Learning sparsity-promoting regularizers for linear inverse problems

• URL: http://arxiv.org/abs/2412.16031v1
• Date: Fri, 20 Dec 2024 16:26:54 GMT
• Title: Learning sparsity-promoting regularizers for linear inverse problems
• Authors: Giovanni S. Alberti, Ernesto De Vito, Tapio Helin, Matti Lassas, Luca Ratti, Matteo Santacesaria,
• Abstract summary: This paper introduces a novel approach to learning sparsity-promoting regularizers for solving linear inverse problems. We develop a bilevel optimization framework to select an optimal synthesis operator, denoted as $B$, which regularizes the inverse problem while promoting sparsity in the solution.
• Score: 5.812129569528998
• License:
• Abstract: This paper introduces a novel approach to learning sparsity-promoting regularizers for solving linear inverse problems. We develop a bilevel optimization framework to select an optimal synthesis operator, denoted as $B$, which regularizes the inverse problem while promoting sparsity in the solution. The method leverages statistical properties of the underlying data and incorporates prior knowledge through the choice of $B$. We establish the well-posedness of the optimization problem, provide theoretical guarantees for the learning process, and present sample complexity bounds. The approach is demonstrated through examples, including compact perturbations of a known operator and the problem of learning the mother wavelet, showcasing its flexibility in incorporating prior knowledge into the regularization framework. This work extends previous efforts in Tikhonov regularization by addressing non-differentiable norms and proposing a data-driven approach for sparse regularization in infinite dimensions.

Related papers

• Generalization Bounds of Surrogate Policies for Combinatorial Optimization Problems [61.580419063416734]
  A recent stream of structured learning approaches has improved the practical state of the art for a range of optimization problems. The key idea is to exploit the statistical distribution over instances instead of dealing with instances separately. In this article, we investigate methods that smooth the risk by perturbing the policy, which eases optimization and improves the generalization error.
  arXiv  Detail & Related papers  (2024-07-24T12:00:30Z)
• A Guide to Stochastic Optimisation for Large-Scale Inverse Problems [4.926711494319977]
  optimisation algorithms are the de facto standard for machine learning with large amounts of data. Handling only a subset of available data in each optimisation step dramatically reduces the per-iteration computational costs. We focus on the potential and the challenges for optimisation that are unique to variational regularisation for inverse imaging problems.
  arXiv  Detail & Related papers  (2024-06-10T15:02:30Z)
• Randomized algorithms and PAC bounds for inverse reinforcement learning in continuous spaces [47.907236421762626]
  This work studies discrete-time discounted Markov decision processes with continuous state and action spaces. We first consider the case in which we have access to the entire expert policy and characterize the set of solutions to the inverse problem.
  arXiv  Detail & Related papers  (2024-05-24T12:53:07Z)
• From Inverse Optimization to Feasibility to ERM [11.731853838892487]
  We study the contextual inverse setting that utilizes additional contextual information to better predict parameters. We experimentally validate our approach on synthetic and real-world problems and demonstrate improved performance compared to existing methods.
  arXiv  Detail & Related papers  (2024-02-27T21:06:42Z)
• Analyzing and Enhancing the Backward-Pass Convergence of Unrolled Optimization [50.38518771642365]
  The integration of constrained optimization models as components in deep networks has led to promising advances on many specialized learning tasks. A central challenge in this setting is backpropagation through the solution of an optimization problem, which often lacks a closed form. This paper provides theoretical insights into the backward pass of unrolled optimization, showing that it is equivalent to the solution of a linear system by a particular iterative method. A system called Folded Optimization is proposed to construct more efficient backpropagation rules from unrolled solver implementations.
  arXiv  Detail & Related papers  (2023-12-28T23:15:18Z)
• Stable Nonconvex-Nonconcave Training via Linear Interpolation [51.668052890249726]
  This paper presents a theoretical analysis of linearahead as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear can help by leveraging the theory of nonexpansive operators.
  arXiv  Detail & Related papers  (2023-10-20T12:45:12Z)
• Backpropagation of Unrolled Solvers with Folded Optimization [55.04219793298687]
  The integration of constrained optimization models as components in deep networks has led to promising advances on many specialized learning tasks. One typical strategy is algorithm unrolling, which relies on automatic differentiation through the operations of an iterative solver. This paper provides theoretical insights into the backward pass of unrolled optimization, leading to a system for generating efficiently solvable analytical models of backpropagation.
  arXiv  Detail & Related papers  (2023-01-28T01:50:42Z)
• Learning to Optimize with Stochastic Dominance Constraints [103.26714928625582]
  In this paper, we develop a simple yet efficient approach for the problem of comparing uncertain quantities. We recast inner optimization in the Lagrangian as a learning problem for surrogate approximation, which bypasses apparent intractability. The proposed light-SD demonstrates superior performance on several representative problems ranging from finance to supply chain management.
  arXiv  Detail & Related papers  (2022-11-14T21:54:31Z)
• Semi-Supervised Learning with Meta-Gradient [123.26748223837802]
  We propose a simple yet effective meta-learning algorithm in semi-supervised learning. We find that the proposed algorithm performs favorably against state-of-the-art methods.
  arXiv  Detail & Related papers  (2020-07-08T08:48:56Z)
• Consistency analysis of bilevel data-driven learning in inverse problems [1.0705399532413618]
  We consider the adaptive learning of the regularization parameter from data by means of optimization. We demonstrate how to implement our framework on linear inverse problems. Online numerical schemes are derived using the gradient descent method.
  arXiv  Detail & Related papers  (2020-07-06T12:23:29Z)

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.