Beyond Procrustes: Balancing-Free Gradient Descent for Asymmetric Low-Rank Matrix Sensing

Low-rank matrix estimation plays a central role in various applications across science and engineering. Recently, nonconvex formulations based on factorization are provably solved by simple gradient descent algorithms with strong computational statistical guarantees. However, when the low-rank matrices asymmetric, existing approaches rely adding regularization term to balance scale of two factors which practice can be removed safely without hurting performance initialized via spectral method. In this paper, we provide theoretical justification for sensing problem, aims recover from small number linear measurements. As long as measurement ensemble satisfies restricted isometry property, -- conjunction initialization converges linearly need explicitly promoting balancedness factors; fact, stay balanced automatically throughout execution algorithm. Our analysis is analyzing evolution new distance metric that directly accounts ambiguity due invertible transforms, might independent interest.