Matrix sensing is a problem in signal processing and machine learning that involves recovering a low-rank matrix from a set of linear measurements. The goal is to reconstruct the original matrix as accurately as possible, given only a set of linear measurements obtained by sensing the matrix [Jain, Netrapalli and Shanghavi, 2013]. In this work, we focus on a particular direction of matrix sensing, which is called rank-$1$ matrix sensing [Zhong, Jain and Dhillon, 2015]. We present an improvement over the original algorithm in [Zhong, Jain and Dhillon, 2015]. It is based on a novel analysis and sketching technique that enables faster convergence rates and better accuracy in recovering low-rank matrices. The algorithm focuses on developing a theoretical understanding of the matrix sensing problem and establishing its advantages over previous methods. The proposed sketching technique allows for efficiently extracting relevant information from the linear measurements, making the algorithm computationally efficient and scalable. Our novel matrix sensing algorithm improves former result [Zhong, Jain and Dhillon, 2015] on in two senses: $\bullet$ We improve the sample complexity from $\widetilde{O}(\epsilon^{-2} dk^2)$ to $\widetilde{O}(\epsilon^{-2} (d+k^2))$. $\bullet$ We improve the running time from $\widetilde{O}(md^2 k^2)$ to $\widetilde{O}(m d^2 k)$. The proposed algorithm has theoretical guarantees and is analyzed to provide insights into the underlying structure of low-rank matrices and the nature of the linear measurements used in the recovery process. It advances the theoretical understanding of matrix sensing and provides a new approach for solving this important problem.

翻译：矩阵感知是信号处理与机器学习领域中的一个问题，旨在从一组线性测量中恢复低秩矩阵。其目标是在仅获得通过感知矩阵所得的线性测量集 [Jain, Netrapalli and Shanghavi, 2013] 的情况下，尽可能准确地重构原始矩阵。本文聚焦于矩阵感知的一个特定方向，即秩-1矩阵感知 [Zhong, Jain and Dhillon, 2015]。我们针对 [Zhong, Jain and Dhillon, 2015] 中的原始算法提出了改进。该改进基于一种新颖的分析与草图技术，能够在低秩矩阵恢复中实现更快的收敛速度和更高的精度。该算法致力于从理论上加深对矩阵感知问题的理解，并确立其相对于先前方法的优势。所提出的草图技术能够从线性测量中高效提取相关信息，使算法具备计算高效性和可扩展性。我们新颖的矩阵感知算法在以下两个方面改进了 [Zhong, Jain and Dhillon, 2015] 的先前结果：$\bullet$ 我们将样本复杂度从 $\widetilde{O}(\epsilon^{-2} dk^2)$ 降低至 $\widetilde{O}(\epsilon^{-2} (d+k^2))$。$\bullet$ 我们将运行时间从 $\widetilde{O}(md^2 k^2)$ 降低至 $\widetilde{O}(m d^2 k)$。所提出的算法具有理论保证，并通过分析揭示了低秩矩阵的底层结构以及恢复过程中所用线性测量的本质。该工作推进了矩阵感知的理论理解，并为解决这一重要问题提供了新方法。