. cs ucr www..duu/ reamonn/t. _ee__srreee es__ita/ and the code for reproducing results is provided in the supplementary material.
REFERENCES
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Marco Cuturi and Mathieu Blondel. Soft-dtw: a differentiable loss function for time-series. In International conference machine learning, pp. 894-903. PMLR, 2017.

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Arthur Mensch and Mathieu Blondel. Differentiable dynamic programming for structured prediction and attention. In International Conference on Machine Learning, pp. 3462-3471. PMLR, 2018.
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A PROOFS AND COMPUTATIONS
A.1 PROOF OF THEOREM 1
We first show the following result.
Lemma 2. Let
M =
max ve(J),el[k]
{uza + Ezkbs
with e following the GEV joint distribution
K2
F(Eu....,EEK.) exp
Σ
Σ k=1
exp
Ezk'
Tj
Then
1
K2
)
E[M] =
Σ
Σ
cxp
j=1
1=
Proof. For any E R.
Pr(M ft = Pr(Ezk St - Hij kj,k)
=((t Mil....,t pJk,)
exp
exp
Kj
Ts
exp
k=1
Kj
cHilk1Ts Js
exp
j=1
Define
K2 c//s//T3
2(x) log
1=
Then
Pr(M < = exp exp( (t ((x))))
so M is Gumbel distributed with location ((n) and scale 1. By property of Gumbel distribution, we
know that
[[M] = 3(p) -c
where e zo 0.5772 is the Euler-Mascheromi constant. the GEV errors are centered to have mean zero (i.e. Esk Ezk- c), then
E
max (uze + Exx] s(u)) j.k
Next, give the proof of Theorem
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Proof. If the GEV errors centered to have mean zero (i.e. Esk Esb c),
{14,0) -
7E]
perturbed-DTW,,(C =
min
AEAAm,
(A,C) + =7]
max
AEAm,,
(-(4-0))
)")
y log
exp
YTE
AEJ
where the last equality comes from Lemma Denote $jke = (A, C) if A is the kith one in group j. The probability distritution of alignment matrix is
8yk
>og
exp
0sik
YTt
1=1
MEJ
T--1
Sim 7T/
m==
exp
8jk YTy
exp
mm== exp
8im 9922
as desired.
A.2 PROOF OF PROPOSITION 2
Proof. For simplicity, we just show the case y The Lagrangian function is defined as
L(p, A) = (p. s(C) + H(p) + X ME>
Σ pem - 1).
(18)
The optimality condition gives
3pm + Te(1 + log (Te 1)(1 E Pem) - 0, J3, m
(19)
Σe Em Pêm = 1.
Equation 19 implies
pbn exp
-88m -
-
log Σ Pem
(20)
Tt
Tt
With a little abuse of notations, we denote pt
Σm Pêm
Then
Pe
Σ Pim
Σ exp
88m 1
X
Te-
log
Σ pim
(21)
TV.
T1
Σ exp
-8fm
1 1 x
T8 1
log Pt
(22)
T2
T2
Stm
1 >
log.r
log
exp(
(23)
TB.
T8.
By leveraging the equality condition, we have
Σ Pr Σ
exp(
8tm
=
Σ
exp(
88m
exxp(1 + +2) (24)
Tt
2
m
m
A= log
Σ
exp(
Stm
1.
(25)
TP
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Published as a conference paper at ICLR 2026
By substituting 23 and 25
into
20 have
Ptm = exp
88m
X (Te
1)lo
Σ
exp
Sim
1+
1
x
+
Tt
m
=xx
Stm + (Te 1) log Σ
exp
8tm
log
Σ
Σ
exp(
5tm
m
m
(26)
The proof completes by plugging (26 into 66
A.3
PROOF OF PROPOSITION 3
The fourth property is straightforward; here, we present only the first three.
Proof.
1. By definition of perturbed-DTW, we have
pertarbed-DTW, (C) Eg-P
min (A,C) C) )8
Eep min { {/ C) - -
perturbed-DTW,
2. The optimal alignment matrix A follows
P(A:C) P (AA, (A. -7 -) (A', 5
75', -T', v'','')
= E arg min { ((A.C))
3. By the Williams Daltums--Dly-Zachar theorem McFadden 1981.
V perturbed-DTW,(C))
E
arg min (A,C)
(A, C) - -
vo(A,
Σ
arg min
((A,C) - -
AEAm,
AEAm,
Σ
P(A:C) A
AEAm.
A.4 GENERAL FORMULATION NS- DTW
As discussed earlier, ns-DTW can be viewed as a generalized perturbed variant soft-DDTW ob- tained by replacing the Gumbel perturbation with GEV perturbation. Note that the GEV distri- bution can be regarded as multivariate generalization of the Gumbel distribution with groupwise correlation. This substitution introduces not only a hyper- apper-parameter but also multiple schemes of direction groupings. The dynamic programming formula for general perturbed-DTW is
ViJ = E min Vi-1.--1 - 76--1.i11 V4-1. 7Ei-1.i V1,j-1 - yEa.-- }| + Ci
When the perturbation vector (E1-1.511, 1.01--,,1-115 tinct grouping schemes need to be considered:
Ei- E1 follows GEV distribution, three dis-
91 = 4), }
= = (1+ >, }
g3 =3 >J {+}}.
It can be revealed that grouping all three directions together-or placing each direction in its group- yields the classical soft-IDTW recursion.
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Published as a conference paper at ICLR 2026
Therefore, the general formulas for dynamic-prorraming recursions for different grouping schemes are
91: Vis lo (e ( +e te Vo--ss/()))) + -V------177 - Cr.y
(27)
92 Vis = Log (e -1, te (o-i----1(171) +e -e-isssh Oaa
(28)
93 Vis log (e-Vinss te 1o---1-1))77) te +e-iss-lly
Osg.
(29)
A.5 NS-DTW COMPUTATION
To streamline the presentation, focus on the ns-DTW computation under the first scheme of grouping 91. The transition probabilities and gradients for the remaining groupings can be derived analogously. The dynamic programming formula of perturbed-DDT under GEV perturbation in Ø1 is
Vi, = E (VinnVVij-11 + Ci. - 1, Vi-1.j + Cis - -7E--,,j V1-15-1 + CCs - -
= --yloo log ((exp( 7 (exp(-Va--)) Viu=t) + exp(- (x ))
+ +exp (-V-zuz) + Ci..
(30)
Therefore, the transition probability
eepp(= Vi--i) ) + exp( Vi-1z)
--1
P(Atj-1=1) Gi,,1 (exp(-
78
75
eepp
(- Vi
Vi expl Vi-tis) + exp
Vi-15-1
77
(exp( - Vi,11 ) + exp(
V/
P(A1-15 =1) Gi,,
78
-exp
Vi-14 79
(exp( Ve.s-1 + exp(
Vi-i.y
Vi-1.-11
+ exx
77
(- Vi-1.-11
exp
P(A--15-1 = Gi,,3
(exp( H Vi,i-1) + exp( - 77
))
+ cxp (- Vi-ia=
Then the expected alignment matrix is computed by
Eaj G. Eij+1 + G.11j2E441, + G.I.HH13BHHHHH
Algorithm presents the pseudocode for computing gradient of ns-DTW.
Algorithm ns-DTW gradient computation
Require: Ge (0. x (Algorithm
1
1: Initialize: Em+1, +- 0, E <- 0, Em+11041 1-1
2: Initialize: Gm+l.g +- 00,0,0), 0), G.n+1. +- (0,0,0), Gm+1.+11, +- (0,1,0)
3: for E [m...., 11.j E [n..... 1), do
4: Eig f- Gr,+1,11 E. + G.114412 E+IHH1 G. E+is
5: end for
6: Return: (CD)-DTW,(C)) E mmxm
Consider the warping cost C E p to compute the value of ns-DTW, Algorithm 1 requires (mm)) operations and 0(mn) storage cost as well. This is as same as the soft-DTW. However, if we consider different groupings, in other words, dividing directions into two groups the computational cost would be three times as soft-DTW (since soft-DTW can be treated as one special grouping of ns-DTW).
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A.6 IMPLEMENTATION DETAILS AND EXPERIMENTAL SETUP
Software Environment. All experiments were implemented in Python 3.9. The core logic relies on tslearn (v0.6.4) for time series operations, numpy (v2.0.2) and pandas (v2.3.3) for data manipulation, and scikit-lears (v1.6.1) for evaluation metrics.
Hyperparameter Configuration. To ensure reproducibility and fair comparison, we standardized the search space and initialization protocols across all datasets. We utilized Euclidean averaging for initialization and maximum budget of 30 iterations for the barycenter computation, observ- ing convergence in most cases within this limit. The specific hyperparameter ranges and grouping schemes for ns-DTW are detailed in Table
Table 2: Summary of experimental settings and hyperparameter search spaces.
Parameter
Value / Definition
Software
Python 3.9, tslearn 0.6.4, numpy 2.0.2, sklearn 1.6.1
Parameter (T)
{0.80, 0.85, 0.90, 0.95}
Smoothing (1)
{0.1, 0.01, 0.001, 0.0001)
91 = ((-,4),
Grouping Schemes
92 (1-,),)1
93 = ({L. (,]+)
