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Christophe Schülke1, Philip Schniter2, Lenka Zdeborová3

  • 1Laboratoire de Physique Statistique, CNRS, PSL Universités et Ecole Normale Supérieure, 75005, Paris, France and Institut de Physique Théorique, CNRS, CEA, Université Paris-Saclay, 91191, Gif-sur-Yvette, France.

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This study analyzes matrix compressed sensing using a Bayes-optimal inference procedure. We reveal phase transitions impacting problem solvability and link matrix recovery to matrix factorization.

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Area of Science:

  • Signal Processing
  • Information Theory
  • Machine Learning

Background:

  • Matrix compressed sensing aims to recover low-rank matrices from limited, noisy linear measurements.
  • Bayes-optimal inference procedures are crucial for accurate matrix recovery in such scenarios.

Purpose of the Study:

  • To analyze the asymptotic performance of a Bayes-optimal inference procedure for low-rank matrix recovery.
  • To investigate the state evolution of the Parametric Bilinear Generalized Approximate Message Passing (P-BiG-AMP) algorithm.
  • To understand the implications of phase transitions on problem solvability in matrix compressed sensing.

Main Methods:

  • Utilizing the replica method to analyze the asymptotic performance of the inference procedure.
  • Describing the state evolution of the P-BiG-AMP algorithm.
  • Comparing theoretical analysis with numerical performance of P-BiG-AMP.

Main Results:

  • Identified two distinct types of phase transitions in matrix compressed sensing.
  • Demonstrated that the asymptotic replica equations for matrix compressed sensing are identical to those for matrix factorization.
  • Showcased the theoretical analysis aligning with the numerical performance of P-BiG-AMP.

Conclusions:

  • The replica method provides a powerful tool for analyzing complex inference problems in signal processing.
  • The identified phase transitions offer insights into the fundamental limits of low-rank matrix recovery.
  • The equivalence of replica equations suggests a unified theoretical framework for matrix recovery and factorization problems.