Multimeasurement Generative Models
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We formally map the problem of sampling from an unknown distribution with density p_X in ℝ^d to the problem of learning and sampling p_𝐘 in ℝ^Md obtained by convolving p_X with a fixed factorial kernel: p_𝐘 is referred to as M-density and the factorial kernel as multimeasurement noise model (MNM). The M-density is smoother than p_X, easier to learn and sample from, yet for large M the two problems are mathematically equivalent since X can be estimated exactly given 𝐘=𝐲 using the Bayes estimator x(𝐲)=𝔼[X|𝐘=𝐲]. To formulate the problem, we derive x(𝐲) for Poisson and Gaussian MNMs expressed in closed form in terms of unnormalized p_𝐘. 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 from p_X are obtained by walk-jump sampling (Saremi Hyvarinen, 2019) via underdamped Langevin MCMC (walk) to sample from p_𝐘 and the multimeasurement Bayes estimation of X (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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