论文标题
从阳性矩阵的混合物中稀疏的非负重恢复
Robust Recovery of Sparse Nonnegative Weights from Mixtures of Positive-Semidefinite Matrices
论文作者
论文摘要
我们考虑一个结构化估计问题，其中假定观察到的矩阵作为$ n $ n $ $ n \ times n $ n $ -semidefinite矩阵生成的$ S $ -SPARSE线性组合。从嘈杂的观测值中恢复未知的$ n $尺寸和$ s $ sparse权重是信号处理的各个领域的重要问题，也是协方差估算中相关的预处理步骤。我们将提供相关的恢复保证，并专注于非负重的情况。该问题是作为凸面程序提出的，可以在不进一步调整的情况下解决。这种坚固的，非拜访和无参数的方法对于先前分布和进一步模型参数未知的应用很重要。由无线通信中的明确应用激励，我们将考虑特定的等级案例，其中已知的矩阵是IID的外部产品。零均值的subgaussian $ n $二维复合体向量。我们表明，对于给定的$ n $和$ n $，一个人可以恢复非负$ s $ - 一旦$ s \ leq o（n^2 / \ log^2（n / n^2）$，使用无参数的凸面程序稀疏权重。我们的错误估计在瞬时噪声范围中，我们的错误估计不需要以前的噪声，即使噪声均不重要。失真取决于未知本身。
We consider a structured estimation problem where an observed matrix is assumed to be generated as an $s$-sparse linear combination of $N$ given $n\times n$ positive-semidefinite matrices. Recovering the unknown $N$-dimensional and $s$-sparse weights from noisy observations is an important problem in various fields of signal processing and also a relevant pre-processing step in covariance estimation. We will present related recovery guarantees and focus on the case of nonnegative weights. The problem is formulated as a convex program and can be solved without further tuning. Such robust, non-Bayesian and parameter-free approaches are important for applications where prior distributions and further model parameters are unknown. Motivated by explicit applications in wireless communication, we will consider the particular rank-one case, where the known matrices are outer products of iid. zero-mean subgaussian $n$-dimensional complex vectors. We show that, for given $n$ and $N$, one can recover nonnegative $s$--sparse weights with a parameter-free convex program once $s\leq O(n^2 / \log^2(N/n^2)$. Our error estimate scales linearly in the instantaneous noise power whereby the convex algorithm does not need prior bounds on the noise. Such estimates are important if the magnitude of the additive distortion depends on the unknown itself.