Initialization
Euclidean Averaging
Max Iterations
30
We clarify the hyperparameter selection procedure for two types tasks: averaging and clustering( unsupervised), and classification (supervisd). Detailed results are presented in Appendix B
Averaging and Clustering Tasks: For these tasks, we use a grid search evaluate combinations of grouping schemes (ys E {91, 92, 93)),, parameter E {0.80, 0.85, 0.90, 0.95] and the parameter E (0.1,0.01,0.000) 0.001, }. other words, we evaluate all datasets for every possible combination of (.)).
To examine the effeet of 9s, we fixed each grouping scheme and then found the best t and y combination within the grid search that yielded the lowest DTW losses. These results presented in the last three columns of Tables 3 and 6.
To examine the effect of r, we collected results for fixed 7 and 9;, then selected the y that achieved the lowest DTW losses. The detailed results for averaging are in Appendix C (Table for gr, Table 8 for 92, and Table 9 for ga). For clustering, results in Appendix E (Table 13 for gr. Table 14 for I2, and Table 15 for ga).
Classification Task: For classification, we still perform a grid search over gi E {g1, T2. 93} and E {0.80, 0.85, 0.90, 0.95). However, for each {9:, T} pair, is selected via cross-validation on the training set. This approach allows us to specifically evaluate the effects of different grouping schemes and while ensuring y is optimally tuned for each combination.
To examine the effect of gi, we fixed each grouping scheme and then found the best 7 (with y de- termined by cross-validation within the grid search that yielded the highest classification accuracy. The results of g; on classification performance are presented in the last three columns of Table To examine the effeec of T, collected results for fixed 7 and 9t, then selected the that achieved the highest classification accuracy. Further results showing the effects of different groupings and t with cross-valian valsdaaaldd 7y are provided in Appendix D (Table 10 for 9r, Table 11 for g2, and Table 12 for gz)
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Published as a conference paper at ICLR 2026
B RESULTS
B.1 AVERAGING
Table 3: UCR Barycenter DTW Losses
Dataset
subgradient
DBA
soft-DTW
ns-DTW (01)
ns-DTW (92)
ns-DTW (os)
Adiac
0.2594
0.2576
0.2552
0.2551
0.2553
0.2552
ArrowHead
1.2413
1.1893
1.1440
1.1646
1.1640
1.1545
Beef
8.4287
4.1666
4.0666
4.1005
4.1025
4.1015
BeetleFly
4.9672
4.5245
4.2886
4.1907
4.1909
4.1905
BirdChicken
4.9910
2.4679
2.3858
2.3930
2.3921
2.3938
CBF
4.3019
3,8861
3,4473
3.7857
3,8222
3.7833
Car
0.6908
0.6703
0.5948
0,5954
0,5964
0.5963
ChlorineConcenntraioo
3.6285
3.4941
3.4297
3.4047
3,4061
3,4091
CinCECGTorso
17.9134
8.8686
8.3582
8.3653
8,3597
8,3599
Coffee
0.6249
0.5950
0.5872
0,5890
0,5892
0.5893
Computers
17.2013
14.8046
14.6065
14.6314
14.6228
14.6203
CricketX
7.9406
5.9852
5.7930
5.6919
5,7399
5.6772
Yricket
6.3425
5.7147
5,5186
5.5646
5,5598
5,5645
CricketZ
6.2965
5.4289
5.3826
5,3563
5,3752
5.3454
DiatomSSizeReduction
0.2529
0.2357
0.2282
0.2278
0.2278
0.2278
DistalPhalannKOutiinnAggger
0.7220
0.7073
0.7043
0.7036
0.7038
0.7038
DistalPallaxxOOtiieCCorre
0.6972
0.6670
0.6658
0.6662
0.6662
0,6662
DistalPhalanXTW
0.3622
0.3478
0.3448
0.3458
0.3458
0.3458
ECG200
2.7862
2.7185
2.6750
2.6470
2.6537
2.6704
ECG5000
2.2571
2.2061
2.2012
2.1968
2.2024
2.2023
ECGFiveDays
2.7898
2.5564
2.5618
2.5275
2.4869
2.5243
Earthquakes
12.2798
10.7766
10.6706
10.7192
10.7052
10.7222
FaceAll
3.2504
2.9970
2.8392
2.8917
2,9055
2.8971
FaceFour
5.6599
5,3963
5,2482
5,3374
5,3145
5,3444
GunPoint
2.2964
1.6673
1.7319
1.6998
1.7036
1.7078
Ham
4.4250
4.1339
4.0861
4.0997
4,0929
4.0984
Medicallmages
2.7714
2.6551
2.6738
2.6693
2.6675
2.6693
Middlle
0.5451
0.5206
0.5138
0.5120
0.5120
0.5143
MiddlePhalanx OutlineCorrect
0.6671
0.6423
0.6343
0.6341
0.6341
0.6334
0.3322
0.3284
0.3190
0.3188
0.3189
0.3187
MoteStrain
4.3593
4.2874
4.2537
4,2381
4,2390
4.2453
ProximaIPhalanXTW
0.3773
0.3402
0.3365
0.3383
0,3378
0.3382
RefrigerattonDevices
11.5572
8.5680
7,4193
7.9655
7,9086
7.8243
ScreenType
13.8207
12.2042
12.1071
12.0787
12.0857
12.0816
ShapeletSim
12.2430
11.3275
11.0399
11.0316
11.0313
11.0834
ShapesAll
1.2198
1.0711
1.0687
1.0652
1,0653
1.0690
SmallKKtthennApplaaane
13.0036
10.3012
9.2726
9.3408
9,3408
9,3398
SonyALBORRbbeSSraacel
1.7197
1.6757
1.6536
1.6075
1,6088
1.6397
SonyAlBOORobosSrriacc
2.7282
2.6203
2.5849
2.5633
2.5633
2.5633
SyntheticControl
4.5651
4.1545
4.1459
4.1092
4.0827
4.0813
Trace
1.8690
0.9323
0.8951
0.8954
0.8499
0.8699
LwoLGadECG
0.9302
0,8615
0,8463
0.8444
0.8442
0.8473
Wine
0.3271
0.3239
0.3257
0.3245
0.3251
0.3240
Worms
14.1987
10.5608
10.0867
9,9822
9,9110
9,9007
Worms TwoClass
9.8468
8.7107
8.3123
8.4709
8.4173
8.4515
Count
0
20
12
6
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Published as a conference paper at ICLR 2026
B.2
NEAREST CENTROID CLASSHICATION
Table 4: UCR nearest centroid classification accuracy
Dataset
subgradient
DBA
soft-DTW
ns-DTW (91)
ns-DTW (02)
ns-DTW (ga)
Adiac
0.4490
0.4388
0.6224
0.6327
0.65
0.6327
ArrowHead
0.1111
0.2222
0.33
0.2222
0.33
0.1111
Beef
0.2500
0,2500
0,2500
0,1250
0.38
0,2500
BeetleFly
0.8000
1.00
1.00
1.00
0,8000
0.8000
BirdChicken
0.8000
0.8000
0.8000
1.00
0,8000
1.00
CBF
0.8750
1.00
1.00
1.00
0,8750
0.8750
Car
0.6000
0.5333
0.73
0.6667
0.6667
0.6667
ChlorineConneentation
0.2735
0.3504
0.38
0.3675
0.3162
0.3419
CinCECGTorso
0.3000
0.4000
0.4000
0.3000
0.3000
0.50
Coffee
0.0000
0.0000
0.0000
0.14
0,0000
0.0000
Computers
0.6349
0.75
0.7143
0.6667
0.6825
0.6508
CricketX
0.3878
0.4592
0.4694
0.5204
0.62
0.5918
CricketY
0.3878
0.4490
0.4694
0.4796
0.5102
0.53
CricketZ
0.3980
0.5408
0.5816
0,5612
0,5510
0.65
DistaPhallaxxOOttieeAggCooo
0.8700
0. 0.7900
0.7900
0.8100
0.91
0.8200
DistalPhllanxOOutinnCCCCoo
0.1000
0.0867
0.0933
0.1067
0.19
0.1400
DistalPhalanxTW
0.04
0.0000
0.0000
0.0000
0.0100
0.0100
ECG200
0. 120000
0,0800
0.0800
0,0800
0.28
0.0000
ECGFiveDays
0.5000
0.5000
0.1667
0.1667
0.67
0.67
Earthquakes
0.0494
0.07
0.07
0.0617
0.0370
0.0617
FaceAll
0.8357
0.8429
0.8714
0,9000
0.93
0.93
FaCesUCR
0.90
0.7800
0.8600
0,8600
0,8800
0.90
GunPoint
0.3846
0.77
0.6923
0.6154
0.6923
0.6154
Ham
0.6429
0.6071
0.7143
0.79
0.6429
0.6429
Medicallmages
0.4583
0.4479
0.4479
0.4375
0,4271
0.52
MiddleePhalannxouiineeeeee
0.7600
0.81
0.81
0.81
0.7600
0.7700
MiddlePhalanx OutlineCorrect
0.2467
0.2067
0.2333
0,2800
0.2267
0.39
MiddlcPhalanXTW
0.0300
0.0200
0.0300
0.0100
0.0300
0.07
MoteStrain
0.6000
0,8000
0,8000
0,8000
1.00
0,6000
ProximalPhallnxTTW
0.07
0.0100
0.0100
0.0000
0.0000
0.0000
RefiigerationDevices
0.61
0.4787
0.4787
0.5000
0.5638
0.5106
ScreenType
0.4255
0.4149
0.4787
0.51
0.3936
0.3936
ShapeletSim
0.2000
0.4000
0.2000
0.4000
0.60
0.2000
ShapesAll
0.5333
0.6600
0.6533
0.67
0.6600
0.6067
SmallKiitchenAApllinness
0.6277
0.6064
0.6170
0.64
0,6064
0.5957
SonyAllBORoboSSuriccl
1.00
0.6000
0.6000
0.6000
1.00
0.6000
SonyAAlBOORobotSurfaaae
0.8571
1.00
1.00
0.8571
0.7143
0.5714
SyntheticControl
1.00
0.9867
1.00
0.9867
1.00
1.00
Trace
0.8800
0.8800
0.9200
0.9200
1.00
0.9600
TwoLeadECG
0.6667
0.8333
0,8333
1.00
0,8333
0,8333
Wine
0.5333
0.3333
0.3333
0.4667
0,5333
0.67
Worms
0.4565
0.4348
0,4783
0.50
0.3478
0,4783
0.5870
0.6739
0.6739
0.6739
0.6522
0.70
Count
6
9
11
13
19

