Solving Inverse Problems with Deep Linear Neural Networks: Global Convergence Guarantees for Gradient Descent with Weight Decay

• URL: http://arxiv.org/abs/2502.15522v1
• Date: Fri, 21 Feb 2025 15:24:34 GMT
• Title: Solving Inverse Problems with Deep Linear Neural Networks: Global Convergence Guarantees for Gradient Descent with Weight Decay
• Authors: Hannah Laus, Suzanna Parkinson, Vasileios Charisopoulos, Felix Krahmer, Rebecca Willett,
• Abstract summary: We show that deep linear networks trained with weight decay automatically adapt to latent subspace structure in the data.<n>This is the first result to rigorously show that deep linear networks trained with weight decay automatically adapt to latent subspace structure in the data.
• Score: 11.619364664070666
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Machine learning methods are commonly used to solve inverse problems, wherein an unknown signal must be estimated from few measurements generated via a known acquisition procedure. In particular, neural networks perform well empirically but have limited theoretical guarantees. In this work, we study an underdetermined linear inverse problem that admits several possible solution mappings. A standard remedy (e.g., in compressed sensing) establishing uniqueness of the solution mapping is to assume knowledge of latent low-dimensional structure in the source signal. We ask the following question: do deep neural networks adapt to this low-dimensional structure when trained by gradient descent with weight decay regularization? We prove that mildly overparameterized deep linear networks trained in this manner converge to an approximate solution that accurately solves the inverse problem while implicitly encoding latent subspace structure. To our knowledge, this is the first result to rigorously show that deep linear networks trained with weight decay automatically adapt to latent subspace structure in the data under practical stepsize and weight initialization schemes. Our work highlights that regularization and overparameterization improve generalization, while overparameterization also accelerates convergence during training.

Related papers

• Convergence Analysis for Learning Orthonormal Deep Linear Neural Networks [27.29463801531576]
  We provide convergence analysis for training orthonormal deep linear neural networks. Our results shed light on how increasing the number of hidden layers can impact the convergence speed.
  arXiv  Detail & Related papers  (2023-11-24T18:46:54Z)
• Convergence Guarantees of Overparametrized Wide Deep Inverse Prior [1.5362025549031046]
  Inverse Priors is an unsupervised approach to transform a random input into an object whose image under the forward model matches the observation. We provide overparametrization bounds under which such network trained via continuous-time gradient descent will converge exponentially fast with high probability. This work is thus a first step towards a theoretical understanding of overparametrized DIP networks, and more broadly it participates to the theoretical understanding of neural networks in inverse problem settings.
  arXiv  Detail & Related papers  (2023-03-20T16:49:40Z)
• Globally Optimal Training of Neural Networks with Threshold Activation Functions [63.03759813952481]
  We study weight decay regularized training problems of deep neural networks with threshold activations. We derive a simplified convex optimization formulation when the dataset can be shattered at a certain layer of the network.
  arXiv  Detail & Related papers  (2023-03-06T18:59:13Z)
• Implicit Bias in Leaky ReLU Networks Trained on High-Dimensional Data [63.34506218832164]
  In this work, we investigate the implicit bias of gradient flow and gradient descent in two-layer fully-connected neural networks with ReLU activations. For gradient flow, we leverage recent work on the implicit bias for homogeneous neural networks to show that leakyally, gradient flow produces a neural network with rank at most two. For gradient descent, provided the random variance is small enough, we show that a single step of gradient descent suffices to drastically reduce the rank of the network, and that the rank remains small throughout training.
  arXiv  Detail & Related papers  (2022-10-13T15:09:54Z)
• Imbedding Deep Neural Networks [0.0]
  Continuous depth neural networks, such as Neural ODEs, have refashioned the understanding of residual neural networks in terms of non-linear vector-valued optimal control problems. We propose a new approach which explicates the network's depth' as a fundamental variable, thus reducing the problem to a system of forward-facing initial value problems.
  arXiv  Detail & Related papers  (2022-01-31T22:00:41Z)
• Path Regularization: A Convexity and Sparsity Inducing Regularization for Parallel ReLU Networks [75.33431791218302]
  We study the training problem of deep neural networks and introduce an analytic approach to unveil hidden convexity in the optimization landscape. We consider a deep parallel ReLU network architecture, which also includes standard deep networks and ResNets as its special cases.
  arXiv  Detail & Related papers  (2021-10-18T18:00:36Z)
• Deep Networks Provably Classify Data on Curves [12.309532551321334]
  We study a model problem that uses a deep fully-connected neural network to classify data drawn from two disjoint smooth curves on the unit sphere. We prove that when (i) the network depth is large to certain properties that set the difficulty of the problem and (ii) the network width and number of samples is intrinsic in the relative depth, randomly-d gradient descent quickly learns to correctly classify all points on the two curves with high probability.
  arXiv  Detail & Related papers  (2021-07-29T20:40:04Z)
• On the Explicit Role of Initialization on the Convergence and Implicit Bias of Overparametrized Linear Networks [1.0323063834827415]
  We present a novel analysis of single-hidden-layer linear networks trained under gradient flow. We show that the squared loss converges exponentially to its optimum. We derive a novel non-asymptotic upper-bound on the distance between the trained network and the min-norm solution.
  arXiv  Detail & Related papers  (2021-05-13T15:13:51Z)
• Solving Sparse Linear Inverse Problems in Communication Systems: A Deep Learning Approach With Adaptive Depth [51.40441097625201]
  We propose an end-to-end trainable deep learning architecture for sparse signal recovery problems. The proposed method learns how many layers to execute to emit an output, and the network depth is dynamically adjusted for each task in the inference phase.
  arXiv  Detail & Related papers  (2020-10-29T06:32:53Z)
• Revisiting Initialization of Neural Networks [72.24615341588846]
  We propose a rigorous estimation of the global curvature of weights across layers by approximating and controlling the norm of their Hessian matrix. Our experiments on Word2Vec and the MNIST/CIFAR image classification tasks confirm that tracking the Hessian norm is a useful diagnostic tool.
  arXiv  Detail & Related papers  (2020-04-20T18:12:56Z)
• Revealing the Structure of Deep Neural Networks via Convex Duality [70.15611146583068]
  We study regularized deep neural networks (DNNs) and introduce a convex analytic framework to characterize the structure of hidden layers. We show that a set of optimal hidden layer weights for a norm regularized training problem can be explicitly found as the extreme points of a convex set. We apply the same characterization to deep ReLU networks with whitened data and prove the same weight alignment holds.
  arXiv  Detail & Related papers  (2020-02-22T21:13:44Z)

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.