In this paper we focus on the issue of quantization.

Dear reviewers, thank you for your time to thoroughly read and review our paper. Y ou said "the addressed problem is relevant and timely" (R4), "useful in practice with many applications" (R1), with Quantifying for the first time the gap between train and test losses in the approach of Ballé et al. (see below) The proposed approach only marginally improves PSNR for hyperprior models. Note that in compression lightweight models are very relevant in practice . Secondly, our empirical results allow us for the first time to quantify the gap between training and test losses . Finally, universal quantization has the potential to lead to much bigger improvements in the future .

Summary and Contributions: Neural network based compressors usually apply additive uniform noise during training as a proxy for the quantization that is performed during test-time. This creates a mismatch between the training and testing phases. This work proposes to instead apply universal quantization at test time thus eliminating the mismatch between training and test phases while maintaining a differentiable loss function. It is based on the fact that adding uniform noise to an input x is equivalent to subtracting a uniform random variable from x, rounding the result and then adding the same uniform random variable back. As a result, by sharing a random seed across the encoder and decoder we can easily implement universal quantization for neural network based compressors.

Stochastic encoders have been used in rate-distortion theory and neural compression because they can be easier to handle. However, in performance comparisons with deterministic encoders they often do worse, suggesting that noise in the encoding process may generally be a bad idea. It is poorly understood if and when stochastic encoders do better than deterministic encoders. In this paper we provide one illustrative example which shows that stochastic encoders can significantly outperform the best deterministic encoders. Our toy example suggests that stochastic encoders may be particularly useful in the regime of "perfect perceptual quality".

A popular approach to learning encoders for lossy compression is to use additive uniform noise during training as a differentiable approximation to test-time quantization. We demonstrate that a uniform noise channel can also be implemented at test time using universal quantization (Ziv, 1985). This allows us to eliminate the mismatch between training and test phases while maintaining a completely differentiable loss function. Implementing the uniform noise channel is a special case of the more general problem of communicating a sample, which we prove is computationally hard if we do not make assumptions about its distribution. However, the uniform special case is efficient as well as easy to implement and thus of great interest from a practical point of view. Finally, we show that quantization can be obtained as a limiting case of a soft quantizer applied to the uniform noise channel, bridging compression with and without quantization.

The recent framework of compressive statistical learning aims at designing tractable learning algorithms that use only a heavily compressed representation-or sketch-of massive datasets. Compressive K-Means (CKM) is such a method: it estimates the centroids of data clusters from pooled, non-linear, random signatures of the learning examples. While this approach significantly reduces computational time on very large datasets, its digital implementation wastes acquisition resources because the learning examples are compressed only after the sensing stage. The present work generalizes the sketching procedure initially defined in Compressive K-Means to a large class of periodic nonlinearities including hardware-friendly implementations that compressively acquire entire datasets. This idea is exemplified in a Quantized Compressive K-Means procedure, a variant of CKM that leverages 1-bit universal quantization (i.e. retaining the least significant bit of a standard uniform quantizer) as the periodic sketch nonlinearity. Trading for this resource-efficient signature (standard in most acquisition schemes) has almost no impact on the clustering performances, as illustrated by numerical experiments.