Homotopy Continuation Made Easy: Regression-based Online Simulation of Starting Problem-Solution Pairs

• URL: http://arxiv.org/abs/2411.03745v1
• Date: Wed, 06 Nov 2024 08:22:00 GMT
• Title: Homotopy Continuation Made Easy: Regression-based Online Simulation of Starting Problem-Solution Pairs
• Authors: Xinyue Zhang, Zijia Dai, Wanting Xu, Laurent Kneip,
• Abstract summary: homotopy continuation has been introduced as a plausible alternative to elimination templates. Our innovation consists of employing a regression network trained in simulation to directly predict a solution from input correspondences. We apply this elegant combination to generalized camera resectioning, and also introduce a new solution to the challenging generalized relative pose and scale problem.
• Score: 17.543457476766367
• License:
• Abstract: While automatically generated polynomial elimination templates have sparked great progress in the field of 3D computer vision, there remain many problems for which the degree of the constraints or the number of unknowns leads to intractability. In recent years, homotopy continuation has been introduced as a plausible alternative. However, the method currently depends on expensive parallel tracking of all possible solutions in the complex domain, or a classification network for starting problem-solution pairs trained over a limited set of real-world examples. Our innovation consists of employing a regression network trained in simulation to directly predict a solution from input correspondences, followed by an online simulator that invents a consistent problem-solution pair. Subsequently, homotopy continuation is applied to track that single solution back to the original problem. We apply this elegant combination to generalized camera resectioning, and also introduce a new solution to the challenging generalized relative pose and scale problem. As demonstrated, the proposed method successfully compensates the raw error committed by the regressor alone, and leads to state-of-the-art efficiency and success rates while running on CPU resources, only.

Related papers

• The Differentiable Feasibility Pump [49.55771920271201]
  This paper shows that the traditional feasibility pump and many of its follow-ups can be seen as gradient-descent algorithms with specific parameters. A central aspect of this reinterpretation is observing that the traditional algorithm differentiates the solution of the linear relaxation with respect to its cost.
  arXiv  Detail & Related papers  (2024-11-05T22:26:51Z)
• Error Feedback under $(L_0,L_1)$-Smoothness: Normalization and Momentum [56.37522020675243]
  We provide the first proof of convergence for normalized error feedback algorithms across a wide range of machine learning problems. We show that due to their larger allowable stepsizes, our new normalized error feedback algorithms outperform their non-normalized counterparts on various tasks.
  arXiv  Detail & Related papers  (2024-10-22T10:19:27Z)
• Solving Inverse Problems with Model Mismatch using Untrained Neural Networks within Model-based Architectures [14.551812310439004]
  We introduce an untrained forward model residual block within the model-based architecture to match the data consistency in the measurement domain for each instance. Our approach offers a unified solution that is less parameter-sensitive, requires no additional data, and enables simultaneous fitting of the forward model and reconstruction in a single pass.
  arXiv  Detail & Related papers  (2024-03-07T19:02:13Z)
• Optimizing Solution-Samplers for Combinatorial Problems: The Landscape of Policy-Gradient Methods [52.0617030129699]
  We introduce a novel theoretical framework for analyzing the effectiveness of DeepMatching Networks and Reinforcement Learning methods. Our main contribution holds for a broad class of problems including Max-and Min-Cut, Max-$k$-Bipartite-Bi, Maximum-Weight-Bipartite-Bi, and Traveling Salesman Problem. As a byproduct of our analysis we introduce a novel regularization process over vanilla descent and provide theoretical and experimental evidence that it helps address vanishing-gradient issues and escape bad stationary points.
  arXiv  Detail & Related papers  (2023-10-08T23:39:38Z)
• Shuffled Autoregression For Motion Interpolation [53.61556200049156]
  This work aims to provide a deep-learning solution for the motion task. We propose a novel framework, referred to as emphShuffled AutoRegression, which expands the autoregression to generate in arbitrary (shuffled) order. We also propose an approach to constructing a particular kind of dependency graph, with three stages assembled into an end-to-end spatial-temporal motion Transformer.
  arXiv  Detail & Related papers  (2023-06-10T07:14:59Z)
• Exploring the solution space of linear inverse problems with GAN latent geometry [23.779985842891705]
  Inverse problems consist in reconstructing signals from incomplete sets of measurements. We propose a method to generate multiple reconstructions that fit both the measurements and a data-driven prior learned by a generative adversarial network.
  arXiv  Detail & Related papers  (2022-07-01T14:33:44Z)
• 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)
• Learning Iterative Robust Transformation Synchronization [71.73273007900717]
  We propose to use graph neural networks (GNNs) to learn transformation synchronization. In this work, we avoid handcrafting robust loss functions, and propose to use graph neural networks (GNNs) to learn transformation synchronization.
  arXiv  Detail & Related papers  (2021-11-01T07:03:14Z)
• Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training [0.0]
  We present a framework for solving problems governed by partial differential equations (PDEs) based on elastic networks. We find that the approach converges to solve problems of porosci, Barry-Scier's injection-production problem, and a two-phase drainage problem.
  arXiv  Detail & Related papers  (2021-10-06T20:09:09Z)
• The Neural Network shifted-Proper Orthogonal Decomposition: a Machine Learning Approach for Non-linear Reduction of Hyperbolic Equations [0.0]
  In this work we approach the problem of automatically detecting the correct pre-processing transformation in a statistical learning framework. The purely data-driven method allowed us to generalise the existing approaches of linear subspace manipulation to non-linear hyperbolic problems with unknown advection fields. The proposed algorithm has been validated against simple test cases to benchmark its performances and later successfully applied to a multiphase simulation.
  arXiv  Detail & Related papers  (2021-08-14T15:13:35Z)
• Residual Gaussian Process: A Tractable Nonparametric Bayesian Emulator for Multi-fidelity Simulations [6.6903363553912305]
  A novel additive structure is introduced in which the highest fidelity solution is written as a sum of the lowest fidelity solution and residuals. The resulting model is equipped with a closed-form solution for the predictive posterior. It is shown how active learning can be used to enhance the model, especially with a limited computational budget.
  arXiv  Detail & Related papers  (2021-04-08T12:57:46Z)

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.