Order-Preserving Wasserstein Distance for Sequence Matching

We present a new distance measure between sequences that can tackle local temporal distortion and periodic sequences with arbitrary starting points. Through viewing the instances of sequences as empirical samples of an unknown distribution, we cast the calculation of the distance between sequences as the optimal transport problem. To preserve the inherent temporal relationships of the instances in sequences, we smooth the optimal transport problem with two novel temporal regularization terms. The inverse difference moment regularization enforces transport with local homogeneous structures, and the KL-divergence with a prior distribution regularization prevents transport between instances with far temporal positions. We show that this problem can be efficiently optimized through the matrix scaling algorithm. Extensive experiments on different datasets with different classifiers show that the proposed distance outperforms the traditional DTW variants and the smoothed optimal transport distance without temporal regularization.