SIAM Journal on Optimization, Volume 31, Issue 3, Page 1722-1747, January 2021.
Fundamental in matrix algebra and its applications, a generalized inverse of a real matrix $A$ is a matrix $H$ that satisfies the Moore--Penrose (M--P) property $AHA=A$. If $H$ also satisfies the additional useful M--P property, $HAH=H$, it is called a reflexive generalized inverse. Reflexivity is equivalent to minimum rank, so we are particularly interested in reflexive generalized inverses. We consider aspects of symmetry related to the calculation of a sparse reflexive generalized inverse of $A$. As is common, and following Fampa and Lee [Oper. Res. Lett., 46 (2018), pp. 605--610] for calculating sparse generalized inverses, we use (vector) 1-norm minimization for inducing sparsity and for keeping the magnitude of entries under control. When $A$ is symmetric, we may naturally desire a symmetric $H$, while generally such a restriction on $H$ may not lead to a 1-norm minimizing reflexive generalized inverse. We investigate a block construction method to produce a symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We provide a theoretically efficient and practical local-search algorithm to block construct an approximate 1-norm minimizing symmetric reflexive generalized inverse. Another aspect of symmetry that we consider relates to another M--P property: $H$ is ah-symmetric if $AH$ is symmetric. The ah-symmetry property is the key one for solving least-squares problems using $H$. Here we do not assume that $A$ is symmetric, and we do not impose symmetry on $H$. We investigate a column block construction method to produce an ah-symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We provide a theoretically efficient and practical local-search algorithm to column block construct an approximate 1-norm minimizing ah-symmetric reflexive generalized inverse.

中文翻译：

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通过局部搜索对各种广义逆进行近似 1 范数最小化和最小秩结构稀疏性

SIAM 优化杂志，第 31 卷，第 3 期，第 1722-1747 页，2021 年 1 月。
在矩阵代数及其应用中，实矩阵 $A$ 的广义逆是满足 Moore--Penrose (M--P) 属性 $AHA=A$ 的矩阵 $H$。如果 $H$ 还满足额外的有用的 M--P 属性 $HAH=H$，则称为自反广义逆。自反性相当于最小秩，因此我们对自反广义逆特别感兴趣。我们考虑与$A$ 的稀疏自反广义逆计算相关的对称性方面。很常见，并遵循 Fampa 和 Lee [Oper。水库。Lett., 46 (2018), pp. 605--610] 用于计算稀疏广义逆，我们使用（向量）1-范数最小化来诱导稀疏性并控制条目的大小。当 $A$ 是对称的时，我们自然会想要一个对称的 $H$，而通常对 $H$ 的这种限制可能不会导致 1 范数最小化自反广义逆。我们研究了一种块构造方法来产生一个对称的自反广义逆，它是结构化的并且保证了稀疏性。我们提供了一种理论上有效且实用的局部搜索算法来块构造一个近似 1 范数最小化对称自反广义逆。我们考虑的对称性的另一个方面与另一个 M--P 属性有关：如果 $AH$ 是对称的，则 $H$ 是 ah 对称的。ah-symmetry 属性是使用 $H$ 解决最小二乘问题的关键属性。这里我们不假设$A$ 是对称的，并且我们不对$H$ 施加对称性。我们研究了一种列块构造方法，以产生一个结构化并保证稀疏性的 ah 对称自反广义逆。我们提供了一种理论上有效且实用的局部搜索算法来构造列块构造一个近似 1-范数最小化 ah 对称自反广义逆。