Equality in the Matrix Entropy-Power Inequality and Blind Separation of Real and Complex sources

The matrix version of the entropy-power inequality for real or complex coefficients and variables is proved using a transportation argument that easily settles the equality case. An application to blind source extraction is given.Comment: 5 pages, in Proc. 2019 IEEE International Symposium on Information Theory (ISIT 2019), Paris, France, July 7-12, 201

On Codes for the Noisy Substring Channel

We consider the problem of coding for the substring channel, in which information strings are observed only through their (multisets of) substrings. Because of applications to DNA-based data storage, due to DNA sequencing techniques, interest in this channel has renewed in recent years. In contrast to existing literature, we consider a noisy channel model, where information is subject to noise \emph{before} its substrings are sampled, motivated by in-vivo storage. We study two separate noise models, substitutions or deletions. In both cases, we examine families of codes which may be utilized for error-correction and present combinatorial bounds. Through a generalization of the concept of repeat-free strings, we show that the added required redundancy due to this imperfect observation assumption is sublinear, either when the fraction of errors in the observed substring length is sufficiently small, or when that length is sufficiently long. This suggests that no asymptotic cost in rate is incurred by this channel model in these cases.Comment: ISIT 2021 version (including all proofs

Slope and generalization properties of neural networks

Neural networks are very successful tools in for example advanced classification. From a statistical point of view, fitting a neural network may be seen as a kind of regression, where we seek a function from the input space to a space of classification probabilities that follows the "general" shape of the data, but avoids overfitting by avoiding memorization of individual data points. In statistics, this can be done by controlling the geometric complexity of the regression function. We propose to do something similar when fitting neural networks by controlling the slope of the network. After defining the slope and discussing some of its theoretical properties, we go on to show empirically in examples, using ReLU networks, that the distribution of the slope of a well-trained neural network classifier is generally independent of the width of the layers in a fully connected network, and that the mean of the distribution only has a weak dependence on the model architecture in general. The slope is of similar size throughout the relevant volume, and varies smoothly. It also behaves as predicted in rescaling examples. We discuss possible applications of the slope concept, such as using it as a part of the loss function or stopping criterion during network training, or ranking data sets in terms of their complexity

Convergence of Smoothed Empirical Measures with Applications to Entropy Estimation

This paper studies convergence of empirical measures smoothed by a Gaussian kernel. Specifically, consider approximating $P\ast\mathcal{N}_\sigma$, for $\mathcal{N}_\sigma\triangleq\mathcal{N}(0,\sigma^2 \mathrm{I}_d)$, by $\hat{P}_n\ast\mathcal{N}_\sigma$, where $\hat{P}_n$ is the empirical measure, under different statistical distances. The convergence is examined in terms of the Wasserstein distance, total variation (TV), Kullback-Leibler (KL) divergence, and $\chi^2$-divergence. We show that the approximation error under the TV distance and 1-Wasserstein distance ($\mathsf{W}_1$) converges at rate $e^{O(d)}n^{-\frac{1}{2}}$ in remarkable contrast to a typical $n^{-\frac{1}{d}}$ rate for unsmoothed $\mathsf{W}_1$ (and $d\ge 3$). For the KL divergence, squared 2-Wasserstein distance ($\mathsf{W}_2^2$), and $\chi^2$-divergence, the convergence rate is $e^{O(d)}n^{-1}$, but only if $P$ achieves finite input-output $\chi^2$ mutual information across the additive white Gaussian noise channel. If the latter condition is not met, the rate changes to $\omega(n^{-1})$ for the KL divergence and $\mathsf{W}_2^2$, while the $\chi^2$-divergence becomes infinite - a curious dichotomy. As a main application we consider estimating the differential entropy $h(P\ast\mathcal{N}_\sigma)$ in the high-dimensional regime. The distribution $P$ is unknown but $n$ i.i.d samples from it are available. We first show that any good estimator of $h(P\ast\mathcal{N}_\sigma)$ must have sample complexity that is exponential in $d$. Using the empirical approximation results we then show that the absolute-error risk of the plug-in estimator converges at the parametric rate $e^{O(d)}n^{-\frac{1}{2}}$, thus establishing the minimax rate-optimality of the plug-in. Numerical results that demonstrate a significant empirical superiority of the plug-in approach to general-purpose differential entropy estimators are provided.Comment: arXiv admin note: substantial text overlap with arXiv:1810.1158

Analytical calculation formulas for capacities of classical and classical-quantum channels

We derive an analytical calculation formula for the channel capacity of a classical channel without any iteration while its existing algorithms require iterations and the number of iteration depends on the required precision level. Hence, our formula is its first analytical formula without any iteration. We apply the obtained formula to examples and see how the obtained formula works in these examples. Then, we extend it to the channel capacity of a classical-quantum (cq-) channel. Many existing studies proposed algorithms for a cq-channel and all of them require iterations. Our extended analytical algorithm have also no iteration and output the exactly optimum values