Soft-MSM: Differentiable Context-Aware Elastic Alignment for Time Series

TL;DR

Soft-MSM introduces a differentiable, context-aware elastic alignment loss for time series, improving clustering and classification performance over existing methods like Soft-DTW.

Contribution

It presents the first smooth relaxation of MSM, enabling gradient-based optimization for context-aware elastic distance in time series analysis.

Findings

Soft-MSM achieves lower MSM barycentre loss than existing MSM methods.

It significantly outperforms Soft-DTW-based methods in clustering and classification.

Experiments on 112 UCR datasets validate its effectiveness.

Abstract

Elastic distances like dynamic time warping (DTW) are central to time series machine learning because they compare sequences under local temporal misalignment. Soft-DTW is an adaptation of DTW that can be used as a gradient-based loss by replacing the hard minimum in its dynamic-programming recursion with a smooth relaxation. However, this approach does not directly extend to elastic distances whose transition costs depend on the local alignment context. Move-Split-Merge (MSM) is one such distance: it uses context-aware split and merge penalties and has often outperformed DTW in supervised and unsupervised time series machine learning tasks such as classification and clustering. We introduce Soft-MSM, a smooth relaxation of MSM and an elastic alignment loss with context-aware transition costs. Central to the formulation is a smooth gated surrogate for MSM's piecewise split/merge cost, which enables gradients through both the dynamic-programming recursion and the local transition structure. We derive the forward recursion, backward recursion, soft alignment matrix, closed-form gradient, limiting behaviour, and divergence-corrected formulation. Experiments on 112 UCR datasets show that Soft-MSM gives lower MSM barycentre loss than existing MSM barycentre methods, and yields significantly better clustering and nearest-centroid classification performance than Soft-DTW-based alternatives. An implementation is available in the open-source \texttt{aeon} toolkit.