A General Algorithm for Solving Rank-one Matrix Sensing
This addresses a more general matrix sensing problem for applications in system control and computer vision, though it appears incremental as it builds on prior work by removing the low-rank restriction.
The paper tackles the matrix sensing problem by relaxing the low-rank assumption of previous work and provides an algorithm that computes a matrix A in time O~(m^{3/2} n^2 δ^{-1}) such that the measurement error is bounded by δ for all measurements.
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 $δ\in (0,1)$, we can compute $A \in \mathbb{R}^{n \times n}$ in $\widetilde{O}(m^{3/2} n^2 δ^{-1} )$, such that $ |u_i^\top A u_i - b_i| \leq δ$ for all $i \in [m]$. We design an efficient algorithm with provable convergence guarantees using stochastic gradient descent for this problem.