# Closed-Form Diffeomorphic Transformations for Time Series Alignment

arXiv: https://arxiv.org/abs/2206.08107 · HTML: https://mlsanity.com/paper/2206.08107/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 51 references that are in the map.
- First posted on arXiv 16 Jun 2022.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Soft-DTW: a Differentiable Loss Function for Time-Series](https://mlsanity.com/paper/1703.01541/) (2017)
- [Unsupervised Learning for Fast Probabilistic Diffeomorphic Registration](https://mlsanity.com/paper/1805.04605/) (2018)
- [Deep Learning in Medical Image Registration: A Review](https://mlsanity.com/paper/1912.12318/) (2019)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [Don't Unroll Adjoint: Differentiating SSA-Form Programs](https://mlsanity.com/paper/1810.07951/) (2018)
- [MALI: A memory efficient and reverse accurate integrator for Neural ODEs](https://mlsanity.com/paper/2102.04668/) (2021)
- [DiffEqFlux.jl - A Julia Library for Neural Differential Equations](https://mlsanity.com/paper/1902.02376/) (2019)
- [Differentiable Divergences Between Time Series](https://mlsanity.com/paper/2010.08354/) (2020)
- [Instead of Rewriting Foreign Code for Machine Learning, Automatically Synthesize Fast Gradients](https://mlsanity.com/paper/2010.01709/) (2020)
- [TensorFlow Eager: A Multi-Stage, Python-Embedded DSL for Machine Learning](https://mlsanity.com/paper/1903.01855/) (2019)
- [Temporal Transformer Networks: Joint Learning of Invariant and Discriminative Time Warping](https://mlsanity.com/paper/1906.05947/) (2019)
- [Explicit Disentanglement of Appearance and Perspective in Generative Models](https://mlsanity.com/paper/1906.11881/) (2019)
- [Learning to Exploit Invariances in Clinical Time-Series Data using Sequence Transformer Networks](https://mlsanity.com/paper/1808.06725/) (2018)
- [Neural Time Warping For Multiple Sequence Alignment](https://mlsanity.com/paper/2006.15753/) (2020)
- [Residual Networks as Flows of Velocity Fields for Diffeomorphic Time Series Alignment](https://mlsanity.com/paper/2106.11911/) (2021)
- [Trainable Time Warping: Aligning Time-Series in the Continuous-Time Domain](https://mlsanity.com/paper/1903.09245/) (2019)
- [SrvfNet: A Generative Network for Unsupervised Multiple Diffeomorphic Shape Alignment](https://mlsanity.com/paper/2104.13449/) (2021)

## Built on by (most cited first, 7 of 7)

- [DiTASK: Multi-Task Fine-Tuning with Diffeomorphic Transformations](https://mlsanity.com/paper/2502.06029/) (2025)
- [Trainable Highly-expressive Activation Functions](https://mlsanity.com/paper/2407.07564/) (2024)
- [Shifting the Paradigm: A Diffeomorphism Between Time Series Data Manifolds for Achieving Shift-Invariancy in Deep Learning](https://mlsanity.com/paper/2502.19921/) (2025)
- [Diffeomorphic Temporal Alignment Nets for Time-series Joint Alignment and Averaging](https://mlsanity.com/paper/2502.06591/) (2025)
- [Synchronization of Multiple Videos](https://mlsanity.com/paper/2510.14051/) (2025)
- [DiGRAF: Diffeomorphic Graph-Adaptive Activation Function](https://mlsanity.com/paper/2407.02013/) (2024)
- [TimePoint: Accelerated Time Series Alignment via Self-Supervised Keypoint and Descriptor Learning](https://mlsanity.com/paper/2505.23475/) (2025)

## Abstract (arXiv)

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.