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.

中文翻译：

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从正半定矩阵的混合中稳健恢复稀疏非负权重

我们考虑一个结构化估计问题，其中假设观察到的矩阵被生成为给定 $n\times n$ 正半定矩阵的 $N$ 的 $s$-稀疏线性组合。从噪声观测中恢复未知的 $N$ 维和 $s$ 稀疏权重是信号处理各个领域中的一个重要问题，也是协方差估计中的相关预处理步骤。我们将介绍相关的恢复保证，并重点关注非负权重的情况。该问题被表述为凸程序，无需进一步调整即可解决。这种稳健、非贝叶斯和无参数的方法对于先验分布和进一步模型参数未知的应用程序非常重要。受无线通信中显式应用的启发，我们将考虑特定的一级情况，其中已知矩阵是 iid 的外积。零均值亚高斯 $n$ 维复数向量。我们表明，对于给定的 $n$ 和 $N$，可以使用无参数凸程序一次 $s\leq O(n^2 / \log^2(N/ n^2)$。我们的误差估计在瞬时噪声功率中线性缩放，因此凸算法不需要噪声的先验界限。如果加性失真的幅度取决于未知数本身，则这种估计很重要。我们的误差估计在瞬时噪声功率中成线性比例，因此凸算法不需要噪声的先验界限。如果加性失真的幅度取决于未知数本身，则此类估计很重要。我们的误差估计在瞬时噪声功率中成线性比例，因此凸算法不需要噪声的先验界限。如果加性失真的幅度取决于未知数本身，则此类估计很重要。