Fundamentals of Computing Continuous Dynamic Time Warping in 2D under Different Norms

TL;DR

This paper investigates the computational properties of Continuous Dynamic Time Warping (CDTW) in 2D under various norms, revealing limitations under the Euclidean norm and proposing algorithms for approximations.

Contribution

It proves that exact CDTW computation under the Euclidean norm is impossible with algebraic operations and provides an exact algorithm for norms approximating the 2-norm.

Findings

Exact CDTW cannot be computed under Euclidean 2-norm with algebraic operations.

An exact algorithm is provided for CDTW under norms approximating the 2-norm.

Results generalize to other norms and related similarity measures.

Abstract

Continuous Dynamic Time Warping (CDTW) measures the similarity of polygonal curves robustly to outliers and to sampling rates, but the design and analysis of CDTW algorithms face multiple challenges. We show that CDTW cannot be computed exactly under the Euclidean 2-norm using only algebraic operations, and we give an exact algorithm for CDTW under norms approximating the 2-norm. The latter result relies on technical fundamentals that we establish, and which generalise to any norm and to related measures such as the partial Fr\'echet similarity.