---

Published as a conference paper at ICLR 2026
B.3 INN CLASSIFICATION
Table 5: UCR 1-nearest neighbor classification accuracy
Dataset
DBA
soft-DTW
ns-DTW (91)
ns- (g2)
ns-DTW (ga)
Adiac
0.5204
0.5714
0.62
0.5714
0.5510
ArrowHead
0.89
0.89
0.6667
0.7778
0.89
Beef
0,2500
0,2500
0.62
0,5000
0,2500
BeetleFly
0.8000
0.8000
0.8000
1.00
0,8000
BirdChicken
0.6000
0.6000
1.00
1.00
0.6000
CBF
1.00
1.00
1.00
1.00
1.00
Car
0.3333
0.5333
0.67
0.4667
0.6000
ChlorineConnentraaion
0,4359
0,4872
0.4701
0.56
0.4957
CinCECGTorso
0.5000
0,5000
0.6000
0.70
0,5000
Coffee
1.00
1.00
1.00
1.00
1.00
Computers
0.76
0.76
0.5556
0.7460
0.76
CricketX
0.7449
0.79
0.7245
0.7449
0.79
CricketY
0.6735
0.6837
0.7041
0.72
0.7041
CricketZ
0.7347
0.7347
0,6735
0.7143
0.74
DistaPhalanxxOnttieAAgGGGio
0.7700
0.7700
0.81
0.7800
0.7700
DistaPPallaxxxOutiinCCreeet
0.7733
0.7733
0.81
0.8000
0.7667
DistalPhalanxTW
0.7200
0.75
0.6900
0.7300
0.7400
ECG200
0,7600
0.7600
0.7200
0.88
0.7600
ECGFiveDays
0.8333
1.00
0.6667
0.5000
1.00
Earthquakes
0.7654
0.7654
0.78
0.7407
0.7654
FaceAll
0,9214
0,9500
0,9214
0.9357
0.96
FacesUCR
0.7600
0.78
0.6667
0.6667
0.78
GunPoint
0.8462
0.92
0.8600
0.9200
0,92
Ham
0.7500
0.7500
0.92
0.7692
0.8214
Medicallmages
0.7083
0.7604
0.82
0.7857
0,7083
MiddtePhallannCutinneAgCCC
0.7000
0.6900
0.6979
0.6979
0.73
0.7267
0.7733
0.85
0.8000
0.7667
MiddIe-halanxTW
0.6200
0,5900
0.7400
0.79
0.6000
MoteStrain
0.80
0.80
0,5900
0.5500
0.80
ProximalPhalanx TW
0.7500
0.7500
0.6000
0.80
0.7600
RefrigerationDeviiess
0.5957
0.6383
0.7100
0.83
0.6383
ScreenType
0.4574
0.5106
0.70
0.6702
0,4574
ShapeletSim
0.2000
0.2000
0.4787
0.52
0.2000
ShapesAll
0.7067
0.7467
0.6000
0.4000
0.75
SmallKiicchenAAppiinnes
0.6596
0.6915
0.7267
0.79
0,6702
SonyALBOORoboSSuraacc
0.80
0,80
0,5851
0.6383
0.80
SonyAAIBORRoozSuraacee
0.86
0.86
0.8000
0.8000
0.86
SyntheticControl
1.00
1.00
0.8571
0.7143
1.00
Trace
1.00
1.00
1.00
1.00
1.00
TwoLeadECG
1.00
1.00
0,9600
1.00
1.00
Wine
0.87
0.87
0.8333
0.6667
0.87
Worms
0,4565
0,4565
0,6667
0.87
0,4565
NormsTwoClass
0.7391
0.76
0.4348
0.5000
0.7391
Count
17
16
19
20

