Learning Meta-Distance for Sequences by Learning a Ground Metric via Virtual Sequence Regression

Distance between sequences is structural by nature because it needs to establish the temporal alignments among the temporally correlated vectors in sequences with varying lengths. Generally, distances for sequences heavily depend on the ground metric between the vectors in sequences to infer the alignments and hence can be viewed as meta-distances upon the ground metric. Learning such meta-distance from multi-dimensional sequences is appealing but challenging. We propose to learn the meta-distance through learning a ground metric for the vectors in sequences. The learning samples are sequences of vectors for which how the ground metric between vectors induces the meta-distance is given. The objective is that the meta-distance induced by the learned ground metric produces large values for sequences from different classes and small values for those from the same class. We formulate the ground metric as a parameter of the meta-distance and regress each sequence to an associated pre-generated virtual sequence w.r.t. the meta-distance, where the virtual sequences for sequences of different classes are well-separated. We develop general iterative solutions to learn both the Mahalanobis metric and the deep metric induced by a neural network for any ground-metric-based sequence distance. Experiments on several sequence datasets demonstrate the effectiveness and efficiency of the proposed methods.