For Gaussian matrices, generalized first-order methods and AMP are close to a conditionally Gaussian comparison process at dimension-free error, under Lipschitz and moment-matching conditions.
Generalized approximate message passing for estimation with random linear mixing,
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Dimension-Free Bounds for Generalized First-Order Methods via Gaussian Coupling
For Gaussian matrices, generalized first-order methods and AMP are close to a conditionally Gaussian comparison process at dimension-free error, under Lipschitz and moment-matching conditions.