---

Published as a conference paper at ICLR 2026
B.4 CLUSTERING
Table UCR Clustering DTW Losses
Dataset
DBA
soft-DTW
ns-DTW (91)
ns- (g2)
ns-DTW (gs)
Adiac
0.0921
0.0878
0.0885
0.0882
0.0885
ArrowHead
1.2090
1.0741
1.0624
1.0433
1.0650
Beef
2.8558
2.2851
2.1169
2.1680
2.1365
BeetleFly
20.6455
19.8514
19.0531
18.9430
19.0679
BirdChicken
9.4679
7.9387
8.0163
8.0368
8.0304
CBF
12.0459
11.8482
11.8576
11.7534
11.7463
Car
0.8623
0.6263
0.5699
0.5669
0.5676
ChlorineConnentration
6,5869
6.4396
6,4328
6.4346
6.4263
CinCECGTorso
669.7685
669.7685
669.7685
669.7685
669.7685
Coffee
0.4366
0.4297
0,4305
0.4304
0,4306
Computers
143.5164
141.2816
137.3022
141.0881
139.3186
CricketX
114.1204
114.1204
114,1204
114.1204
114.1204
CricketY
103.0182
103.0182
103.0182
103.0182
103.0182
CricketZ
124.6247
124.6247
124.6247
124.6247
124.6247
DiatomSSizeReduction
0.0712
0.0756
0.0628
0.0632
0.0633
DisalPhallaxxOOttineAAGGGoo
0.3678
0.3633
0.3636
0.3637
0.3637
DistaPhallanxOutiinCCorre
0.8450
0.7544
0.7631
0.7635
0.7645
DistalPhalanxTW
0.3325
0.3319
0,3303
0.3292
0,3305
ECG200
4.5978
4.5201
4.5334
4.5394
4.5334
ECG5000
6.4048
6.6103
6.5432
6,4040
6.5932
BCGFiveDays
5,7908
5.6721
5.5425
5,5447
5,6063
Earthquakes
437.7200
437.7200
437.7200
437.7200
437.7200
FaceAll
34.8896
34.8896
34.8896
34.8896
34.8896
FaceFour
21.2485
20.2836
20.1979
20.1897
20.2069
FacesUCR
27.0649
27.0649
27.0649
27.0649
27.0649
GunPoint
1.1022
1.0099
0.9990
0.9987
0.9989
Ham
17.1259
17.0784
17.0572
17.0463
17.0749
MedicalImages
4.1345
3.5272
3,4652
3,4744
3,4197
MiddlePhailnnxXutinnAeeeee
0.2368
0.2329
0,2338
0,2335
0,2338
MiddlePPallnxXCuliiCCrrre
0.3350
0.3348
0.3348
0.3348
0.3348
MiddlePhalanxTTW
0.2193
0.2134
0.2171
0.2152
0.2172
MoteStrain
18.8957
19.0558
19.0490
19.0497
18.8197
ProximalPhalanx
0.1591
0.1584
0.1585
0.1584
0.1585
RefrigerationDevvie Devices
122.9185
115.9638
113.0523
116.9189
114.3597
ScreenType
108.5347
104.8087
104.5366
103.5914
104.1625
ShapeletSim
127.3930
122.4791
120.5546
120.9429
120.4151
ShapesAAI
25.5056
25.5057
25.5056
25,5056
25.5056
SmallKincchenAppliance
123.9963
123.7793
122.3944
123.1039
123.3957
SonyAlBORobodSuraaee
4.9519
4.7078
4.6157
4.6127
4.6157
SonyAAIBORobotSrracce
8,9998
8,8919
8.8860
8,8879
8,8808
SyntheticControl
8.7978
8.7625
8.6648
8.6736
8.6778
Trace
3.5626
3,4207
2.9248
2.9365
2.9483
TwoLeadECG
0.9103
0.8642
0.8644
0.8645
0.8644
Wine
0.0697
0.0649
0.0702
0.0691
0.0678
WordSynonyms
27.5408
27.5408
27.5408
27.5408
27.5408
Worms
451.2960
451.2960
451.2960
451.2960
451.2960
WormsTwoClass
81.4757
78.6740
76.7911
78.2881
78.3617
Count
10
20
19
21
16
21

---

Published as a conference paper at ICLR 2026
C
ABLATION STUDY OF AVERAGING
Table Ablation Study Averaging (Grouping 91): Comparison of UCR Barycenter DTW Losses across different parameters Bold indicates the lowest loss.
ns-DTW (91)
Dataset
0.95
0.90
T 0.85
0.80
Adiac
0.2551
0.2550
0.2549
0.2549
ArrowHead
1.1772
1.1771
1.1772
1.1774
Beef
4.1005
4.1014
4.1068
4.1015
BeetleFly
4.3015
4.3096
4.3031
4.3037
BirdChicken
2.4647
2.4646
2.4649
2.4575
CBF
3.7857
3.7857
3.7857
3.7857
Car
0.6489
0.6493
0.6577
0.6575
ChlorineConcentraaion
3,4750
3.4750
3,4750
3,4750
8.3677
8,5311
8.3574
8,5775
Coffee
0.5890
0.5890
0.5889
0.5889
Computers
14.6314
14.6176
14.6282
14.6000
CricketX
5.8424
5.8500
5.8272
5,8537
Cricket
5,5948
5,5975
5.5974
5,5950
CricketZ
5.3866
5.3843
5.3875
5.3840
DiatomSizzRReduction
0.2278
0.2278
0.2291
0.2289
DistalPhalanxOOttiinAgggng
0.7059
0.7059
0.7059
0.7059
DisalPPalannxOutiieCCrrer
0.6666
0.6666
0.6666
0.6666
DistalPhalanxTW
0.3458
0.3458
0.3458
0.3458
ECG200
2.6704
2.6704
2.6703
2.6703
ECG5000
2.2057
2.2060
2.2063
2.2057
ECGFiveDays
2.5275
2.4832
2.5278
2.5324
Earthquakes
10.8445
10.8446
10.8446
10.8446
FaceAll
3.0059
3.0109
3.0110
3,0104
FaceFour
5,3563
5.3563
5.3563
5,3563
FCCesUCR
3.2074
3.2074
3.2074
3.2074
GunPoint
1.7460
1.7440
1.7202
1.7458
Ham
4.0997
4.1004
4.1003
4.1019
Medicallmages
2.6887
2.6888
2.6887
2.6886
MiddiePhallanxOutiinnAggge
0.5152
0,5152
0.5152
0.5152
MiddlePPaaaaxxnniinnCCCreee
0.6389
0.6389
0.6389
0.6389
MiddlcPhalanxTW
0.3282
0.3282
0,3282
0,3282
MoteStrain
4.2600
4.2588
4,2589
4,2589
ProximalPhallnxTTV
0.3390
0.3398
0.3396
0.3395
RefrigerationDevicess
8.2907
8.2763
8.3036
8.3232
ScreenType
12.0787
12.0795
12.1121
12.0785
ShapeletSim
11,0850
11.0820
11.1077
11.1077
ShapesAll
1.0652
1.0652
1.0638
1.0657
SmallKiicchenAAplliannes
9.9110
9,9131
9.3478
9.3585
SonyAllBORRoosSurffcel
1.6774
1.6774
1.6774
1.6774
Sony:AlIOORobotSurfeeee
2.5914
2.5914
2.5914
2.5914
SyntheticControl
4.1615
4.1615
4.1615
4.1615
Trace
0.8954
0.8949
0.8969
0.8962
TwoLeadECG
0.8555
0.8554
0.8554
0.8553
Wine
0.3245
0.3245
0.3245
0.3245
WordSynonyms
2.1832
2.1731
2.1836
2.1744
Worms
10.2557
10.4207
10.4068
10.4219
Worms'
WormsTwoClass TwoClass
8.4709
8.4596
8.4805
8.4603
Count
25
20
17
21
22

---

Published as a conference paper at ICLR 2026
Table Ablation Study of Averaging (Grouping 92): Comparison of UCR Barycenter DTW Losses across different parameters 7. Bold indicates the lowest loss.
ns-DTW (ga)
Dataset
0.95
0.90
T 0.85
0.80
Adiac
0.2553
0.2553
0.2553
0.2552
ArrowHead
1.1663
1.1651
1.1652
1.1846
Beef
4.1025
4.1114
4.1036
4.0924
BeetleFly
4.3052
4.3067
4.3170
4.3045
BirdChicken
2.4589
2.4597
2.4540
2.4633
CBF
3.8330
3.8132
3,8132
3.8140
Car
0.6492
0.6493
0.6571
0.6576
ChlorineConcentration
3,4750
3,4750
3,4749
3,4750
CinCECGTorso
8.3597
8,4085
8.5551
8.4076
Coffee
0.5892
0.5894
0.5906
0.5901
Computers
14.6228
14.6119
14.6227
14.6075
CricketX
5.8423
5.8244
5.8738
5.8396
CricketY
5.5944
5.5968
5,5980
5,5974
CricketZ
5.3852
5.3829
5.3826
5.3826
DiatomSizeRReduction
0.2278
0.2275
0.2274
0.2271
DistuPPhllannxOutiiinAAggg
0.7059
0.7058
0.7059
0.7058
DnstalPhalannOuutineCCCooo
0.6666
0.6666
0.6666
0.6666
DistalPhalanxTW
0.3458
0.3458
0.3458
0.3459
ECG200
2.6726
2.6718
2.6738
2.6720
ECG5000
2.2080
2.2074
2.2065
2.2011
ECGFiveDays
2.5185
2.5265
2.5293
2.5237
Earthquakes
10.8293
10.8420
10.8365
10.8365
FaceAll
3.0224
3.0080
3.0065
3.0178
FaceFour
5.3563
5.3564
5.3564
5.3565
GunPoint
1.7464
1.7436
1.7441
1.7451
Ham
4.0929
4.0802
4.1005
4.1018
Medicallmages
2.6858
2.6864
2.6860
2.6870
0.5145
0.5142
0.5143
0.5143
0.6389
0.6389
0.6388
0.6388
MiddlePhalanxTTW
0.3282
0.3282
0.3282
0,3282
MoteStrain
4.2591
4,2565
4.2546
4.2567
ProximalPhalanxTI TW
0.3399
0.3397
0.3399
0.3399
RefrigerationDevices
8.9489
8,9519
8.9514
8.8522
ScreenType
12.0857
12.0816
12.0815
12,0838
ShapeletSim
11.0833
11.0836
11.1077
11.1078
ShapesAll
1.0653
1.0740
1.0662
1.0680
SmallKKitchenApplinnne
9,4743
9.4160
9.4293
9,9123
SonyAlBORRoooSSuficce
1.6774
1.6774
1.6774
1.6774
SonyAlBORRoboSurracce2
2.5914
2,5914
2,5914
2.5914
SyntheticControl
4.1615
4.1615
4.1615
4.1615
Trace
0.8949
0.9133
0.8964
0.9030
TwoLeadECG
0.8554
0.8549
0.8551
0.8547
Wine
0.3251
0.3253
0.3264
0.3260
Worms
10.4246
10.4022
10.4252
10.4123
Worms TwoClass
8.5204
8.4643
8,4992
8.5186
Count
17
15
10
13
23

