Iterative regularization for low complexity regularizers

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS
Cesare Molinari, Mathurin Massias, Lorenzo Rosasco, Silvia Villa
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Abstract

Iterative regularization exploits the implicit bias of optimization algorithms to regularize ill-posed problems. Constructing algorithms with such built-in regularization mechanisms is a classic challenge in inverse problems but also in modern machine learning, where it provides both a new perspective on algorithms analysis, and significant speed-ups compared to explicit regularization. In this work, we propose and study the first iterative regularization procedure with explicit computational steps able to handle biases described by non smooth and non strongly convex functionals, prominent in low-complexity regularization. Our approach is based on a primal-dual algorithm of which we analyze convergence and stability properties, even in the case where the original problem is unfeasible. The general results are illustrated considering the special case of sparse recovery with the \(\ell _1\) penalty. Our theoretical results are complemented by experiments showing the computational benefits of our approach.

Abstract Image

低复杂度正则的迭代正则化
迭代正则化利用优化算法的隐含偏差来正则化问题。与显式正则化相比,迭代正则化不仅为算法分析提供了新的视角,而且显著提高了速度。在这项工作中,我们提出并研究了第一个具有明确计算步骤的迭代正则化程序,该程序能够处理非平滑和非强凸函数描述的偏差,这在低复杂度正则化中非常突出。我们的方法基于一种基元-二元算法,我们分析了该算法的收敛性和稳定性,即使在原始问题不可行的情况下也是如此。考虑到具有 \(\ell _1\) 惩罚的稀疏恢复的特殊情况,我们对一般结果进行了说明。我们的理论结果得到了实验的补充,实验显示了我们方法的计算优势。
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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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