论文标题
1-符号最小化和对称和AH对称性对称的最小级结构稀疏性：等级一和二
1-norm minimization and minimum-rank structured sparsity for symmetric and ah-symmetric generalized inverses: rank one and two
论文作者
论文摘要
广义倒置在统计和应用矩阵代数的其他领域很重要。真正矩阵$ a $的A \ emph {概括性逆}是满足Moore-Penrose（M-p）属性$ aha = a $的矩阵$ h $。如果$ h $也满足M-p属性$ hah = h $，则称为\ emph {fulexive}。广义逆的反射性等同于最低等级，这是一种非常可取的特性。我们考虑了与$ a $的各种\ emph {稀疏}反射概括的反相的计算相关的对称方面。通常，我们使用（矢量）1-norm最小化来诱导稀疏性和保持所控制的条目的幅度。 当$ a $是对称的时，对称$ h $是非常需要的，但是通常对$ h $的限制不会导致1-符号最大程度地减少反身推广倒数。我们研究了一种块构造方法，以产生对称反射性概括性逆，该逆向构造并保证了稀疏性。让$ a $ a $ a $ r $的等级确定，当（i）$ r = 1 $和（ii）$ r = 2 $ and $ a $是非负责任时，1-norm最小化这种类型的逆向这种类型的倒数是一种1-核对对称的逆向倒数。 我们认为的对称性的另一个方面与另一个M-P属性有关：如果$ ah $是对称的，则$ h $是\ emph {ah-Memmetric}。 AH-AMMETRY属性足以使广义逆将用于解决最小二乘问题$ \ min \ {\ | ax-b \ | _2：〜x \ in \ Mathbb {r}^n \} $，使用$ h $，通过$ x：= hb $。我们研究了一种柱块构造方法，以产生结构化并保证稀疏性的AH对称反射概括。我们确定，当（i）$ r = 1 $和（ii）$ r = 2 $ and $ a $满足技术状况时，我们确定了这种类型的1核对AH-对称性概括为此类型的倒数倒数，将AH-对称性逆向倒数最小化。
Generalized inverses are important in statistics and other areas of applied matrix algebra. A \emph{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 M-P property $HAH=H$, then it is called \emph{reflexive}. Reflexivity of a generalized inverse is equivalent to minimum rank, a highly desirable property. We consider aspects of symmetry related to the calculation of various \emph{sparse} reflexive generalized inverses of $A$. As is common, we use (vector) 1-norm minimization for both inducing sparsity and for keeping the magnitude of entries under control. When $A$ is symmetric, a symmetric $H$ is highly desirable, but generally such a restriction on $H$ will 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. Letting the rank of $A$ be $r$, we establish that the 1-norm minimizing generalized inverse of this type is a 1-norm minimizing symmetric generalized inverse when (i) $r=1$ and when (ii) $r=2$ and $A$ is nonnegative. Another aspect of symmetry that we consider relates to another M-P property: $H$ is \emph{ah-symmetric} if $AH$ is symmetric. The ah-symmetry property is sufficient for a generalized inverse to be used to solve the least-squares problem $\min\{\|Ax-b\|_2:~x\in\mathbb{R}^n\}$ using $H$, via $x:=Hb$. We investigate a column block construction method to produce an ah-symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We establish that the 1-norm minimizing ah-symmetric generalized inverse of this type is a 1-norm minimizing ah-symmetric generalized inverse when (i) $r=1$ and when (ii) $r=2$ and $A$ satisfies a technical condition.
