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

Amer Krivošija, Alexander Munteanu, André Nusser, Chris Schwiegelshohn

TL;DR: We introduce $k$-DTW, a novel distance for polygonal curves that generalizes and provides the best of the gold standards: DTW and Fréchet distances.

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

Lay Summary: This paper introduces $k$-Dynamic Time Warping ($k$-DTW), a new way to measure how different polygonal curves are. It sums only $k$ large values and ignores smaller ones. It is more robust than traditional methods like Fréchet distance and captures the geometry better than Dynamic Time Warping. The choice of the parameter $k$ interpolates between the two classic distances. The paper presents two algorithms for $k$-DTW, one exact and one faster approximate. It shows that $k$-DTW allows for learning with fewer samples compared to DTW. The new distance excels in tasks like clustering and classifying curves. These findings are illustrated via experiments.

Primary Area: Theory->Learning Theory

Keywords: polygonal curves, dynamic time warping, learning theory, clustering, classification

Link To Code: https://github.com/akrivosija/kDTW/

Submission Number: 12988