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

• URL: http://arxiv.org/abs/2505.23431v1
• Date: Thu, 29 May 2025 13:25:45 GMT
• Title: Improved Learning via k-DTW: A Novel Dissimilarity Measure for Curves
• Authors: Amer Krivošija, Alexander Munteanu, André Nusser, Chris Schwiegelshohn,
• Abstract summary: We introduce a novel dissimilarity measure for polygonal curves called $k$-Dynamic Time Warping ($k$-DTW)<n>$k$-DTW has stronger metric properties than Dynamic Time Warping (DTW) and is more robust to outliers than the Fr'echet distance.<n>We show interesting properties of $k$-DTW and give an exact algorithm as well as a $(1+varepsilon)$-approximation algorithm for $k$-DTW.
• Score: 48.84313029960523
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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