Approximating Length-Restricted Means under Dynamic Time Warping

TL;DR

This paper investigates the computational problem of approximating length-restricted means under the p-Dynamic Time Warping distance, providing polynomial-time algorithms for fixed-length means and approximation methods with linear scalability.

Contribution

It introduces polynomial-time algorithms for fixed-length means under p-DTW and develops scalable approximation algorithms with theoretical guarantees.

Findings

Exact polynomial-time algorithm for constant-length means.

Approximation algorithms with linear dependency on input size.

Application to clustering with theoretical guarantees.

Abstract

We study variants of the mean problem under the $p$-Dynamic Time Warping ($p$-DTW) distance, a popular and robust distance measure for sequential data. In our setting we are given a set of finite point sequences over an arbitrary metric space and we want to compute a mean point sequence of given length that minimizes the sum of $p$-DTW distances, each raised to the $q$\textsuperscript{th} power, between the input sequences and the mean sequence. In general, the problem is $\mathrm{NP}$-hard and known not to be fixed-parameter tractable in the number of sequences. On the positive side, we show that restricting the length of the mean sequence significantly reduces the hardness of the problem. We give an exact algorithm running in polynomial time for constant-length means. We explore various approximation algorithms that provide a trade-off between the approximation factor and the running time. Our approximation algorithms have a running time with only linear dependency on the number of input sequences. In addition, we use our mean algorithms to obtain clustering algorithms with theoretical guarantees.