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

TL;DR

This paper introduces $k$-DTW, a new dissimilarity measure for polygonal curves that improves learning efficiency and robustness over traditional measures like DTW and Fréchet distance, with theoretical and experimental validation.

Contribution

The paper proposes $k$-DTW, a novel metric for curves with better theoretical properties and practical performance, including algorithms and learning bounds that outperform existing measures.

Findings

$k$-DTW has stronger metric properties than DTW.

$k$-DTW is more robust to outliers than Fréchet distance.

Experimental results show $k$-DTW improves clustering and classification tasks.

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 $\tilde\Omega(\sqrt{m})$ 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.