---

Published as a conference paper at ICLR 2026
Table Ablation Study of Averaging (Grouping ga): Comparison of UCR Barycenter DTW Losses across different parameters 7. Bold indicates the lowest loss.
ns-DTW (gs)
Dataset
0.95
0.90
7 0.85
0.80
Adiac
0.2552
0,2553
0.2554
0.2556
ArrowHead
1.1643
1.1774
1.1789
1.1775
Beef
4.1015
4.1018
4.1054
4.1238
BeetleFly
4.3045
4.3085
4.3055
4.3070
BirdChicken
2.4642
2.4654
2.4548
2.4548
CBF
3.8332
3.8331
3,8132
3.8140
Car
0.6489
0.6486
0.6446
0.6485
ChlorineConcentration
3,4751
3,4746
3,4745
3,4746
CinCECGTorso
8.3599
8,5016
8.3633
8,5583
Coffee
0.5893
0.5903
0.5904
0.5905
Computers
14.6203
14.6059
14.6578
14.6558
CricketX
5.8534
5,9529
5.8229
6.0129
CricketY
5.5951
5.5954
5,5947
5,5952
CricketZ
5.3862
5.3891
5.3829
5.3817
DiatomSizeRReduttion
0.2278
0.2280
0.2285
0,2279
DistulPhalannxOuttinnngggg
0.7057
0.7060
0.7055
0.7047
DistalPhalanxOurtiieCCComo
0.6666
0.6666
0.6666
0.6666
DistalPhalanxTW
0.3458
0.3459
0.3458
0.3458
ECG200
2.6704
2.6704
2.6704
2.6704
ECG5000
2.2060
2.2061
2.2056
2.2000
ECGFiveDays
2.5252
2.5248
2.5262
2.4842
Earthquakes
10.8283
10.8446
10.8447
10.8446
FaceAll
3.0104
3.0102
3.0108
3.0096
FaceFour
5.3562
5.3562
5.3562
5.3562
FCCesUCR
3.2074
3.2074
3.2074
3,2074
GunPoint
1.7408
1.7427
1.7440
1.7423
Ham
4.0984
4.0993
4.0993
4.1070
Medicallmages
2.6887
2.6896
2.6896
2.6886
MiddlePhallnnxxottinneeeeee
0.5143
0.5141
0.5141
0.5141
0.6389
0.6390
0.6390
0,6390
MiddlePhalanxTW
0.3281
0.3281
0.3281
0.3281
MoteStrain
4.2582
4.2565
4.2572
4,2571
ProximalPhalanxTW
0.3399
0.3400
0.3398
0.3397
ReliicerationDvvices
8,8975
8,9557
8.9514
8,9532
ScreenType
12.0816
12.0858
12.0848
12,0858
ShapeletSim
11.0834
11.1077
11.1077
11.1077
ShapesAll
1.0690
1.0688
1.0683
1.0641
SmallKiichhenAppllannes
9,4851
9.3029
9,7840
9.7895
SonyALlBORobosSuraeee
1.6774
1.6773
1.6773
1.6773
SonyAlBORRobtSurficce
2.5914
2.5914
2.5914
2.5914
SymtheticControl
4.1615
4.1615
4.1615
4.1615
Trace
0.8970
0.8961
0.9118
0.8960
TwoLeadECCG
0.8554
0.8554
0.8553
0,8553
Wine
0.3240
0.3238
0.3242
0.3240
WordSynonyms
2.1858
2.1513
2.1593
2.1863
Worms
10.4158
10.2059
10.4138
10.4293
WormsTwoClie TwoClass
8.4515
8.4789
8.4532
8.4484
Count
20
11
12
20
24

---

Published as a conference paper at ICLR 2026
D ABLATION STUDY OF CLASSIFICATION
Table 10: Ablation Study of Classification (Grouping 91): Comparison of nearest centroid classifi-
cation accuracy across different parameters Bold indicates the highest accuracy.
ns-DTW (g1)
Dataset
0.95
0.90
T 0.85
0.80
Adiac
0.63
0.5306
0.5918
0.5408
ArrowHead
0.2222
0.2222
0.33
0.2222
Beef
0,1250
0.50
0.2500
0.3750
BeetleFly
1.00
0.8000
0.8000
0.8000
BirdChicken
1.00
0.6000
0.6000
0.6000
CBF
1.00
1.00
0.8750
1.00
Car
0.6667
0.7333
0.7333
0.80
ChlorineConcentraaioo
0.37
0.3162
0.2137
0.3504
ConCECGTorso
0.3000
0.5000
0.60
0,3000
Coffee
0.14
0.0000
0.0000
0.14
Computers
0.6667
0.5397
0.68
0,5714
CricketX
0.5204
0.6122
0.5000
0.63
Cricket
0,4796
0,5612
0.4490
0.57
CricketZ
0.5612
0.5816
0.5408
0.59
DistaPhallmxOOuttineeeegeooo
0.8100
0.8400
0.8300
0.85
0.1067
0.2200
0.23
0.0867
DistalPhalanxTW
0.0000
0.0200
0.0300
0.06
ECG200
0.0800
0.1600
0.1600
0.28
ECGFiveDays
0.1667
0.67
0.5000
0,5000
Earthquakes
0.06
0.0123
0.0123
0.06
FaceAll
0.9000
0.92
0.9143
0.9143
FacesUCR
0.86
0.86
0.86
0,8000
GunPoint
0.6154
0.6154
0.6154
0.85
Ham
0.79
0,6429
0.7500
0.6429
Medicallmages
0.4375
0,4271
0.47
0.47
MildeePPhllmnoOOnineeeeGGGro
0.81
0.7300
0.7700
0,8000
0.28
0,2533
0.2267
0.2067
PiddlePhalanxTWW
0.0100
0.05
0.05
0.05
MoteStrain
0.80
0.80
0.80
0.80
ProximalPhalanXTW.
0.00
0.00
0.00
0.00
RefiigerationDevices
0.5000
0.5426
0.60
0,5106
ScreenType
0.51
0,4149
0.4043
0,4574
ShapeletSim
0.40
0.0000
0.2000
0.40
ShapesAll
0.67
0.6133
0.6600
0.6600
SmallKiichennApliaanes
0.64
0.5745
0.5745
0.64
SonyAllBORooosSurface
0.6000
1,00
0.8000
1.00
SonyAlBBRRobotSrriacce
0,8571
0.8571
0.5714
1.00
SyntheticControl
0.9867
1.00
1.00
1.00
Trace
0.9200
1.00
0.8400
1.00
TwoLeadECG
1.00
0.6667
0.6667
0.8333
Wine
0.4667
0.3333
0.73
0,4667
Worms
0.50
0.2826
0.3913
0.4130
Worms TwoClass
0.67
0.5652
0.5435
0.5652
Count
20
12
21
25

