Publications by Michael Christoph Gastpar

A Unified Discretization Approach to Compute–Forward: From Discrete to Continuous Inputs

Chen Feng, Michael Christoph Gastpar, Sung Hoon Lim, Adriano Pastore

Compute–forward is a coding technique that enables receiver(s) in a network to directly decode one or more linear combinations of the transmitted codewords. Initial efforts focused on Gaussian channels and derived achievable rate regions via nested lattice codes and single-user (lattice) decoding as well as sequential (lattice) decoding. Recently, these results have been generalized to discrete memoryless channels via nested linear codes and joint typicality coding, culminating in a simultaneous-decoding rate region for recovering one or more linear combinations from K users. Using a discretization approach, this paper translates this result into a simultaneous-decoding rate region for a wide class of continuous memoryless channels, including the important special case of Gaussian channels. Additionally, this paper derives a single, unified expression for both discrete and continuous rate regions via an algebraic generalization of Rényi’s information dimension.

2022

Alpha-NML Universal Predictors

Marco Bondaschi, Michael Christoph Gastpar

Inspired by Sibson’s alpha-mutual information, we introduce a new parametric class of universal predictors. This class interpolates two well-known predictors, the mixture estimator, that includes the Laplace and the Krichevsky-Trofimov predictors, and the Normalized Maximum Likelihood (NML) estimator. We point out some advantages of this class of predictors and study its performance in terms of known regret measures under logarithmic loss, in particular for the well-studied case of discrete memoryless sources.

2022

Differential Entropy of the Conditional Expectation Under Additive Gaussian Noise

Ahmet Arda Atalik, Michael Christoph Gastpar, Alper Köse

The conditional mean is a fundamental and important quantity whose applications include the theories of estimation and rate-distortion. It is also notoriously difficult to work with. This paper establishes novel bounds on the differential entropy of the conditional mean in the case of finite-variance input signals and additive Gaussian noise. The main result is a new lower bound in terms of the differential entropies of the input signal and the noisy observation. The main results are also extended to the vector Gaussian channel and to the natural exponential family. Various other properties such as upper bounds, asymptotics, Taylor series expansion, and connection to Fisher Information are obtained. Two applications of the lower bound in the remote-source coding and CEO problem are discussed.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2022

Finite Littlestone Dimension Implies Finite Information Complexity

Michael Christoph Gastpar, Ido Nachum, Aditya Pradeep

We prove that every online learnable class of functions of Littlestone dimension d admits a learning algorithm with finite information complexity. Towards this end, we use the notion of a globally stable algorithm. Generally, the information complexity of such a globally stable algorithm is large yet finite, roughly exponential in d. We also show there is room for improvement; for a canonical online learnable class, indicator functions of affine subspaces of dimension d, the information complexity can be upper bounded logarithmically in d.

2022

From Generalisation Error to Transportation-cost Inequalities and Back

Amedeo Roberto Esposito, Michael Christoph Gastpar

In this work, we connect the problem of bounding the expected generalisation error with transportation-cost inequalities. Exposing the underlying pattern behind both approaches we are able to generalise them and go beyond Kullback- Leibler Divergences/Mutual Information and sub-Gaussian measures. In particular, we are able to provide a result showing the equivalence between two families of inequalities: one involving functionals and one involving measures. This result generalises the one proposed by Bobkov and Götze that connects transportation-cost inequalities with concentration of measure. Moreover, it allows us to recover all standard generalisation error bounds involving mutual information and to introduce new, more general bounds, that involve arbitrary divergence measures.

2022

Lower-bounds on the Bayesian Risk in Estimation Procedures via f–Divergences

Amedeo Roberto Esposito, Michael Christoph Gastpar, Adrien Vandenbroucque

We consider the problem of parameter estimation in a Bayesian setting and propose a general lower-bound that includes part of the family of f-Divergences. The results are then applied to specific settings of interest and compared to other notable results in the literature. In particular, we show that the known bounds using Mutual Information can be improved by using, for example, Maximal Leakage, Hellinger divergence, or generalizations of the Hockey-Stick divergence.

2022

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2022

Shannon Bounds on Lossy Gray-Wyner Networks

Michael Christoph Gastpar, Erixhen Sula

The Gray-Wyner network subject to a fidelity criterion is studied. Upper and lower bounds for the trade-offs between the private sum-rate and the common rate are obtained for arbitrary sources subject to mean-squared error distortion. The bounds meet exactly, leading to the computation of the rate region, when the source is jointly Gaussian. They meet partially when the sources are modeled via an additive Gaussian “channel”. The bounds are inspired from the Shannon bounds on the rate-distortion problem.

2022

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This paper presents explicit solutions for two related non-convex information extremization problems due to Gray and Wyner in the Gaussian case. The first problem is the Gray-Wyner network subject to a sum-rate constraint on the two private links. Here, our argument establishes the optimality of Gaussian codebooks and hence, a closed-form formula for the optimal rate region. The second problem is Wyner's common information and a generalization thereof, where conditional independence is generalized to a limit on the conditional mutual information. We present full explicit solutions for the scalar as well as the vector case.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2022

The Price of Distributed: Rate Loss in the CEO Problem

Ahmet Arda Atalik, Michael Christoph Gastpar, Alper Köse

In the distributed remote (CEO) source coding problem, many separate encoders observe independently noisy copies of an underlying source. The rate loss is the difference between the rate required in this distributed setting and the rate that would be required in a setting where the encoders can fully cooperate. In this sense, the rate loss characterizes the price of distributed processing. We survey and extend the known results on the rate loss in various settings, with a particular emphasis on the case where the noise in the observations is Gaussian, but the underlying source is general.

2022