An Efficient Transport-Based Dissimilarity Measure for Time Series Classification under Warping Distortions

TL;DR

The paper addresses warp-invariant time series classification by formulating a deformation-based class model and introducing two efficient divergences. It defines Continuous Dynamic Time Warping (CDTW) to capture exact equivalence classes under nondecreasing reparametrizations and derives a transport-based divergence $d_T$ that inherits the classification guarantees with significantly lower compute cost. Theoretical results establish equivalence-class structure, CDTW/DTW relationships, and zero-cost conditions, while experiments on both synthetic and UCR data demonstrate high accuracy in low-sample regimes and a dramatic speed-up of $d_T$ over DTW. The approach promises practical, scalable warp-aware TSC with rigorous guarantees and accessible implementations via 1-NN with CDTW or $d_T$.

Abstract

Time Series Classification (TSC) is an important problem with numerous applications in science and technology. Dissimilarity-based approaches, such as Dynamic Time Warping (DTW), are classical methods for distinguishing time series when time deformations are confounding information. In this paper, starting from a deformation-based model for signal classes we define a problem statement for time series classification problem. We show that, under theoretically ideal conditions, a continuous version of classic 1NN-DTW method can solve the stated problem, even when only one training sample is available. In addition, we propose an alternative dissimilarity measure based on Optimal Transport and show that it can also solve the aforementioned problem statement at a significantly reduced computational cost. Finally, we demonstrate the application of the newly proposed approach in simulated and real time series classification data, showing the efficacy of the method.

Figures (8)

• Figure 1: Nearest neighbor classification using different distances on 2 distinct simulated classification problems. The top row contains examples of atomic classes. The bottom row contains the accuracy obtained by different distances as a function of the number of samples in the training set. Left: 1 atomic class per class. Right: 5 atomic classes per class.
• Figure 2: Time requirements for $d_T$, DTW, and Euclidean 1-NN classification with $1,2,4,8,16,32$ samples per class in the training set. Each signal is discretized using 150 evaluation points. The full dataset consists of 250 signals.
• Figure 3: Examples of signals within some datasets from the UCR time series classification archive dau2019ucr. Each color represents a sample from a different class.
• Figure 4: Each point of the plot corresponds to a single dataset for which a 1NN-Classification algorithm has been run using $d_T$ divergence and DTW divergence. The coordinates of each point are the respective accuracies for $d_T$ and DTW.
• Figure 5: Classification accuracy of nearest neighbor classifiers using different distances.

Theorems & Definitions (36)

• Definition 2.1.1: DTW
• Definition 2.2.1: Generalized inverse - Comments and properties in Section \ref{['app: gen inv']}
• Proposition 3.1.2: Informal statement - Proof in Section \ref{['app: Equivalence relation']}
• Definition 3.2.1: Informal definition CDTW
• Theorem 3.2.2: Informal statement - Proof in Section \ref{['app: CDTW']}
• Theorem 3.2.3: CDTW -- DTW - Proof in Section \ref{['app: CDTW-DTW']}
• Definition 3.3.1
• Lemma 3.3.2: Informal statement - Proof in Section \ref{['app: d_T']}
• Remark 3.3.3
• Theorem 4.0.1: Informal statement - Proof in Section \ref{['app: classif']}