---

Published as a conference paper at ICLR 2026
Table 11: Ablation Study of Classification (Grouping 92):
Comparison of nearest centroid classifi-
cation accuracy across different parameters r. Bold indicates the highest accuracy.
ns-DTW (g2)
Dataset
0.95
0.90
T 0.85
0.80
Adiac
0.65
0.5510
0.6122
0.5714
ArrowHead
0.33
0.1111
0.0000
0.1111
Beef
0.3750
0.50
0.50
0.3750
BeetleFly
0.80
0.80
0.6000
0.6000
BirdChicken
0.8000
1.00
0.8000
0.8000
CBF
0.88
0.88
0.88
0.88
Car
0.67
0.67
0.67
0.67
ChlorineConcentration
0.3162
0.2906
0.3675
0.38
CinCECGTorso
0.3000
0.3000
0.4000
0.50
Coffee
0.00
0.00
0.00
0.00
Computers
0.68
0.6508
0.6508
0.6349
CricketX
0.6224
0.5204
0,5714
0.65
CricketY
0.5102
0.61
0,5612
0.5918
CricketZ
0.5510
0.61
0.4592
0.4796
DistuPPaalannxOuttnnkgGruuuuu
0.91
0.8500
0.8800
0,8400
AstuuPhallanxQuttieCCroro
0.1867
0.1400
0.1667
0.29
DistalPhalanx
0.0100
0.0100
0.0000
0.02
ECG200
0.28
0.2400
0.2000
0.1200
ECGFiveDays
0.6667
0.5000
0.6667
0.83
Earthquakes
0.0370
0.0247
0.05
0.0370
FaceAll
0.93
0.9143
0.8643
0,9000
FacesUCR
0.8800
0.94
0.8200
0.9000
GunPoint
0.69
0.69
0.6154
0.69
Ham
0.6429
0.6071
0.7500
0.86
Medicallmages
0.4271
0.4271
0.3646
0.49
MiddlePPalaaxxOunnneeeGGrruuu
0.7600
0.78
0.7500
0,7500
MiddllePhalanxOOuiinCCCCrrr
0.2267
0.2600
0.3600
0.41
MiddlePhalanxTWW
0.0300
0.0500
0.07
0.0600
MoteStrain
1.00
0.8000
0.8000
0.6000
ProximaiPhalanxTVV
0.0000
0.01
0.0000
0,0000
RefiiieratioonDevicess
0.5638
0.68
0.5426
0.5426
ScreenType
0.3936
0.48
0.4149
0,4255
ShapeletSim
0.60
0.2000
0.2000
0.4000
ShapesAll
0.6600
0.6533
0.6467
0.68
SmallKitchen/Apllianees
0.6064
0.73
0.6383
0.5957
SonyAllBORRoosSurfaee
1.00
0.8000
1.00
1.00
SongAlBBORobotSrrfacee
0.7143
0.7143
1.00
1.00
SyntheticControl
1.00
0.9867
1.00
0.9867
Trace
1.00
1.00
0,9600
1.00
TwoLeadECG
0.83
0.6667
0.6667
0.6667
Wine
0.53
0.53
0.53
0,2667
Worms
0.3478
0.5000
0.3696
0.52
Worms TwoClass
0.6522
0.5217
0.67
0.6087
Count
18
17
11
18
26

---

Published as a conference paper at ICLR 2026
Table 12:
Ablation Study of Classification (Grouping 93):
Comparison of nearest centroid classifi-
cation accuracy across different parameters r. Bold indicates the highest accuracy.
ns-DTW (gs)
Dataset
0.95
0.90
T 0.85
0.80
Adiac
0.63
0.6020
0.5918
0.6020
ArrowHead
0.1111
0.33
0.33
0.2222
Beef
0.25
0.1250
0.1250
0.25
BeetleFly
0.80
0.6000
0.6000
0.80
BirdChicken
1.00
0.6000
0.4000
0.8000
CBF
0.8750
1.00
0.8750
0.8750
Car
0.6667
0.80
0.7333
0.80
ChlorineConcentration
0.34
0.2564
0.3077
0.3333
CinCECGTorso
0.50
0.50
0.4000
0,2000
Coffee
0.0000
0.0000
0.0000
0.14
Computers
0.65
0.65
0.5873
0.6190
CricketX
0.59
0.5612
0.5612
0,5510
CricketY
0.5306
0,5102
0.56
0.5408
CricketZ
0.65
0.5306
0.5000
0.5204
DistuPPaalannOOuttnnAgGGuuuuu
0.8200
0.86
0.8100
0.8500
SotulPhalanxOuutineCorree
0.1400
0.26
0.0867
0.1933
DistalPhalanx
0.0100
0.03
0.0100
0.0200
ECG200
0.0000
0.20
0.1200
0.1200
ECGFiveDays
0.6667
0,5000
1.00
0,6667
Earthquakes
0.06
0.0247
0.0247
0.0247
FaceAll
0.93
0.9143
0.9143
0.8857
FacesUCR
0.9000
0.7600
0.9200
0.94
GunPoint
0.6154
0.77
0.77
0,4615
Ham
0.6429
0.7143
0.75
0.75
Medicallmages
0.52
0.4271
0.3542
0.3542
MiddiePPalaaxOOttineeggGuuuuu
0.7700
0.7500
0.80
0,7900
MiddlePhalnnxuutineCCrooo
0.3867
0.3667
0.43
0.2867
MiddlePhalanxTWW
0.07
0.0600
0.0400
0.0400
MoteStrain
0.6000
0.80
0.6000
0.80
ProximaiPhalanxTV
0.00
0.00
0.00
0.00
RefiiieraaioonDevicess
0.5106
0.5426
0.61
0.5319
ScreenType
0.3936
0.44
0,4255
0,4149
ShapeletSim
0.2000
0.40
0,2000
0.40
ShapesAll
0.6067
0.6467
0.72
0,6533
SmallKitchenAApllianees
0.5957
0.66
0.6489
0.6277
SonyAllBORRoosSurfeee
0.6000
0.8000
1.00
0.8000
SongAlBBORobotSrrfacee
0.5714
1.00
1.00
0.7143
SyntheticControl
1.00
1.00
0.9867
0.9867
Trace
0.96
0.96
0.9200
0.96
TwoLeadECG
0.8333
0.8333
0.3333
1.00
Wine
0.67
0,5333
0.4667
0,4000
Worms
0.48
0.4348
0.4130
0.3696
Worms TwoClass
0.70
0.6522
0.6304
0.5870
Count
19
18
12
11
27

---

Published as a conference paper at ICLR 2026
E ABLATION STUDY FOR CLUSTERING
Table 13: Ablation Study of Clustering (Grouping 91): Comparison of UCR Clustering DTW Losses across different parameters Bold indicates the lowest loss.
ns-DTW (91)
Dataset
0.95
0.90
T 0.85
0.80
Adiac
0.0933
0.0887
0.0887
0.0885
ArrowHead
1.0624
1.0781
1.0750
1.0723
Beef
2.6543
2.3194
2.1169
2.6488
BeetleFly
19.0531
19.0865
19.0850
19.1989
BirdChicken
8.0163
8.1141
8.0221
8.0705
CBF
11.9793
11.8627
11.8566
11.8919
Car
0.5828
0.6079
0.5699
0,5820
ChlorineConcentraaioo
6,4328
6.4504
6.4373
6,4452
ConCECGTorso
669.7685
669.7685
669.7685
669.7685
Coffee
0.4340
0,4313
0.4305
0.4305
Computers
141.2125
142.1296
142.4221
137.3022
CricketX
114.1204
114.1204
114.1204
114.1204
Cricket
103.0182
103.0182
103.0182
103.0182
CricketZ
124.6247
124.6247
124.6247
124.6247
DiatomSizzRReduction
0.1008
0.0634
0.0631
0.0628
DssaalPhalannOOttiinAAGGGGo
0.3636
0.3646
0.3639
0.3639
DisalPhalannxOutineCCrrer
0.7662
0.7631
0.7662
0.7644
DistalPhalanxTW
0.3305
0.3318
0.3305
0.3303
ECG200
4.5388
4.5473
4.5334
4.5376
ECG5000
6.7068
6.6826
6,5888
6,5432
ECGFiveDays
5.7248
5,5425
5.5456
5,6963
Earthquakes
437.7200
437.7200
437.7200
437.7200
FaceAll
34.8896
34.8896
34.8896
34,8896
FaceFour
20.2089
20.2031
20.1979
20.3103
FCCesUCR
27.0649
27.0649
27.0649
27.0649
GunPoint
1.0274
0.9990
1.0012
1.0023
Ham
17.0643
17.0764
17.0641
17.0572
Medicallmages
3.4652
3.5714
3.5776
3.6044
MiddiePhallanxOutiinnAggge
0.2338
0.2349
0.2342
0.2340
MiddlePhallnxOuuinnnCrreee
0.3353
0.3349
0.3349
0.3348
MiddlcPhalanxTW
0.2171
0,2180
0,2171
0.2178
MoteStrain
19.0490
19.0853
19.0503
19.0503
ProximalPhallnxTTV
0.1595
0.1585
0.1585
0.1585
RefrigerationDeviiess
113.0523
114.4612
114.9686
117.9007
ScreenType
104.5366
105.8296
105.6790
104.8913
ShapeletSim
121.8188
120.5682
120.5546
120.7565
ShapesAll
25.5065
25.5056
25.5056
25,5056
SmallKiicchenAAplliannes
124.1084
122.3944
123.6107
123.2592
SonyAllBORRoosSuufacel
4.6157
4,6442
4.6401
4,6435
Sony:AlIOORobotSurfeeee
8.8879
8.8915
8.8879
8.8860
SyntheticControl
8.6648
8.6900
8.6810
8.6787
Trace
2.9443
2.9413
2,9340
2.9248
TwoLeadECG
0.8646
0.8769
0.8648
0.8644
Wine
0.0704
0.0702
0.0702
0.0702
WordSynonyms
27.5408
27.5408
27.5408
27.5408
Worms
451.2960
451.2960
451.2960
451.2960
Worms'
WormsTwoClass TwoClass
78.2919
78.5687
76.7911
78.7067
Count
21
14
19
21
28

