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

Kevin Buchin, Maike Buchin, Jan Erik Swiadek, Sampson Wong

TL;DR

It is shown that CDTW cannot be computed exactly under the Euclidean 2-norm using only algebraic operations, and an exact algorithm is given for CDTW under norms approximating the 2-norm.

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 Fréchet similarity.

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

TL;DR

It is shown that CDTW cannot be computed exactly under the Euclidean 2-norm using only algebraic operations, and an exact algorithm is given for CDTW under norms approximating the 2-norm.

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 Fréchet similarity.

Paper Structure

This paper contains 7 sections, 7 theorems, 3 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Geometry of Parameter Space Cells
  5. Optimal Paths through Cell Terrain
  6. Existence and Computation of Valleys
  7. Unsolvability within ACMQ under the 2-Norm

Key Result

theorem thmcountertheorem

Let $\overline{P},\overline{Q}$ be polygonal segments with a valley $\ell$ of positive slope under a norm $\hbox{$\| \cdot \|$}$, and let $x,y \in \mathbb{R}^2$ with $\Xi := [x_1,y_1] \times [x_2,y_2] \neq \varnothing$ be two points in parameter space. If $\ell \cap \Xi \neq \varnothing$, then the $

Figures (3)

  • Figure 1: Simple example of optimal matchings for our CDTW variant
  • Figure 2: Connection between optimal paths through cell terrain and geometric shape of cell terrain, as provided by valleys
  • Figure 3: Valley characterisation under a norm with regular hexagons as sublevel sets

Theorems & Definitions (14)

  • definition thmcounterdefinition: Buchi2007
  • definition thmcounterdefinition: BuchiNW2022
  • definition thmcounterdefinition: BuchiNW2022
  • definition thmcounterdefinition: Brank2022
  • theorem thmcountertheorem: Brank2022
  • lemma thmcounterlemma: AltG1995
  • lemma thmcounterlemma
  • proof
  • theorem thmcountertheorem
  • proof
  • ...and 4 more