Closed-Form Diffeomorphic Transformations for Time Series Alignment

• URL: http://arxiv.org/abs/2206.08107v1
• Date: Thu, 16 Jun 2022 12:02:12 GMT
• Title: Closed-Form Diffeomorphic Transformations for Time Series Alignment
• Authors: I\~nigo Martinez, Elisabeth Viles, Igor G. Olaizola
• Abstract summary: We present a closed-form expression for the ODE solution and its gradient under continuous piecewise-affine velocity functions. Results show significant improvements both in terms of efficiency and accuracy.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• A Constant Velocity Latent Dynamics Approach for Accelerating Simulation of Stiff Nonlinear Systems [0.0]
  Solving stiff ordinary differential equations (StODEs) requires sophisticated numerical solvers, which are often computationally expensive. In this work, we embark on a different path which involves learning a latent dynamics for StODEs, in which one completely avoids numerical integration. In other words, the solution of the original dynamics is encoded into a sequence of straight lines which can be decoded back to retrieve the actual solution as and when required.
  arXiv  Detail & Related papers  (2025-01-14T20:32:31Z)
• Finite Operator Learning: Bridging Neural Operators and Numerical Methods for Efficient Parametric Solution and Optimization of PDEs [0.0]
  We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. We can parametrically solve partial differential equations in a data-free manner and provide accurate sensitivities. Our study focuses on the steady-state heat equation within heterogeneous materials.
  arXiv  Detail & Related papers  (2024-07-04T21:23:12Z)
• Diffeomorphic Transformations for Time Series Analysis: An Efficient Approach to Nonlinear Warping [0.0]
  The proliferation and ubiquity of temporal data across many disciplines has sparked interest for similarity, classification and clustering methods. Traditional distance measures such as the Euclidean are not well-suited due to the time-dependent nature of the data. This thesis proposes novel elastic alignment methods that use parametric & diffeomorphic warping transformations.
  arXiv  Detail & Related papers  (2023-09-25T10:51:47Z)
• Semi-supervised Learning of Partial Differential Operators and Dynamical Flows [68.77595310155365]
  We present a novel method that combines a hyper-network solver with a Fourier Neural Operator architecture. We test our method on various time evolution PDEs, including nonlinear fluid flows in one, two, and three spatial dimensions. The results show that the new method improves the learning accuracy at the time point of supervision point, and is able to interpolate and the solutions to any intermediate time.
  arXiv  Detail & Related papers  (2022-07-28T19:59:14Z)
• AutoIP: A United Framework to Integrate Physics into Gaussian Processes [15.108333340471034]
  We propose a framework that can integrate all kinds of differential equations into Gaussian processes. Our method shows improvement upon vanilla GPs in both simulation and several real-world applications.
  arXiv  Detail & Related papers  (2022-02-24T19:02:14Z)
• Measuring dissimilarity with diffeomorphism invariance [94.02751799024684]
  We introduce DID, a pairwise dissimilarity measure applicable to a wide range of data spaces. We prove that DID enjoys properties which make it relevant for theoretical study and practical use.
  arXiv  Detail & Related papers  (2022-02-11T13:51:30Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Last-Iterate Convergence of Saddle-Point Optimizers via High-Resolution Differential Equations [83.3201889218775]
  Several widely-used first-order saddle-point optimization methods yield an identical continuous-time ordinary differential equation (ODE) when derived naively. However, the convergence properties of these methods are qualitatively different, even on simple bilinear games. We adopt a framework studied in fluid dynamics to design differential equation models for several saddle-point optimization methods.
  arXiv  Detail & Related papers  (2021-12-27T18:31:34Z)
• DiffPD: Differentiable Projective Dynamics with Contact [65.88720481593118]
  We present DiffPD, an efficient differentiable soft-body simulator with implicit time integration. We evaluate the performance of DiffPD and observe a speedup of 4-19 times compared to the standard Newton's method in various applications.
  arXiv  Detail & Related papers  (2021-01-15T00:13:33Z)
• SLEIPNIR: Deterministic and Provably Accurate Feature Expansion for Gaussian Process Regression with Derivatives [86.01677297601624]
  We propose a novel approach for scaling GP regression with derivatives based on quadrature Fourier features. We prove deterministic, non-asymptotic and exponentially fast decaying error bounds which apply for both the approximated kernel as well as the approximated posterior.
  arXiv  Detail & Related papers  (2020-03-05T14:33:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.