Phase Diagram and Approximate Message Passing for Blind Calibration and Dictionary Learning
This work addresses the problem of signal recovery and calibration in high-dimensional settings for researchers in machine learning and signal processing, but it appears incremental as it builds on existing methods like replica analysis and message passing.
The authors tackled the problem of dictionary learning and blind calibration for random signals and matrices by analyzing the mean-squared error and phase transitions in large dimensions, and introduced an approximate message passing algorithm that matches theoretical performance and performs well in numerical tests for tractable system sizes.
We consider dictionary learning and blind calibration for signals and matrices created from a random ensemble. We study the mean-squared error in the limit of large signal dimension using the replica method and unveil the appearance of phase transitions delimiting impossible, possible-but-hard and possible inference regions. We also introduce an approximate message passing algorithm that asymptotically matches the theoretical performance, and show through numerical tests that it performs very well, for the calibration problem, for tractable system sizes.