Matrix sensing has many real-world applications in science and engineering, such as system control, distance embedding, and computer vision. The goal of matrix sensing is to recover a matrix $A_\star \in \mathbb{R}^{n \times n}$, based on a sequence of measurements $(u_i,b_i) \in \mathbb{R}^{n} \times \mathbb{R}$ such that $u_i^\top A_\star u_i = b_i$. Previous work [ZJD15] focused on the scenario where matrix $A_{\star}$ has a small rank, e.g. rank-$k$. Their analysis heavily relies on the RIP assumption, making it unclear how to generalize to high-rank matrices. In this paper, we relax that rank-$k$ assumption and solve a much more general matrix sensing problem. Given an accuracy parameter $\delta \in (0,1)$, we can compute $A \in \mathbb{R}^{n \times n}$ in $\widetilde{O}(m^{3/2} n^2 \delta^{-1} )$, such that $ |u_i^\top A u_i - b_i| \leq \delta$ for all $i \in [m]$. We design an efficient algorithm with provable convergence guarantees using stochastic gradient descent for this problem.

翻译：矩阵感知在科学与工程领域具有广泛的实际应用，例如系统控制、距离嵌入和计算机视觉。其目标是基于一组测量值序列 $(u_i,b_i) \in \mathbb{R}^{n} \times \mathbb{R}$（满足 $u_i^\top A_\star u_i = b_i$）来恢复矩阵 $A_\star \in \mathbb{R}^{n \times n}$。先前的工作 [ZJD15] 主要关注矩阵 $A_{\star}$ 具有低秩（如秩-$k$）的情况。其分析严重依赖于 RIP 假设，使得该方法的推广到高秩矩阵变得困难。本文放宽了这一秩-$k$ 假设，并解决了一个更通用的矩阵感知问题。给定精度参数 $\delta \in (0,1)$，我们可以在 $\widetilde{O}(m^{3/2} n^2 \delta^{-1})$ 时间内计算出矩阵 $A \in \mathbb{R}^{n \times n}$，使得对所有 $i \in [m]$ 满足 $ |u_i^\top A u_i - b_i| \leq \delta$。我们为此问题设计了一种基于随机梯度下降的高效算法，并给出了其可证明的收敛性保证。