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We study matrix sensing, which is the problem of reconstructing a low-rankmatrix from a few linear measurements. It can be formulated as anoverparameterized regression problem, which can be solved by factorizedgradient descent when starting from a small random initialization.  Linear neural networks, and in particular matrix sensing by factorizedgradient descent, serve as prototypical models of non-convex problems in modernmachine learning, where complex phenomena can be disentangled and studied indetail. Much research has been devoted to studying special cases of asymmetricmatrix sensing, such as asymmetric matrix factorization and symmetric positivesemi-definite matrix sensing.  Our key contribution is introducing a continuous differential equation thatwe call the $\textit{perturbed gradient flow}$. We prove that the perturbedgradient flow converges quickly to the true target matrix whenever theperturbation is sufficiently bounded. The dynamics of gradient descent formatrix sensing can be reduced to this formulation, yielding a novel proof ofasymmetric matrix sensing with factorized gradient descent. Compared todirectly analyzing the dynamics of gradient descent, the continuous formulationallows bounding key quantities by considering their derivatives, oftensimplifying the proofs. We believe the general proof technique may prove usefulin other settings as well.

Asymmetric matrix sensing by gradient descent with small random initialization