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In the theory of lossy compression, the rate-distortion (R-D) function R(D) describes how much a data source can be compressed (in bit-rate) at any given level of fidelity (distortion). Obtaining R(D) for a given data source establishes the fundamental performance limit for all compression algorithms. We propose a new method to estimate R(D) from the perspective of optimal transport. Unlike the classic Blahut--Arimoto algorithm which fixes the support of the reproduction distribution in advance, our Wasserstein gradient descent algorithm learns the support of the optimal reproduction distribution by moving particles. We prove its local convergence and analyze the sample complexity of our R-D estimator based on a connection to entropic optimal transport. Experimentally, we obtain comparable or tighter bounds than state-of-the-art neural network methods on low-rate sources while requiring considerably less tuning and computation effort. We also highlight a connection to maximum-likelihood deconvolution and introduce a new class of sources that can be used as test cases with known solutions to the R-D problem. 

Abstract In many scientific fields which rely on statistical inference, simulations are often used to map from theoretical models to experimental data, allowing scientists to test model predictions against experimental results. Experimental data is often reconstructed from indirect measurements causing the aggregate transformation from theoretical models to experimental data to be poorly-described analytically. Instead, numerical simulations are used at great computational cost. We introduce Optimal-Transport-based Unfolding and Simulation (OTUS), a fast simulator based on unsupervised machine-learning that is capable of predicting experimental data from theoretical models. Without the aid of current simulation information, OTUS trains a probabilistic autoencoder to transform directly between theoretical models and experimental data. Identifying the probabilistic autoencoder’s latent space with the space of theoretical models causes the decoder network to become a fast, predictive simulator with the potential to replace current, computationally-costly simulators. Here, we provide proof-of-principle results on two particle physics examples, Z -boson and top-quark decays, but stress that OTUS can be widely applied to other fields. 

Recent work by Marino et al. (2020) showed improved performance in sequential density estimation by combining masked autoregressive flows with hierarchical latent variable models. We draw a connection between such autoregressive generative models and the task of lossy video compression. Specifically, we view recent neural video compression methods (Lu et al., 2019; Yang et al., 2020b; Agustsson et al., 2020) as instances of a generalized stochastic temporal autoregressive transform, and propose avenues for enhancement based on this insight. Comprehensive evaluations on large-scale video data show improved rate-distortion performance over both state-of-the-art neural and conventional video compression methods. 

We consider the problem of lossy image compression with deep latent variable models. State-of-the-art methods [Ballé et al., 2018, Minnen et al., 2018, Lee et al., 2019] build on hierarchical variational autoencoders (VAEs) and learn inference networks to predict a compressible latent representation of each data point. Drawing on the variational inference perspective on compression [Alemi et al., 2018], we identify three approximation gaps which limit performance in the conventional approach: an amortization gap, a discretization gap, and a marginalization gap. We propose remedies for each of these three limitations based on ideas related to iterative inference, stochastic annealing for discrete optimization, and bits-back coding, resulting in the first application of bits-back coding to lossy compression. In our experiments, which include extensive baseline comparisons and ablation studies, we achieve new state-of-the-art performance on lossy image compression using an established VAE architecture, by changing only the inference method. 

We propose a novel algorithm for quantizing continuous latent representations in trained models. Our approach applies to deep probabilistic models, such as variational autoencoders (VAEs), and enables both data and model compression. Unlike current end-to-end neural compression methods that cater the model to a fixed quantization scheme, our algorithm separates model design and training from quantization. Consequently, our algorithm enables “plug-and-play” compression with variable rate-distortion trade-off, using a single trained model. Our algorithm can be seen as a novel extension of arithmetic coding to the continuous domain, and uses adaptive quantization accuracy based on estimates of posterior uncertainty. Our experimental results demonstrate the importance of taking into account posterior uncertainties, and show that image compression with the proposed algorithm outperforms JPEG over a wide range of bit rates using only a single standard VAE. Further experiments on Bayesian neural word embeddings demonstrate the versatility of the proposed method. 

Lifted inference algorithms exploit model symmetry to reduce computational cost in probabilistic inference. However, most existing lifted inference algorithms operate only over discrete domains or continuous domains with restricted potential functions. We investigate two approximate lifted variational approaches that apply to domains with general hybrid potentials, and are expressive enough to capture multi-modality. We demonstrate that the proposed variational methods are highly scalable and can exploit approximate model symmetries even in the presence of a large amount of continuous evidence, outperforming existing message-passing-based approaches in a variety of settings. Additionally, we present a sufficient condition for the Bethe variational approximation to yield a non-trivial estimate over the marginal polytope.