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The information-theoretic framework promises to explain the predictive power of neural networks. In particular, the information plane analysis, which measures mutual information (MI) between input and representation as well as representation and output, should give rich insights into the training process. This approach, however, was shown to strongly depend on the choice of estimator of the MI. The problem is amplified for deterministic networks if the MI between input and representation is infinite. Thus, the estimated values are defined by the different approaches for estimation, but do not adequately represent the training process from an information-theoretic perspective. In this work, we show that dropout with continuously distributed noise ensures that MI is finite. We demonstrate in a range of experiments that this enables a meaningful information plane analysis for a class of dropout neural networks that is widely used in practice., Comment: Published as a conference paper at ICLR2023

We survey information-theoretic approaches to the reduction of Markov chains. Our survey is structured in two parts: The first part considers Markov chain coarse graining, which focuses on projecting the Markov chain to a process on a smaller state space that is informative}about certain quantities of interest. The second part considers Markov chain model reduction, which focuses on replacing the original Markov model by a simplified one that yields similar behavior as the original Markov model. We discuss the practical relevance of both approaches in the field of knowledge discovery and data mining by formulating problems of unsupervised machine learning as reduction problems of Markov chains. Finally, we briefly discuss the concept of lumpability, the phenomenon when a coarse graining yields a reduced Markov model., Comment: 16 pages, 3 figures; survey paper

We derive two sufficient conditions for a function of a Markov random field (MRF) on a given graph to be a MRF on the same graph. The first condition is information-theoretic and parallels a recent information-theoretic characterization of lumpability of Markov chains. The second condition, which is easier to check, is based on the potential functions of the corresponding Gibbs field. We illustrate our sufficient conditions at the hand of several examples and discuss implications for practical applications of MRFs. As a side result, we give a partial characterization of functions of MRFs that are information-preserving., Comment: 7 pages, submitted to IEEE Information Theory Workshop

We review the current literature concerned with information plane analyses of neural network classifiers. While the underlying information bottleneck theory and the claim that information-theoretic compression is causally linked to generalization are plausible, empirical evidence was found to be both supporting and conflicting. We review this evidence together with a detailed analysis of how the respective information quantities were estimated. Our survey suggests that compression visualized in information planes is not necessarily information-theoretic, but is rather often compatible with geometric compression of the latent representations. This insight gives the information plane a renewed justification. Aside from this, we shed light on the problem of estimating mutual information in deterministic neural networks and its consequences. Specifically, we argue that even in feed-forward neural networks the data processing inequality need not hold for estimates of mutual information. Similarly, while a fitting phase, in which the mutual information between the latent representation and the target increases, is necessary (but not sufficient) for good classification performance, depending on the specifics of mutual information estimation such a fitting phase need not be visible in the information plane., Comment: 12 pages, 3 figures; accepted for publication in IEEE Transactions on Neural Networks and Learning Systems. (c) 2021 IEEE

In this draft, which reports on work in progress, we 1) adapt the information bottleneck functional by replacing the compression term by class-conditional compression, 2) relax this functional using a variational bound related to class-conditional disentanglement, 3) consider this functional as a training objective for stochastic neural networks, and 4) show that the latent representations are learned such that they can be used in a naive Bayes classifier. We continue by suggesting a series of experiments along the lines of Nonlinear In-formation Bottleneck [Kolchinsky et al., 2018], Deep Variational Information Bottleneck [Alemi et al., 2017], and Information Dropout [Achille and Soatto, 2018]. We furthermore suggest a neural network where the decoder architecture is a parameterized naive Bayes decoder., Comment: draft; work in progress

This short note presents results about the symmetric Jensen-Shannon divergence between two discrete mixture distributions $p_1$ and $p_2$. Specifically, for $i=1,2$, $p_i$ is the mixture of a common distribution $q$ and a distribution $\tilde{p}_i$ with mixture proportion $\lambda_i$. In general, $\tilde{p}_1\neq \tilde{p}_2$ and $\lambda_1\neq\lambda_2$. We provide experimental and theoretical insight to the behavior of the symmetric Jensen-Shannon divergence between $p_1$ and $p_2$ as the mixture proportions or the divergence between $\tilde{p}_1$ and $\tilde{p}_2$ change. We also provide insight into scenarios where the supports of the distributions $\tilde{p}_1$, $\tilde{p}_2$, and $q$ do not coincide., Comment: four-page tech note

