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Multi-sample, importance-weighted variational autoencoders (IWAE) givetighter bounds and more accurate uncertainty estimates than variationalautoencoders (VAE) trained with a standard single-sample objective. However,IWAEs scale poorly: as the latent dimensionality grows, they requireexponentially many samples to retain the benefits of importance weighting.While sequential Monte-Carlo (SMC) can address this problem, it isprohibitively slow because the resampling step imposes sequential structurewhich cannot be parallelised, and moreover, resampling is non-differentiablewhich is problematic when learning approximate posteriors. To address theseissues, we developed tensor Monte-Carlo (TMC) which gives exponentially manyimportance samples by separately drawing $K$ samples for each of the $n$ latentvariables, then averaging over all $K^n$ possible combinations. While the sumover exponentially many terms might seem to be intractable, in many cases itcan be computed efficiently as a series of tensor inner-products. We show thatTMC is superior to IWAE on a generative model with multiple stochastic layerstrained on the MNIST handwritten digit database, and we show that TMC can becombined with standard variance reduction techniques.

Tensor Monte Carlo: particle methods for the GPU era