On the number of iterations of the DBA algorithm
CoRR(2024)
摘要
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 ℝ^d 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
n=2. 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 polynomial in k, n and d, and for constant n, it is also
polynomial in m. For our analysis, we adapt the set of techniques that were
developed for analysing k-means and observe that this set of techniques is
not sufficient to obtain tight bounds for general n.
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