---

Published as a conference paper at ICLR 2026
Table 14: Ablation Study of Clustering (Grouping g2): Comparison of UCR Clustering DTW Losses across different parameters 7. Bold indicates the lowest loss.
ns-DTW (ga)
Dataset
0.95
0.90
T 0.85
0.80
Adiac
0.0889
0,0886
0.0882
0.0887
ArrowHead
1.0626
1.0754
1.0447
1.0433
Beef
2.6517
2.6512
2.2552
2.1680
BeetleFly
19.1110
19.1173
18,9430
19.0810
BirdChicken
8.0838
8.0368
8.0901
8.0716
CBF
11.8548
11.8228
11.7534
11.8266
Car
0.5669
0.5728
0.5735
0.5785
ChlorineConcentration
6.4346
6,4438
6,4461
6,4400
CinCECGTorso
669.7685
669.7685
669.7685
669.7685
Coffee
0.4305
0.4304
0.4307
0,4314
Computers
143,9456
141.0881
142.3675
141.7750
CricketX
114.1204
114.1204
114.1204
114.1204
CricketY
103.0182
103.0182
103.0182
103.0182
CricketZ
124.6247
124.6247
124.6247
124.6247
DiatomSizeRReduction
0.0632
0.0635
0.0633
0.0636
DistulPhalannxOOttinnnnggg
0.3641
0.3640
0.3637
0.3639
DnstalPhatanxOuutineCCCoom
0.7654
0.7648
0.7641
0.7635
DistalPhalanxTW
0.3305
0.3305
0.3307
0.3292
ECG200
4.5394
4.5605
4.5595
4,5452
ECG5000
6.7502
6.5538
6.4040
6,5952
ECGFiveDays
5,6538
5.6400
5.5714
5.5447
Earthquakes
437.7200
437.7200
437.7200
437.7200
FaceAll
34.8896
34.8896
34.8896
34,8896
FaceFour
20. 2089
20.2024
20.1946
20.1897
FCCesUCR
27. 27.0649
27.0649
27.0649
27.0649
GunPoint
0.9987
1.0007
1.0040
1.0036
Ham
17.0729
17.0593
17.0463
17.1152
Medicallmages
3.4744
3.5939
3,5652
3.5336
MiddlePhalannxxutiineeeeeee
0.2336
0.2335
0.2337
0.2336
0.3349
0.3348
0.3348
0,3349
MiddlePhalanxTW
0.2152
0.2169
0.2169
0.2173
MoteStrain
19.0497
19.0506
19.0507
19,0507
ProximalPhalanxTW
0.1584
0.1584
0.1585
0.1586
RefiicerationDvvices
119,7300
116.9189
119,4748
119.4159
ScreenType
104.6065
104.5283
103.5914
104.3206
ShapeletSim
121.4127
121.6850
122.1467
120.9429
ShapesAll
25.5056
25.5056
25.5056
25.5056
SmallKiichhenAppllannes
123.1039
123.6502
123.9712
124.0442
SonyAllBORobotSurfice
4.6127
4.6506
4.6475
4.6339
SonyAlBORRoboSurficce
8,8879
8.8879
8.8879
8.8879
SymtheticControl
8.6874
8.6919
8.6829
8.6736
Trace
2,9486
2.9450
2.9467
2,9365
TwoLeadECCG
0.8646
0.8645
0.8646
0.8652
Wine
0.0699
0.0694
0.0694
0.0691
WordSynonyms
27.5408
27.5408
27.5408
27.5408
Worms
451.2960
451.2960
451.2960
451.2960
WormsTwoClie TwoClass
78.6962
78.2881
78.6430
78.4662
Count
22
17
18
20
29

---

Published as a conference paper at ICLR 2026
Table 15: Ablation Study of Clustering (Grouping ga): Comparison of UCR Clustering DTW Losses across different parameters 7. Bold indicates the lowest loss.
ns-DTW (gs)
Dataset
0.95
0.90
0.85
0.80
Adiac
0.0885
0,0888
0.0888
0.0885
ArrowHead
1.0650
1.0663
1.0664
1.0665
Beef
2.2629
2.2551
2.2545
2.1365
BeetleFly
19.0952
19.0988
19.0856
19.0679
BirdChicken
8.0704
8.0366
8.0304
8.1060
CBF
11.7832
11.7463
11.8067
11.8216
Car
0.5676
0.5790
0.5970
0.5846
ChlorineConcentration
6.4263
6,4330
6,4319
6,4301
CinCECGTorso
669.7685
669.7685
669.7685
669.7685
Coffee
0,4306
0.4306
0.4307
0,4318
Computers
139.4716
140.3803
139.3186
142.6158
CricketX
114.1204
114.1204
114.1204
114.1204
CricketY
103.0182
103.0182
103.0182
103.0182
CricketZ
124.6247
124.6247
124.6247
124.6247
DiatomSizeRReduction
0.0634
0.0634
0.0633
0.0635
OsstuPhalannxOuttinnegggge
0.3640
0.3638
0.3640
0.3637
DnstalPhalannOuutineCCComo
0.7659
0.7645
0.7688
0.7692
DistalPhalanxTW
0.3305
0.3306
0.3318
0.3317
ECG200
4.5356
4.5382
4,5334
4,5380
ECG5000
6.6823
6,5932
6.7333
6.6061
ECGFiveDays
5.6063
5.6822
5.6325
5.7113
Earthquakes
437.7200
437.7200
437.7200
437.7200
FaceAll
34.8896
34.8896
34.8896
34,8896
FaceFour
20.2126
20.2069
20.2224
20.3481
FCCesUCR
27.0649
27.0649
27.0649
27.0649
GunPoint
1.0012
0.9989
1.0012
0,9996
Ham
17.0756
17.1056
17.0749
17.1139
Medicallmages
3.4197
3.6215
3.5611
3,4369
MiddlePhalannxxutiineeeeeee
0.2339
0.2338
0.2339
0.2339
0.3349
0.3348
0.3348
0.3348
MiddlePhalanxTW
0.2172
0.2174
0.2173
0.2173
MoteStrain
18.8197
19.0501
19.0500
19.0499
ProximalPhalanxTW
0.1585
0.1585
0.1586
0.1586
RefiicerationDevices
120.0953
114.3597
118.9731
115.0863
ScreenType
104.6496
104.7925
105.0297
104.1625
ShapeletSim
121.2170
120.8727
120.6381
120.4151
ShapesAll
25.5056
25.5056
25.5056
25.5056
SmallKiichhenAppllannes
123.8676
123.7506
123.3957
123.5583
SonyALlBORobotSurfice
4.6162
4.6157
4.6566
4.6330
SonyAlBORRobtSurficce
8.8879
8,8856
8.8813
8,8808
SymtheticControl
8.6781
8.7188
8.6778
8.6984
Trace
2,9488
2.9490
2.9518
2,9483
TwoLeadECCG
0.8646
0.8646
0.8646
0.8644
Wine
0.0686
0.0682
0.0680
0.0678
WordSynonyms
27.5408
27.5408
27.5408
27.5408
Worms
451.2960
451.2960
451.2960
451.2960
WormsTwoClie TwoClass
78.5920
78.4135
78.7172
78.3617
Count
20
19
17
21
30

