l1/核范数最小化的一阶算法

IF 16.3 1区数学 Q1 MATHEMATICS

Acta Numerica Pub Date : 2013-04-02 DOI:10.1017/S096249291300007X

Y. Nesterov, A. Nemirovski

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引用次数: 75

摘要

近十年来，l1/核范数最小化问题在信号处理、机器学习和优化领域引起了广泛关注。在本文中，我们将l1/核范数最小化作为“优化野兽”，详细描述了解决这类问题的两种有吸引力的一阶优化技术。第一个主要针对套索型问题，包括应用于复合最小化公式的快速梯度方法。第二种方法，针对dantzig -选择器型问题，利用鞍点一阶算法和将感兴趣的问题重新表述为广义双线性鞍点问题。对于这两种方法，我们都给出了完整和详细的复杂性分析，并讨论了应用领域。

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On first-order algorithms for l1/nuclear norm minimization

In the past decade, problems related to l1/nuclear norm minimization have attracted much attention in the signal processing, machine learning and optimization communities. In this paper, devoted to l1/nuclear norm minimization as ‘optimization beasts’, we give a detailed description of two attractive first-order optimization techniques for solving problems of this type. The first one, aimed primarily at lasso-type problems, comprises fast gradient methods applied to composite minimization formulations. The second approach, aimed at Dantzig-selector-type problems, utilizes saddle-point first-order algorithms and reformulation of the problem of interest as a generalized bilinear saddle-point problem. For both approaches, we give complete and detailed complexity analyses and discuss the application domains.

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来源期刊

Acta Numerica MATHEMATICS-

CiteScore

26.00

自引率

0.70%

发文量

期刊介绍： Acta Numerica stands as the preeminent mathematics journal, ranking highest in both Impact Factor and MCQ metrics. This annual journal features a collection of review articles that showcase survey papers authored by prominent researchers in numerical analysis, scientific computing, and computational mathematics. These papers deliver comprehensive overviews of recent advances, offering state-of-the-art techniques and analyses. Encompassing the entirety of numerical analysis, the articles are crafted in an accessible style, catering to researchers at all levels and serving as valuable teaching aids for advanced instruction. The broad subject areas covered include computational methods in linear algebra, optimization, ordinary and partial differential equations, approximation theory, stochastic analysis, nonlinear dynamical systems, as well as the application of computational techniques in science and engineering. Acta Numerica also delves into the mathematical theory underpinning numerical methods, making it a versatile and authoritative resource in the field of mathematics.