Automatic Segmentation of Continuous Trajectories with Invariance to Nonlinear Warpings of Time

We study the classification problem that arises when two variables---one continuous (x), one discrete (s)---evolve jointly in time. We suppose that the vector x traces out a smooth multidimensional curve, to each point of which the variable s attaches a discrete label. The trace of s thus partitions the curve into different segments whose boundaries occur where s changes value. We consider how to learn the mapping between x and s from examples of segmented curves. Our approach is to model the conditional random process that generates segments of constant s along the curve of x. We suppose that the variable s evolves stochastically as a function of the arc length traversed by x. Since arc length does not depend on the rate at which a curve is traversed, this gives rise to a family of Markov processes whose predictions, Pr[s j x], are invariant to nonlinear warpings (or reparameterizations) of time. We show how to learn the parameters of these Markov processes from labeled and/or un..