A Reduction Theorem for the Sample Mean in Dynamic Time Warping Spaces

Though the concept of sample mean in dynamic time warping (DTW) spaces is used in pattern recognition applications, its existence has neither been proved nor called into question. This article shows that a sample mean exists under general conditions that cover common variations of different DTW-spaces mentioned in the literature. The existence proofs are based on a Reduction Theorem that bounds the length of the candidate solutions we need to consider. The proposed results place the concept of sample mean in DTW-spaces on a sound mathematical foundation and serves as a first step towards a statistical theory of DTW-spaces.