aiqbench analysis
This research extends the universality of state evolution for Approximate Message Passing (AMP) algorithms beyond Gaussian assumptions by establishing conditions under which non-separable non-linearities admit universal guarantees for general data distributions. By identifying a Bounded Composition Property, the authors demonstrate that common classes of denoisers and spectral methods maintain precise mean-field characterizations even with non-Gaussian entries. This theoretical advance strengthens the foundational understanding of iterative learning dynamics in high-dimensional statistical inference.
On Universality of Non-Separable Approximate Message Passing Algorithms
arXiv:2506.23010v2 Announce Type: replace-cross Abstract: Mean-field characterizations of first-order iterative algorithms -- including Approximate Message Passing (AMP), stochastic and proximal gradient descent, and Langevin diffusions -- have enabled a precise understanding of learning dynamics in many statistical applications. For algorithms whose non-linearities have a coordinate-separable form, it is known that such characterizations enjoy a degree of universality with respect to the underlying data distribution. However, mean-field characterizations of non-separable algorithm dynamics have largely remained restricted to i.i.d. Gaussian or rotationally-invariant data. In this work, we initiate a study of universality for non-separable AMP algorithms. We identify a general condition for AMP with polynomial non-linearities, in terms of a Bounded Composition Property (BCP) for their representing tensors, to admit a state evolution that holds universally for matrices with non-Gaussian entries. We then formalize a condition of BCP-approximability for Lipschitz AMP algorithms to enjoy a similar universal guarantee. We demonstrate that many common classes of non-separable non-linearities are BCP-approximable, including local denoisers, spectral denoisers for generic signals, and compositions of separable functions with generic linear maps, implying the universality of state evolution for AMP algorithms employing these non-linearities.
Abstract:Mean-field characterizations of first-order iterative algorithms -- including Approximate Message Passing (AMP), stochastic and proximal gradient descent, and Langevin diffusions -- have enabled a precise understanding of learning dynamics in many statistical applications. For algorithms whose non-linearities have a coordinate-separable form, it is known that such characterizations enjoy a degree of universality with respect to the underlying data distribution. However, mean-field characterizations of non-separable algorithm dynamics have largely remained restricted to i.i.d. Gaussian or rotationally-invariant data. In this work, we initiate a study of universality for non-separable AMP algorithms. We identify a general condition for AMP with polynomial non-linearities, in terms of a Bounded Composition Property (BCP) for their representing tensors, to admit a state evolution that holds universally for matrices with non-Gaussian entries. We then formalize a condition of BCP-approximability for Lipschitz AMP algorithms to enjoy a similar universal guarantee. We demonstrate that many common classes of non-separable non-linearities are BCP-approximable, including local denoisers, spectral denoisers for generic signals, and compositions of separable functions with generic linear maps, implying the universality of state evolution for AMP algorithms employing these non-linearities.
Statistics Theory (math.ST); Information Theory (cs.IT); Machine Learning (cs.LG); Probability (math.PR)
arXiv:2506.23010 [math.ST]
(or arXiv:2506.23010v2 [math.ST] for this version)
https://doi.org/10.48550/arXiv.2506.23010 Focus to learn more arXiv-issued DOI via DataCite
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