@article{JainFroeseSchultz, author = {Jain, Brijnesh and Froese, Vincent and Schultz, David}, title = {An average-compress algorithm for the sample mean problem under dynamic time warping}, series = {Journal of Global Optimization}, journal = {Journal of Global Optimization}, publisher = {Springer}, doi = {10.1007/s10898-023-01294-9}, abstract = {Computing a sample mean of time series under dynamic time warping is NP-hard. Consequently, there is an ongoing research effort to devise efficient heuristics. The majority of heuristics have been developed for the constrained sample mean problem that assumes a solution of predefined length. In contrast, research on the unconstrained sample mean problem is underdeveloped. In this article, we propose a generic average-compress (AC) algorithm to address the unconstrained problem. The algorithm alternates between averaging (A-step) and compression (C-step). The A-step takes an initial guess as input and returns an approximation of a sample mean. Then the C-step reduces the length of the approximate solution. The compressed approximation serves as initial guess of the A-step in the next iteration. The purpose of the C-step is to direct the algorithm to more promising solutions of shorter length. The proposed algorithm is generic in the sense that any averaging and any compression method can be used. Experimental results show that the AC algorithm substantially outperforms current state-of-the-art algorithms for time series averaging.}, language = {en} } @article{BrillFluschnikFroeseetal., author = {Brill, Markus and Fluschnik, Till and Froese, Vincent and Jain, Brijnesh and Niedermeier, Rolf and Schultz, David}, title = {Exact mean computation in dynamic time warping spaces}, series = {Data Mining and Knowledge Discovery}, volume = {33}, journal = {Data Mining and Knowledge Discovery}, publisher = {Springer Nature}, doi = {10.1007/s10618-018-0604-8}, pages = {252 -- 291}, abstract = {Averaging time series under dynamic time warping is an important tool for improving nearest-neighbor classifiers and formulating centroid-based clustering. The most promising approach poses time series averaging as the problem of minimizing a Fr{\´e}chet function. Minimizing the Fr{\´e}chet function is NP-hard and so far solved by several heuristics and inexact strategies. Our contributions are as follows: we first discuss some inaccuracies in the literature on exact mean computation in dynamic time warping spaces. Then we propose an exponential-time dynamic program for computing a global minimum of the Fr{\´e}chet function. The proposed algorithm is useful for benchmarking and evaluating known heuristics. In addition, we present an exact polynomial-time algorithm for the special case of binary time series. Based on the proposed exponential-time dynamic program, we empirically study properties like uniqueness and length of a mean, which are of interest for devising better heuristics. Experimental evaluations indicate substantial deficits of state-of-the-art heuristics in terms of their output quality.}, language = {en} }