论文标题
在矩阵矢量产物的期望方面表达矩阵决定因素的两个平等性
Two equalities expressing the determinant of a matrix in terms of expectations over matrix-vector products
论文作者
论文摘要
我们介绍了两个方程式，以表达完整级别矩阵$ \ mathbf {a} \ in \ mathbb {r}^{n \ times n} $的反向决定因素。第一个关系是$ | \ MATHRM {det}（\ MathBf {a}）|^{ - 1} = \ Mathbb {e} _ {\ Mathbf {\ MathBf {\ MathBf {s} \ sim \ sim \ Mathcal {s} \ bigr] $，期望在$ n $维度半径One hypersphere的表面上均匀地绘制的向量。第二个关系是$ | \ MATHRM {det}（\ MathBf {a}）|^{ - 1} = \ Mathbb {e} _ {\ MathBf {\ Mathbf {x} \ sim q} \ sim q} [\，p（p（\ mathbf {ax}） $ Q $有全部支持。
We introduce two equations expressing the inverse determinant of a full rank matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ in terms of expectations over matrix-vector products. The first relationship is $|\mathrm{det} (\mathbf{A})|^{-1} = \mathbb{E}_{\mathbf{s} \sim \mathcal{S}^{n-1}}\bigl[\, \Vert \mathbf{As}\Vert^{-n} \bigr]$, where expectations are over vectors drawn uniformly on the surface of an $n$-dimensional radius one hypersphere. The second relationship is $|\mathrm{det}(\mathbf{A})|^{-1} = \mathbb{E}_{\mathbf{x} \sim q}[\,p(\mathbf{Ax}) /\, q(\mathbf{x})]$, where $p$ and $q$ are smooth distributions, and $q$ has full support.
