Papers › Multimeasurement Generative Models

Multimeasurement Generative Models

18 Dec 2021ICLR 2022 4arXiv:2112.09822archive 2025-07-28

Saeed Saremi, Rupesh Kumar Srivastava

We formally map the problem of sampling from an unknown distribution with a density in ℝᵈ to the problem of learning and sampling a smoother density in ℝᴹᵈ obtained by convolution with a fixed factorial kernel: the new density is referred to as M-density and the kernel as multimeasurement noise model (MNM). The M-density in ℝᴹᵈ is smoother than the original density in ℝᵈ, easier to learn and sample from, yet for large M the two problems are mathematically equivalent since clean data can be estimated exactly given a multimeasurement noisy observation using the Bayes estimator. To formulate the problem, we derive the Bayes estimator for Poisson and Gaussian MNMs in closed form in terms of the unnormalized M-density. This leads to a simple least-squares objective for learning parametric energy and score functions. We present various parametrization schemes of interest including one in which studying Gaussian M-densities directly leads to multidenoising autoencoders--this is the first theoretical connection made between denoising autoencoders and empirical Bayes in the literature. Samples in ℝᵈ are obtained by walk-jump sampling (Saremi & Hyvarinen, 2019) via underdamped Langevin MCMC (walk) to sample from M-density and the multimeasurement Bayes estimation (jump). We study permutation invariant Gaussian M-densities on MNIST, CIFAR-10, and FFHQ-256 datasets, and demonstrate the effectiveness of this framework for realizing fast-mixing stable Markov chains in high dimensions.

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