In this work, we investigate the use of three information-theoretic quantities -- entropy, mutual information with the class variable, and a class selectivity measure based on Kullback-Leibler divergence -- to understand and study the behavior of already trained fully-connected feed-forward neural networks. We analyze the connection between these information-theoretic quantities and classification performance on the test set by cumulatively ablating neurons in networks trained on MNIST, FashionMNIST, and CIFAR-10. Our results parallel those recently published by Morcos et al., indicating that class selectivity is not a good indicator for classification performance. However, looking at individual layers separately, both mutual information and class selectivity are positively correlated with classification performance, at least for networks with ReLU activation functions. We provide explanations for this phenomenon and conclude that it is ill-advised to compare the proposed information-theoretic quantities across layers. Furthermore, we show that cumulative ablation of neurons with ascending or descending information-theoretic quantities can be used to formulate hypotheses regarding the joint behavior of multiple neurons, such as redundancy and synergy, with comparably low computational cost. We also draw connections to the information bottleneck theory for neural networks., Comment: 12 pages; accepted for publication in IEEE Transactions on Neural Networks and Learning Systems

In this theory paper, we investigate training deep neural networks (DNNs) for classification via minimizing the information bottleneck (IB) functional. We show that the resulting optimization problem suffers from two severe issues: First, for deterministic DNNs, either the IB functional is infinite for almost all values of network parameters, making the optimization problem ill-posed, or it is piecewise constant, hence not admitting gradient-based optimization methods. Second, the invariance of the IB functional under bijections prevents it from capturing properties of the learned representation that are desirable for classification, such as robustness and simplicity. We argue that these issues are partly resolved for stochastic DNNs, DNNs that include a (hard or soft) decision rule, or by replacing the IB functional with related, but more well-behaved cost functions. We conclude that recent successes reported about training DNNs using the IB framework must be attributed to such solutions. As a side effect, our results indicate limitations of the IB framework for the analysis of DNNs. We also note that rather than trying to repair the inherent problems in the IB functional, a better approach may be to design regularizers on latent representation enforcing the desired properties directly., Comment: 16 pages, to appear in IEEE Trans. Pattern Analysis and Machine Intelligence

We present an information-theoretic cost function for co-clustering, i.e., for simultaneous clustering of two sets based on similarities between their elements. By constructing a simple random walk on the corresponding bipartite graph, our cost function is derived from a recently proposed generalized framework for information-theoretic Markov chain aggregation. The goal of our cost function is to minimize relevant information loss, hence it connects to the information bottleneck formalism. Moreover, via the connection to Markov aggregation, our cost function is not ad hoc, but inherits its justification from the operational qualities associated with the corresponding Markov aggregation problem. We furthermore show that, for appropriate parameter settings, our cost function is identical to well-known approaches from the literature, such as Information-Theoretic Co-Clustering of Dhillon et al. Hence, understanding the influence of this parameter admits a deeper understanding of the relationship between previously proposed information-theoretic cost functions. We highlight some strengths and weaknesses of the cost function for different parameters. We also illustrate the performance of our cost function, optimized with a simple sequential heuristic, on several synthetic and real-world data sets, including the Newsgroup20 and the MovieLens100k data sets., Comment: accepted for publication in IEEE Trans. on Knowledge and Data Engineering; (c) 2018 IEEE

The authors have recently defined the R\'enyi information dimension rate $d(\{X_t\})$ of a stationary stochastic process $\{X_t,\,t\in\mathbb{Z}\}$ as the entropy rate of the uniformly-quantized process divided by minus the logarithm of the quantizer step size $1/m$ in the limit as $m\to\infty$ (B. Geiger and T. Koch, "On the information dimension rate of stochastic processes," in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Aachen, Germany, June 2017). For Gaussian processes with a given spectral distribution function $F_X$, they showed that the information dimension rate equals the Lebesgue measure of the set of harmonics where the derivative of $F_X$ is positive. This paper extends this result to multivariate Gaussian processes with a given matrix-valued spectral distribution function $F_{\mathbf{X}}$. It is demonstrated that the information dimension rate equals the average rank of the derivative of $F_{\mathbf{X}}$. As a side result, it is shown that the scale and translation invariance of information dimension carries over from random variables to stochastic processes., Comment: This work will be presented in part at the 2018 International Zurich Seminar on Information and Communication

This paper proposes an information-theoretic cost function for aggregating a Markov chain via a (possibly stochastic) mapping. The cost function is motivated by two objectives: 1) The process obtained by observing the Markov chain through the mapping should be close to a Markov chain, and 2) the aggregated Markov chain should retain as much of the temporal dependence structure of the original Markov chain as possible. We discuss properties of this parameterized cost function and show that it contains the cost functions previously proposed by Deng et al., Xu et al., and Geiger et al. as special cases. We moreover discuss these special cases providing a better understanding and highlighting potential shortcomings: For example, the cost function proposed by Geiger et al. is tightly connected to approximate probabilistic bisimulation, but leads to trivial solutions if optimized without regularization. We furthermore propose a simple heuristic to optimize our cost function for deterministic aggregations and illustrate its performance on a set of synthetic examples., Comment: 12 pages, 3 figures; submitted to a journal

