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Abstract We study a class of Approximate Message Passing (AMP) algorithms for symmetric and rectangular spiked random matrix models with orthogonally invariant noise. The AMP iterates have fixed dimension $$K \geq 1$$, a multivariate non-linearity is applied in each AMP iteration, and the algorithm is spectrally initialized with $$K$$ super-critical sample eigenvectors. We derive the forms of the Onsager debiasing coefficients and corresponding AMP state evolution, which depend on the free cumulants of the noise spectral distribution. This extends previous results for such models with $K=1$ and an independent initialization. Applying this approach to Bayesian principal components analysis, we introduce a Bayes-OAMP algorithm that uses as its non-linearity the posterior mean conditional on all preceding AMP iterates. We describe a practical implementation of this algorithm, where all debiasing and state evolution parameters are estimated from the observed data, and we illustrate the accuracy and stability of this approach in simulations.
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This content will become publicly available on December 9, 2025
Diffusion Models With Learned Adaptive Noise
Diffusion models have gained traction as powerful algorithms for synthesizing high-quality images. Central to these algorithms is the diffusion process, a set of equations which maps data to noise in a way that can significantly affect performance. In this paper, we explore whether the diffusionprocess can be learned from data.Our work is grounded in Bayesian inference and seeks to improve log-likelihood estimation by casting the learned diffusion process as an approximate variational posterior that yields a tighter lower bound (ELBO) on the likelihood.A widely held assumption is that the ELBO is invariant to the noise process: our work dispels this assumption and proposes multivariate learned adaptive noise (MuLAN), a learned diffusion process that applies noise at different rates across an image. Our method consists of three components: a multivariate noise schedule, adaptive input-conditional diffusion, and auxiliary variables; these components ensure that the ELBO is no longer invariant to the choice of the noise schedule as in previous works. Empirically, MuLAN sets a new state-of-the-art in density estimation on CIFAR-10 and ImageNet while matching the performance of previous state-of-the-art models with 50% fewer steps. We provide the code, along with a blog post and video tutorial on the project page: https://s-sahoo.com/MuLAN
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- Award ID(s):
- 2046760
- PAR ID:
- 10577287
- Publisher / Repository:
- NeurIPS 2024
- Date Published:
- Format(s):
- Medium: X
- Location:
- Vancouver, BC, Canada
- Sponsoring Org:
- National Science Foundation
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