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Register a new userOn the advantages of stochastic encoders
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Added by
Lucas Theis
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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.
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We derive the capacity region of the Gaussian version of Willemss two-user MAC with conferencing encoders. This setting differs from the classical MAC in that, prior to each transmission block, the two transmitters can communicate with each other over noise-free bit-pipes of given capacities. The derivation requires a new technique for proving the optimality of Gaussian input distributions in certain mutual information maximizations under a Markov constraint. We also consider a Costa-type extension of the Gaussian MAC with conferencing encoders. In this extension, the channel can be described as a two-user MAC with Gaussian noise and Gaussian interference where the interference is known non-causally to the encoders but not to the decoder. We show that as in Costas setting the interference sequence can be perfectly canceled, i.e., that the capacity region without interference can be achieved.
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A single-letter characterization is provided for the capacity region of finite-state multiple-access channels, when the channel state process is an independent and identically distributed sequence, the transmitters have access to partial (quantized) state information, and complete channel state information is available at the receiver. The partial channel state information is assumed to be asymmetric at the encoders. As a main contribution, a tight converse coding theorem is presented. The difficulties associated with the case when the channel state has memory are discussed and connections to decentralized stochastic control theory are presented.
In 1959, Renyi proposed the information dimension and the $d$-dimensional entropy to measure the information content of general random variables. This paper proposes a generalization of information dimension to stochastic processes by defining the information dimension rate as the entropy rate of the uniformly-quantized stochastic process divided by minus the logarithm of the quantizer step size $1/m$ in the limit as $mtoinfty$. It is demonstrated that the information dimension rate coincides with the rate-distortion dimension, defined as twice the rate-distortion function $R(D)$ of the stochastic process divided by $-log(D)$ in the limit as $Ddownarrow 0$. It is further shown that, among all multivariate stationary processes with a given (matrix-valued) spectral distribution function (SDF), the Gaussian process has the largest information dimension rate, and that the information dimension rate of multivariate stationary Gaussian processes is given by the average rank of the derivative of the SDF. The presented results reveal that the fundamental limits of almost zero-distortion recovery via compressible signal pursuit and almost lossless analog compression are different in general.
Compressed sensing (CS) is about recovering a structured signal from its under-determined linear measurements. Starting from sparsity, recovery methods have steadily moved towards more complex structures. Emerging machine learning tools such as generative functions that are based on neural networks are able to learn general complex structures from training data. This makes them potentially powerful tools for designing CS algorithms. Consider a desired class of signals $cal Q$, ${cal Q}subset{R}^n$, and a corresponding generative function $g:{cal U}^krightarrow {R}^n$, ${cal U}subset {R}$, such that $sup_{{bf x}in {cal Q}}min_{{bf u}in{cal U}^k}{1over sqrt{n}}|g({bf u})-{bf x}|leq delta$. A recovery method based on $g$ seeks $g({bf u})$ with minimum measurement error. In this paper, the performance of such a recovery method is studied, under both noisy and noiseless measurements. In the noiseless case, roughly speaking, it is proven that, as $k$ and $n$ grow without bound and $delta$ converges to zero, if the number of measurements ($m$) is larger than the input dimension of the generative model ($k$), then asymptotically, almost lossless recovery is possible. Furthermore, the performance of an efficient iterative algorithm based on projected gradient descent is studied. In this case, an auto-encoder is used to define and enforce the source structure at the projection step. The auto-encoder is defined by encoder and decoder (generative) functions $f:{R}^nto{cal U}^k$ and $g:{cal U}^kto{R}^n$, respectively. We theoretically prove that, roughly, given $m>40klog{1over delta}$ measurements, such an algorithm converges to the vicinity of the desired result, even in the presence of additive white Gaussian noise. Numerical results exploring the effectiveness of the proposed method are presented.
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