🤖 AI Summary
This study addresses the failure of classical state evolution in approximate message passing (AMP) when employing discontinuous denoisers. We propose a modified Onsager coefficient incorporating a jump-density-weighted boundary term. Through Lipschitz approximation analysis and iterative optimization over Gaussian matrices, this work overcomes traditional smoothness assumptions and rigorously establishes that the modified AMP still adheres to state evolution theory. Simulations demonstrate that the proposed coefficient enables hard-thresholding AMP to approach Bayes-optimal performance, effectively correcting both the analytical discrepancies and algorithmic failures induced by standard coefficients in non-smooth settings.
📝 Abstract
We consider approximate message passing (AMP) with discontinuous denoisers, such as hard slicers, quantizers and hard thresholding, for which the classical state evolution (SE) analysis does not apply. It is first shown that the standard Onsager coefficient; i.e., the average pointwise derivative of the denoiser, misses a boundary term, namely, the sum over the jumps of the jump size times the density of the denoiser input at the jump. Then, for real independent and identically distributed (i.i.d.) Gaussian matrices, a positive noise variance, a fixed number of iterations and a prior with finite second moment, we transfer the Lipschitz SE theorem directly to denoisers, fixed in advance, that are continuously differentiable with bounded derivative except at finitely many jumps, by bounding the distance between the iterates of AMP and of AMP on Lipschitz ramps of the denoisers through the fraction of inputs that cross a jump. It is thereby proven that AMP follows SE, also for test functions such as symbol errors, if its Onsager coefficient contains the boundary term, evaluated either along SE or, as we propose, at the noise level estimated from the residual, with the density computed from the known prior. Simulation results with 4- and 8-level pulse amplitude modulation (PAM) demonstrate that AMP with a hard slicer, whose standard coefficient is identically zero, fails with this coefficient, while with the proposed one, it follows SE and comes close to Bayes-optimal AMP.