Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations

• URL: http://arxiv.org/abs/2104.14072v1
• Date: Thu, 29 Apr 2021 01:54:05 GMT
• Title: Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations
• Authors: Anthony Gruber, Max Gunzburger, Lili Ju, Yuankai Teng, Zhu Wang
• Abstract summary: "Nonlinear Level set Learning" (NLL) approach is presented for the pointwise prediction of functions which have been sparsely sampled. The proposed algorithm effectively reduces the input dimension to the theoretical lower bound with minor accuracy loss. Experiments and applications are presented which compare this modified NLL with the original NLL and the Active Subspaces (AS) method.
• Score: 6.184270985214254
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A dimension reduction method based on the "Nonlinear Level set Learning" (NLL) approach is presented for the pointwise prediction of functions which have been sparsely sampled. Leveraging geometric information provided by the Implicit Function Theorem, the proposed algorithm effectively reduces the input dimension to the theoretical lower bound with minor accuracy loss, providing a one-dimensional representation of the function which can be used for regression and sensitivity analysis. Experiments and applications are presented which compare this modified NLL with the original NLL and the Active Subspaces (AS) method. While accommodating sparse input data, the proposed algorithm is shown to train quickly and provide a much more accurate and informative reduction than either AS or the original NLL on two example functions with high-dimensional domains, as well as two state-dependent quantities depending on the solutions to parametric differential equations.

Related papers

• Convergence of Implicit Gradient Descent for Training Two-Layer Physics-Informed Neural Networks [4.313136216120379]
  In this paper, we provide convergence analysis for the implicit gradient descent for training over-parametrized two-layer PINNs. We show that the randomly IGD converges a globally optimal solution at a linear convergence rate.
  arXiv  Detail & Related papers  (2024-07-03T06:10:41Z)
• A Mean-Field Analysis of Neural Stochastic Gradient Descent-Ascent for Functional Minimiax Optimization [90.87444114491116]
  This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparametricized two-layer neural networks. We address (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural networks. Results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $O(alpha-1)$, measured in terms of the Wasserstein distance.
  arXiv  Detail & Related papers  (2024-04-18T16:46:08Z)
• Mapping-to-Parameter Nonlinear Functional Regression with Novel B-spline Free Knot Placement Algorithm [12.491024918270824]
  We propose a novel approach to nonlinear functional regression. The model is based on the mapping of function data from an infinite-dimensional function space to a finite-dimensional parameter space. The performance of our knot placement algorithms is shown to be robust in both single-function approximation and multiple-function approximation contexts.
  arXiv  Detail & Related papers  (2024-01-26T16:35:48Z)
• An Accelerated Doubly Stochastic Gradient Method with Faster Explicit Model Identification [97.28167655721766]
  We propose a novel doubly accelerated gradient descent (ADSGD) method for sparsity regularized loss minimization problems. We first prove that ADSGD can achieve a linear convergence rate and lower overall computational complexity.
  arXiv  Detail & Related papers  (2022-08-11T22:27:22Z)
• Laplacian-based Cluster-Contractive t-SNE for High Dimensional Data Visualization [20.43471678277403]
  We propose LaptSNE, a new graph-based dimensionality reduction method based on t-SNE. Specifically, LaptSNE leverages the eigenvalue information of the graph Laplacian to shrink the potential clusters in the low-dimensional embedding. We show how to calculate the gradient analytically, which may be of broad interest when considering optimization with Laplacian-composited objective.
  arXiv  Detail & Related papers  (2022-07-25T14:10:24Z)
• Level set learning with pseudo-reversible neural networks for nonlinear dimension reduction in function approximation [8.28646586439284]
  We propose a new method of Dimension Reduction via Learning Level Sets (DRiLLS) for function approximation. Our method contains two major components: one is the pseudo-reversible neural network (PRNN) module that effectively transforms high-dimensional input variables to low-dimensional active variables. The PRNN not only relaxes the invertibility constraint of the nonlinear transformation present in the NLL method due to the use of RevNet, but also adaptively weights the influence of each sample and controls the sensitivity the function to the learned active variables.
  arXiv  Detail & Related papers  (2021-12-02T17:25:34Z)
• Inverse Problem of Nonlinear Schr\"odinger Equation as Learning of Convolutional Neural Network [5.676923179244324]
  It is shown that one can obtain a relatively accurate estimate of the considered parameters using the proposed method. It provides a natural framework in inverse problems of partial differential equations with deep learning.
  arXiv  Detail & Related papers  (2021-07-19T02:54:37Z)
• Optimal oracle inequalities for solving projected fixed-point equations [53.31620399640334]
  We study methods that use a collection of random observations to compute approximate solutions by searching over a known low-dimensional subspace of the Hilbert space. We show how our results precisely characterize the error of a class of temporal difference learning methods for the policy evaluation problem with linear function approximation.
  arXiv  Detail & Related papers  (2020-12-09T20:19:32Z)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)
• Effective Dimension Adaptive Sketching Methods for Faster Regularized Least-Squares Optimization [56.05635751529922]
  We propose a new randomized algorithm for solving L2-regularized least-squares problems based on sketching. We consider two of the most popular random embeddings, namely, Gaussian embeddings and the Subsampled Randomized Hadamard Transform (SRHT)
  arXiv  Detail & Related papers  (2020-06-10T15:00:09Z)
• Deep Dimension Reduction for Supervised Representation Learning [51.10448064423656]
  We propose a deep dimension reduction approach to learning representations with essential characteristics. The proposed approach is a nonparametric generalization of the sufficient dimension reduction method. We show that the estimated deep nonparametric representation is consistent in the sense that its excess risk converges to zero.
  arXiv  Detail & Related papers  (2020-06-10T14:47:43Z)

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.