Improved Learning via k-DTW: A Novel Dissimilarity Measure for Curves
This work addresses the challenge of curve comparison in machine learning, offering a more efficient and robust method for tasks like clustering and classification, though it is incremental relative to existing measures.
The paper tackles the problem of measuring dissimilarity between polygonal curves by introducing k-DTW, a novel measure that improves metric properties over DTW and robustness over Fréchet distance, resulting in smaller sample sizes and complexity bounds by factors like replacing curve complexity m with k and achieving a separation factor of Ω̃(√m) when k≪m.
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.