We study matrix sensing, which is the problem of reconstructing a low-rank
matrix from a few linear measurements. It can be formulated as an
overparameterized regression problem, which can be solved by factorized
gradient descent when starting from a small random initialization.
Linear neural networks, and in particular matrix sensing by factorized
gradient descent, serve as prototypical models of non-convex problems in modern
machine learning, where complex phenomena can be disentangled and studied in
detail. Much research has been devoted to studying special cases of asymmetric
matrix sensing, such as asymmetric matrix factorization and symmetric positive
semi-definite matrix sensing.
Our key contribution is introducing a continuous differential equation that
we call the $\textit{perturbed gradient flow}$. We prove that the perturbed
gradient flow converges quickly to the true target matrix whenever the
perturbation is sufficiently bounded. The dynamics of gradient descent for
matrix sensing can be reduced to this formulation, yielding a novel proof of
asymmetric matrix sensing with factorized gradient descent. Compared to
directly analyzing the dynamics of gradient descent, the continuous formulation
allows bounding key quantities by considering their derivatives, often
simplifying the proofs. We believe the general proof technique may prove useful
in other settings as well.

矩阵感知是从少量线性测量中重建低秩矩阵的问题，我们引入了连续微分方程，称其为 “扰动梯度流”，通过边界足够有界的累计误差，证明扰动梯度流迅速收敛到真实目标矩阵，从而提供了一种基于梯度下降的非对称矩阵感知的新证明方法。