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Improved Learning via k-DTW: A Novel Dissimilarity Measure for Curves

Amer Krivošija, Alexander Munteanu, André Nusser, Chris Schwiegelshohn

TL;DR

The paper introduces $k$-DTW, a novel dissimilarity for polygonal curves that interpolates between the Fréchet distance and DTW, with improved robustness to outliers and stronger metric-like properties. It provides an exact algorithm and a $(1+\varepsilon)$-approximation for computing $d_{\textit{k-DTW}}$, plus a dimension-free learning-theoretic framework that yields tighter generalization bounds and a complexity separation from DTW. The authors prove a relaxed triangle inequality for $k$-DTW, establish robustness guarantees via the breakdown point of the top-$k$ median, and present empirical evidence showing advantages in clustering and nearest-neighbor tasks on synthetic and real datasets. The work suggests practical benefits for learning on curve-structured data and identifies open questions around computational efficiency and further theoretical refinements.

Abstract

This paper introduces $k$-Dynamic Time Warping ($k$-DTW), a novel dissimilarity measure for polygonal curves. $k$-DTW has stronger metric properties than Dynamic Time Warping (DTW) and is more robust to outliers than the Fréchet distance, which are the two gold standards of dissimilarity measures for polygonal curves. We show interesting properties of $k$-DTW and give an exact algorithm as well as a $(1+\varepsilon)$-approximation algorithm for $k$-DTW by a parametric search for the $k$-th largest matched distance. We prove the first dimension-free learning bounds for curves and further learning theoretic results. $k$-DTW not only admits smaller sample size than DTW for the problem of learning the median of curves, where some factors depending on the curves' complexity $m$ are replaced by $k$, but we also show a surprising separation on the associated Rademacher and Gaussian complexities: $k$-DTW admits strictly smaller bounds than DTW, by a factor $\tildeΩ(\sqrt{m})$ when $k\ll m$. We complement our theoretical findings with an experimental illustration of the benefits of using $k$-DTW for clustering and nearest neighbor classification.

Improved Learning via k-DTW: A Novel Dissimilarity Measure for Curves

TL;DR

The paper introduces -DTW, a novel dissimilarity for polygonal curves that interpolates between the Fréchet distance and DTW, with improved robustness to outliers and stronger metric-like properties. It provides an exact algorithm and a -approximation for computing , plus a dimension-free learning-theoretic framework that yields tighter generalization bounds and a complexity separation from DTW. The authors prove a relaxed triangle inequality for -DTW, establish robustness guarantees via the breakdown point of the top- median, and present empirical evidence showing advantages in clustering and nearest-neighbor tasks on synthetic and real datasets. The work suggests practical benefits for learning on curve-structured data and identifies open questions around computational efficiency and further theoretical refinements.

Abstract

This paper introduces -Dynamic Time Warping (-DTW), a novel dissimilarity measure for polygonal curves. -DTW has stronger metric properties than Dynamic Time Warping (DTW) and is more robust to outliers than the Fréchet distance, which are the two gold standards of dissimilarity measures for polygonal curves. We show interesting properties of -DTW and give an exact algorithm as well as a -approximation algorithm for -DTW by a parametric search for the -th largest matched distance. We prove the first dimension-free learning bounds for curves and further learning theoretic results. -DTW not only admits smaller sample size than DTW for the problem of learning the median of curves, where some factors depending on the curves' complexity are replaced by , but we also show a surprising separation on the associated Rademacher and Gaussian complexities: -DTW admits strictly smaller bounds than DTW, by a factor when . We complement our theoretical findings with an experimental illustration of the benefits of using -DTW for clustering and nearest neighbor classification.

Paper Structure

This paper contains 22 sections, 21 theorems, 50 equations, 5 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Other Related Work
  4. Definitions
  5. Triangle inequality
  6. Robustness
  7. Construction Algorithm
  8. Dimension-Free and Improved Learning Theory via k-DTW
  9. Experimental Illustration
  10. Proofs
  11. Algorithmic Part
  12. $k$-DTW:
  13. DTW:
  14. Learning Theory
  15. Robustness
  16. ...and 7 more sections

Key Result

Lemma 2.1

Given $1\leq q<\infty$. There exists no constant $c$ that is independent of the complexities of the input curves such that $d_{\mathrm{DTW}^{(q)}\xspace}\!\left({\sigma,\tau}\right) \leq c \cdot \left( d_{\mathrm{DTW}^{(q)}\xspace}\!\left({\sigma,\upsilon}\right) + d_{\mathrm{DTW}^{(q)}\xspace}\!\le

Figures (5)

  • Figure 1: Curves of type $A_2$ (left); type $B$ (middle); type $C$ (right)
  • Figure 2: Single (top) and complete (bottom) linkage clustering; DTW (left), $k$-DTW (middle), Fréchet distance (right); synthetic data.
  • Figure 3: $K$-gadget (left); $D$-gadget (right); traversals realizing $k$-DTW (blue); traversals realizing DTW (red).
  • Figure 4: Example curves used to prove Lemma \ref{['lem:short_kdtw_long_dtw']}. An example instance that has a long DTW traversal (length $2m-5$) but a short $k$-DTW traversal (length $m+1$)
  • Figure 5: Single (top) and complete (bottom) linkage clustering; DTW (left), $k$-DTW (middle), Fréchet distance (right); synthetic data.

Theorems & Definitions (37)

  • Lemma 2.1: Lemma 3 and Theorem 2 of Lemire09
  • Definition 2.2: $k$-DTW distance
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Lemma 3.1: Short version of \ref{['lem:different:kdtw']}
  • Theorem 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Theorem 4.1
  • ...and 27 more