Convex and Nonconvex Optimization Are Both Minimax-Optimal for Noisy Blind Deconvolution under Random Designs
This provides foundational theoretical insights for optimization methods in signal processing and machine learning, addressing a key bottleneck in noisy blind deconvolution.
The paper tackled the problem of solving bilinear systems of equations under random designs with noise, showing that both convex relaxation and a two-stage nonconvex algorithm achieve minimax-optimal statistical accuracy, significantly improving upon prior theoretical guarantees.
We investigate the effectiveness of convex relaxation and nonconvex optimization in solving bilinear systems of equations under two different designs (i.e.$~$a sort of random Fourier design and Gaussian design). Despite the wide applicability, the theoretical understanding about these two paradigms remains largely inadequate in the presence of random noise. The current paper makes two contributions by demonstrating that: (1) a two-stage nonconvex algorithm attains minimax-optimal accuracy within a logarithmic number of iterations. (2) convex relaxation also achieves minimax-optimal statistical accuracy vis-à-vis random noise. Both results significantly improve upon the state-of-the-art theoretical guarantees.