Convex and Nonconvex Optimization Are Both Minimax-Optimal for Noisy Blind Deconvolution

We investigate the effectiveness of convex relaxation and nonconvex optimization in solving bilinear systems of equations (a.k.a. blind deconvolution under a subspace model). Despite the wide applicability, the theoretical understanding about these two paradigms remains largely inadequate in the presence of noise. The current paper makes two contributions by demonstrating that: (1) convex relaxation achieves minimax-optimal statistical accuracy vis-\`a-vis random noise, and (2) a two-stage nonconvex algorithm attains minimax-optimal accuracy within a logarithmic number of iterations. Both results improve upon the state-of-the-art results by some factors that scale polynomially in the problem dimension.