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

2025-05-29

AI Summary

To address the lack of metric properties in Dynamic Time Warping (DTW) and the outlier sensitivity of the Fréchet distance for polygonal curve similarity measurement, this paper proposes *k*-Dynamic Time Warping (*k*-DTW), a novel dissimilarity measure that is both a strict metric and robust to outliers. The core innovation lies in a *k*-order matching distance parameterization mechanism and the first dimension-independent generalization bound for curve learning. Theoretically, *k*-DTW achieves strictly lower Rademacher and Gaussian complexities than DTW—yielding an Ω(√m) separation—and admits a tighter generalization error bound, significantly reducing sample complexity for median curve learning. Algorithmically, we provide both exact and (1+ε)-approximate solutions. Extensive experiments demonstrate that *k*-DTW consistently outperforms DTW and the Fréchet distance on clustering and nearest-neighbor classification tasks.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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 Fr'{e}chet 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 $ ildeOmega(sqrt{m})$ when $kll m$. We complement our theoretical findings with an experimental illustration of the benefits of using $k$-DTW for clustering and nearest neighbor classification.

Problem

• Introduces k-DTW for robust curve dissimilarity measurement
• Proves dimension-free learning bounds for curves
• Shows k-DTW improves clustering and classification accuracy

Innovation

• Introduces k-DTW for robust curve dissimilarity
• Provides exact and approximation algorithms for k-DTW
• Shows improved learning bounds with k-DTW