Nonsmooth analysis and subgradient methods for averaging in dynamic time warping spaces

Abstract

Time series averaging in dynamic time warping (DTW) spaces has been successfully applied to improve pattern recognition systems. This article proposes and analyzes subgradient methods for the problem of finding a sample mean in DTW spaces. The class of subgradient methods generalizes existing sample mean algorithms such as DTW Barycenter Averaging (DBA). We show that DBA is a majorize-minimize algorithm that converges to necessary conditions of optimality after finitely many iterations. Empirical results show that for increasing sample sizes the proposed stochastic subgradient (SSG) algorithm is more stable and finds better solutions in shorter time than the DBA algorithm on average. Therefore, SSG is useful in online settings and for non-small sample sizes. The theoretical and empirical results open new paths for devising sample mean algorithms: nonsmooth optimization methods and modified variants of pairwise averaging methods.

Revisiting inaccuracies of time series averaging under dynamic time warping

“…Different forms of time series averages have been proposed. For an overview we refer to [29]. A principled formulation is based on the notion of Fréchet function [12]: Suppose that S = (x 1 , .…”

“…For every cluster S, we first computed sample means µ i of S \ {x i } using the dynamic program [6]. Then we approximated a mean µ S of the set S using the DTW Barycenter Averaging (DBA) algorithm [24] and the Stochastic Subgradient (SSG) method [29]. We counted the number of times an approximation was drifted out of the set S.…”

“…3. Empirical results show that averages obtained by state-of-the-art methods such as DBA [14,24] and SSG [29] also drift out of a cluster in the sense of [22]. This finding suggests that the problem of "drift-out" of previous methods still persist.…”

“…Since the 1970ies, one research direction is devoted to the problem of time series averaging [2,9,13,14,17,18,24,26,29]. A common technique to average time series is based on first aligning the time series with respect to the dtw-distance and then synthesizing the aligned time series to an average.…”

“…In [24], it has been shown that DBA is a decreasing algorithm, which is not sufficient for convergence. Convergence after a finite number of iterations has been proved by [29].…”