欠定系统的唯一“非负”解：从向量到矩阵
A Unique "Nonnegative" Solution to an Underdetermined System: from Vectors to Matrices
摘要：
本文研究欠定线性方程组非负向量解的唯一性和半正定矩阵解的唯一性。一个向量解是欠定线性方程组的唯一解，仅当测量矩阵有一个与正正交线相交的行跨度。针对两种类型的二元度量矩阵，Bernoulli 0-1矩阵和一般扩张图的邻接矩阵，我们证明了在这两种情况下，唯一非负解的支持度可以线性增长，即O（n），问题维数为n.我们还提供了封闭形式的表征，这个支持大小的信号尺寸的比率。对于矩阵情形，我们证明了在线性压缩观测算子的一个充要条件下，压缩线性观测算子存在唯一的半正定矩阵解.我们进一步证明了随机生成的高斯线性压缩观测算子将以压倒性的高概率满足这一条件。
英文摘要：
This paper investigates the uniqueness of a nonnegative vector solution and the uniqueness of a positive semidefinite matrix solution to underdetermined linear systems. A vector solution is the unique solution to an underdetermined linear system only if the measurement matrix has a row-span intersecting the positive orthant. Focusing on two types of binary measurement matrices, Bernoulli 0-1 matrices and adjacency matrices of general expander graphs, we show that, in both cases, the support size of a unique nonnegative solution can grow linearly, namely O(n), with the problem dimension n. We also provide closed-form characterizations of the ratio of this support size to the signal dimension. For the matrix case, we show that under a necessary and sufficient condition for the linear compressed observations operator, there will be a unique positive semidefinite matrix solution to the compressed linear observations. We further show that a randomly generated Gaussian linear compressed observations operator will satisfy this condition with overwhelmingly high probability.
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