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.
翻译:时间序列对齐方法要求高度直观、差异化和不可逆的扭曲功能,这些功能可以保存时间的表层学,即二光伏变形学。二光伏变形功能可以通过普通差异方程式(ODE)管理的速度字段的整合产生。含有二光伏变异变的渐进式优化框架要求根据模型参数(即敏感度分析)计算不同方程式解决方案的衍生物,即敏感度分析。不幸的是,深深学习框架通常缺乏自动差异性兼容性敏感度分析方法;隐含功能,如ODE的解决方案,需要特别小心。当前对联合敏感方法、自动热数字解答器或ResNet的Eulerian离散化的解决方案具有吸引力。在这项工作中,我们用封闭式表达OD溶剂及其在连续的片段反射速度功能下的梯度。我们对CPU和GPU的结果进行高度优化的落实。此外,我们对一些数据设置进行了广泛的实验,以验证我们模型与无形数据的一般化能力,从而实现时间序列联合一致。