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

Area of Science:

• Optimization Theory
• Statistical Inference
• Applied Mathematics

Background:

• Bilinear systems of equations are widely applicable but lack sufficient theoretical understanding, especially with random noise.
• Existing methods for solving these systems have limitations in their theoretical guarantees.

Purpose of the Study:

• To investigate the effectiveness of convex relaxation and nonconvex optimization for solving bilinear systems with random noise.
• To provide improved theoretical guarantees for these optimization paradigms.

Main Methods:

• Analysis of a two-stage nonconvex algorithm.
• Evaluation of convex relaxation techniques.
• Consideration of two distinct random designs: random Fourier and Gaussian designs.

Main Results:

• A two-stage nonconvex algorithm achieves minimax-optimal accuracy in a logarithmic number of iterations.
• Convex relaxation also attains minimax-optimal statistical accuracy in the presence of random noise.
• Both methods offer significant improvements over current state-of-the-art theoretical guarantees.

Conclusions:

• The study provides enhanced theoretical understanding for solving noisy bilinear systems.
• Both convex relaxation and nonconvex optimization are effective and statistically accurate methods for this problem.
• The findings pave the way for more robust applications of these techniques.