---

Published as a conference paper at ICLR 2026
THE USE OF LARGE LANGUAGE MODELS
Large Language Models (LLMS) were employed for proofreading and typographical error correction in this study.
31


Reviews:
[{"review_id": "uBb4LsijC2", "reviewer": "Reviewer_mXos", "summary": "The author proposes perturbed-DTW, that adds random noise to the alignmetn costs, and then taking average over the noise distribution. With this modification, the DTW optimal solution became a distribution over paths, rather than a single-path one. This also naturally makes the DTW differentiable like soft-DTW. The author also found, under Gumbel noise distribution, the perturbed-DTW reduces to soft-DTW. Therefore, the perturbed-DTW is a differentiable relaxation of the soft-DTW. In the experiments, the authors demonstrate the effectiveness of ns-DTW on diverse time-series tasks, showing that it captures meaningful alignment structures while remaining computationally tractable.", "questions": "How much computation cost in terms of big O notation, would be invovled when using perturbaed-DTW ?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. The paper has strong theoretical contibution. It provides a new probabilistic interpretation for soft-DTW, and extends it to a new family with GEV.\n2. The proposed method also remains practical and computationally tractable.\n3. Experiments shows good representation of time series can be captured using the proposed ns-DTW.", "weaknesses": "1. Some hyper-params might needs tunning.\n2. The ns-DTW might exhibit high computation cost.", "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": "D1xYAp3J3V", "reviewer": "Reviewer_VWnD", "summary": "The paper reframes soft-DTW as the expected DTW under i.i.d. Gumbel perturbations and generalizes to GEV-perturbed alignments, yielding a nested/“skewable” variant (ns-DTW). It preserves dynamic-programming structure and complexity, with a DP recursion that mirrors soft-DTW (noting the nested DP is not identical to the closed-form nested objective). On UCR subsets, ns-DTW improves barycenters and clustering and is competitive for nearest-centroid classification. Overall, it clarifies theory and adds a useful control knob for shaping alignments.", "questions": "The paper is well-motivated and, I believe, will stimulate further study and applications. I was left wondering \n1. how forward/backward runtimes and memory compare to soft-DTW for fixed (m, n) on the same hardware, and \n2. how sensitive the results are to γ and τ, including any simple rule of thumb for selecting them, an ablation over grouping choices would also be helpful.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Clean probabilistic interpretation of soft-DTW (soft-DTW = E[DTW with Gumbel-perturbed costs]). I believe  many practitioners will find such an intuitive interpretation of soft-DTW clarifying.\n- Principled GEV-based extension that induces tunable skewness/correlation across alignment moves.\n- Same order of computation as soft-DTW, makes it easy drop-in to existing pipelines.\n- Consistent gains for barycenters and clustering across many datasets.", "weaknesses": "- Little guidance on choosing γ and τ; limited sensitivity and compute profiling.\n- Classification baselines could be stronger (e.g., 1-NN DTW and 1-NN soft-DTW).\n- Limited sensitivity and runtime/memory analysis, \n- The link between the nested objective and its DP instantiation could be spelled out more clearly.", "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": "AlPkf53wYw", "reviewer": "Reviewer_mNjZ", "summary": "This paper proposes Perturbed-DTW, a probabilistic and differentiable extension of traditional Dynamic Time Warping (DTW) that injects random perturbations into the alignment cost and takes the expectation of the minimum cost. Under Gumbel noise the formulation recovers the well-known soft‑DTW, providing a probabilistic interpretation of its smoothing effect. The authors further generalize to the generalized extreme value (GEV) family of distributions, giving rise to a nested-soft-DTW (ns-DTW) variant with tunable skewness in the alignment structure. They validate the method across barycenter computation, clustering, and classification on several UCR time-series datasets, showing improved performance relative to standard DTW and existing differentiable DTW variants.", "questions": "See Weakness.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Presents a clear probabilistic interpretation of soft-DTW through perturbed optimization, which is conceptually elegant.\n\n2. Extends the formulation to the generalized extreme-value (GEV) family, yielding a tunable nested-soft-DTW variant with adjustable skewness — a novel theoretical contribution.\n\n3. The mathematical exposition is generally sound and the experimental evaluation spans several UCR datasets.", "weaknesses": "1. The supplementary archive hides the core implementation and scripts, making the experiments effectively non-reproducible. This lack of transparency raises serious doubts about the validity and reproducibility of the reported results.\n\n2. Empirical analysis is largely numerical and does not provide insight into runtime cost or robustness; ablations and qualitative visualizations are limited.\n\n3. Presentation could be tightened: notation is occasionally heavy, and experimental details (parameter settings, initialization, convergence) are insufficient for independent verification.", "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": "bSwC4gUAW9", "reviewer": "Reviewer_CEqh", "summary": "This paper provides a new probabilistic perturbation-based interpretation of soft-DTW, formulated using a concept similar to the Gumbel-Max (softmax) trick—specifically, its softmin counterpart (Lemma 1).\nUnder this formulation, the authors define perturbed-DTW and show that when the perturbation noise follows a $\\mathrm{Gumbel}(-c, 1)$ distribution (where $c$ is the Euler–Mascheroni constant, the perturbed-DTW becomes exactly equivalent to soft-DTW.\n\nFurthermore, the paper introduces nested-soft-DTW (ns-DTW), which employs the generalized extreme value (GEV) distribution—a generalization of the Gumbel distribution—as the perturbation noise model.\nTheoretical analysis (Theorem 1) derives a closed-form solution for this generalized case.\nIn ns-DTW, the set of alignment matrices is partitioned into $J$ groups, each associated with a parameter $\\tau_\\ell \\in (0,1]$. Groups with smaller $\\tau_\\ell$ values correspond to higher correlations among the perturbation samples within that group.\nLike soft-DTW, ns-DTW can be efficiently solved via dynamic programming (Algorithm 1).\n\nExperiments using the UCR Time Series Classification Archive compare ns-DTW with existing methods (e.g., DBA, soft-DTW) on barycenter estimation, classification, and clustering tasks.\nResults show that ns-DTW achieves comparable or slightly superior performance to soft-DTW in certain parameter settings.", "questions": "* ll. 045–046: The paper refers to “through the lens of perturbed optimizer (Blondel et al., 2021)”, but according to Blondel et al. (2021), no discussion of perturbed optimizers appears. Is this citation incorrect?\n* Eq. (8): Since $\\langle A, C \\rangle$ is a scalar while $\\epsilon$ is described as a vector, the subtraction seems ill-defined.\n* In the introduction of ns-DTW, $\\mathcal{A}_{m,n}$ is said to be divided into $J$ groups, but the description around line 312 (and Algorithm 1) divides the transition vectors into two groups. The correspondence between these explanations is unclear.\n* In Fig. 3, it is not clearly explained how varying $\\tau_1$ affects the skewness of the resulting distribution.\n* Eq. (16): The text states that this equation assumes two groups “to ease the illustration,” yet the same form is used in Algorithm 1. Shouldn’t the algorithm adopt the general formulation instead? The same issue applies to Eq. (17).\n* As shown in Table 1, ns-DTW can produce smaller DTW losses by adjusting the skewness of the distribution, but the underlying reason for this effect is not well explained.\n* l. 374: “nd-DTW” -> “ns-DTW”?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "* The paper provides a probabilistic perturbation-based interpretation of soft-DTW and extends it to a more general formulation, ns-DTW.\n* The proposed ns-DTW demonstrates improved performance over soft-DTW under some experimental conditions.", "weaknesses": "* The method introduces $\\tau_1$ as a hyperparameter, but it is unclear how this value was selected in the experiments. Since setting $\\tau_1 = 1$ makes ns-DTW equivalent to soft-DTW, this choice is crucial for a fair comparison.\n* The performance improvement over soft-DTW appears limited (e.g., Fig. 5(c)), and in some settings (e.g., Table 1 with $\\gamma = 0.1$), ns-DTW performs worse than soft-DTW.", "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)