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

Yuxin Chen1, Jianqing Fan2, Bingyan Wang2

  • 1Department of Electrical and Computer Engineering, Princeton University.

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Summary

This study shows that both convex relaxation and nonconvex optimization methods can achieve optimal accuracy for solving noisy bilinear systems. These findings improve theoretical understanding and offer better guarantees for these widely applicable techniques.

Keywords:
bilinear systems of equationsblind deconvolutionconvex relaxationleave-one-out analysisnonconvex optimization

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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.