Closed-Form Diffeomorphic Transformations for Time Series Alignment
This work addresses the need for more efficient and accurate time series alignment methods, particularly for applications requiring differentiable and invertible warping functions, though it appears incremental as it builds on existing ODE-based frameworks.
The authors tackled the problem of time series alignment by developing a closed-form solution for diffeomorphic transformations under continuous piecewise-affine velocity functions, resulting in significant improvements in efficiency and accuracy on several datasets.
Time series alignment methods call for highly expressive, differentiable and invertible warping functions which preserve temporal topology, i.e diffeomorphisms. Diffeomorphic warping functions can be generated from the integration of velocity fields governed by an ordinary differential equation (ODE). Gradient-based optimization frameworks containing diffeomorphic transformations require to calculate derivatives to the differential equation's solution with respect to the model parameters, i.e. sensitivity analysis. Unfortunately, deep learning frameworks typically lack automatic-differentiation-compatible sensitivity analysis methods; and implicit functions, such as the solution of ODE, require particular care. Current solutions appeal to adjoint sensitivity methods, ad-hoc numerical solvers or ResNet's Eulerian discretization. In this work, we present a closed-form expression for the ODE solution and its gradient under continuous piecewise-affine (CPA) velocity functions. We present a highly optimized implementation of the results on CPU and GPU. Furthermore, we conduct extensive experiments on several datasets to validate the generalization ability of our model to unseen data for time-series joint alignment. Results show significant improvements both in terms of efficiency and accuracy.