On the number of iterations of the DBA algorithm

• Published in: Data mining and knowledge discovery Vol. 39; no. 5; p. 52
• Main Authors: Brüning, Frederik, Driemel, Anne, Ergür, Alperen, Röglin, Heiko
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2025; Springer Nature B.V
• Subjects: Algorithms; Artificial Intelligence; Center of gravity; Chemistry and Earth Sciences; Computation; Computer Science; Data Mining and Knowledge Discovery; Dynamic programming; Information Storage and Retrieval; Perturbation; Physics; Random variables; Run time (computers); Sequences; Statistics for Engineering; Time series; Upper bounds; Smoothed analysis; Clustering; Dynamic time warping; Time series
• ISSN: 1384-5810; 1573-756X
• DOI: 10.1007/s10618-025-01116-4

The DTW Barycenter Averaging (DBA) algorithm is a widely used algorithm for estimating the mean of a given set of point sequences. In this context, the mean is defined as a point sequence that minimises the sum of dynamic time warping distances (DTW). The algorithm is similar to the k -means algorithm in the sense that it alternately repeats two steps: (1) computing an optimal assignment to the points of the current mean, and (2) computing an optimal mean under the current assignment. The popularity of DBA can be attributed to the fact that it works well in practice, despite any theoretical guarantees to be known. In our paper, we aim to initiate a theoretical study of the number of iterations that DBA performs until convergence. We assume the algorithm is given n sequences of m points in and a parameter k that specifies the length of the mean sequence to be computed. We show that, in contrast to its fast running time in practice, the number of iterations can be exponential in k in the worst case — even if the number of input sequences is . We complement these findings with experiments on real-world data that suggest this worst-case behaviour is likely degenerate. To better understand the performance of the algorithm on non-degenerate input, we study DBA in the model of smoothed analysis, upper-bounding the expected number of iterations in the worst case under random perturbations of the input. Our smoothed upper bound is , where is the variance of the perturbation and the -notation omits logarithmic factors. For our analysis, we adapt the set of techniques that were developed for analysing the k -means method and observe that this set of techniques is not sufficient to obtain tight bounds for general n .