Distribution matching is the process of invertibly mapping a uniformly distributed input sequence onto sequences that approximate the output of a desired discrete memoryless source. The special case of a binary output alphabet and one-to-one mapping is studied. A fixed-length distribution matcher is proposed that is optimal in the sense of minimizing the unnormalized informational divergence between its output distribution and a binary memoryless target distribution. Upper and lower bounds on the unnormalized divergence are computed that increase logarithmically in the output block length $n$. It follows that a recently proposed constant composition distribution matcher performs within a constant gap of the minimal achievable informational divergence., Comment: 5 pages, 1 figure; Lemma 6 updated; This work will be presented at ISIT 2017

This paper takes a rate-distortion approach to understanding the information-theoretic laws governing cache-aided communications systems. Specifically, we characterise the optimal tradeoffs between the delivery rate, cache capacity and reconstruction distortions for a single-user problem and some special cases of a two-user problem. Our analysis considers discrete memoryless sources, expected- and excess-distortion constraints, and separable and f-separable distortion functions. We also establish a strong converse for separable-distortion functions, and we show that lossy versions of common information (G\'{a}cs-K\"{o}rner and Wyner) play an important role in caching. Finally, we illustrate and explicitly evaluate these laws for multivariate Gaussian sources and binary symmetric sources.

In this paper, we investigate mutual information as a cost function for clustering, and show in which cases hard, i.e., deterministic, clusters are optimal. Using convexity properties of mutual information, we show that certain formulations of the information bottleneck problem are solved by hard clusters. Similarly, hard clusters are optimal for the information-theoretic co-clustering problem that deals with simultaneous clustering of two dependent data sets. If both data sets have to be clustered using the same cluster assignment, hard clusters are not optimal in general. We point at interesting and practically relevant special cases of this so-called pairwise clustering problem, for which we can either prove or have evidence that hard clusters are optimal. Our results thus show that one can relax the otherwise combinatorial hard clustering problem to a real-valued optimization problem with the same global optimum.

We consider the problem of reducing a first-order Markov chain on a large alphabet to a higher-order Markov chain on a small alphabet. We present information-theoretic cost functions that are related to predictability and lumpability, show relations between these cost functions, and discuss heuristics to minimize them. Our experiments suggest that the generalization to higher orders is useful for model reduction in reliability analysis and natural language processing.

The approximation of a discrete probability distribution $\mathbf{t}$ by an $M$-type distribution $\mathbf{p}$ is considered. The approximation error is measured by the informational divergence $\mathbb{D}(\mathbf{t}\Vert\mathbf{p})$, which is an appropriate measure, e.g., in the context of data compression. Properties of the optimal approximation are derived and bounds on the approximation error are presented, which are asymptotically tight. It is shown that $M$-type approximations that minimize either $\mathbb{D}(\mathbf{t}\Vert\mathbf{p})$, or $\mathbb{D}(\mathbf{p}\Vert\mathbf{t})$, or the variational distance $\Vert\mathbf{p}-\mathbf{t}\Vert_1$ can all be found by using specific instances of the same general greedy algorithm., Comment: 5 pages

We use results from zero-error information theory to determine the set of non-injective functions through which a Markov chain can be projected without losing information. These lumping functions can be found by clique partitioning of a graph related to the Markov chain. Lossless lumping is made possible by exploiting the (sufficiently sparse) temporal structure of the Markov chain. Eliminating edges in the transition graph of the Markov chain trades the required output alphabet size versus information loss, for which we present bounds., Comment: 6 pages

The generator matrices of polar codes and Reed-Muller codes are obtained by selecting rows from the Kronecker product of a lower-triangular binary square matrix. For polar codes, the selection is based on the Bhattacharyya parameter of the row, which is closely related to the error probability of the corresponding input bit under sequential decoding. For Reed-Muller codes, the selection is based on the Hamming weight of the row. This work investigates the properties of the index sets pointing to those rows in the infinite blocklength limit. In particular, the Lebesgue measure, the Hausdorff dimension, and the self-similarity of these sets will be discussed. It is shown that these index sets have several properties that are common to fractals., Comment: 9 pages, one figure

An alternative parametric description for discrete random variables, called muculants, is proposed. In contrast to cumulants, muculants are based on the Fourier series expansion, rather than on the Taylor series expansion, of the logarithm of the characteristic function. We utilize results from cepstral theory to derive elementary properties of muculants, some of which demonstrate behavior superior to those of cumulants. For example, muculants and cumulants are both additive. While the existence of cumulants is linked to how often the characteristic function is differentiable, all muculants exist if the characteristic function satisfies a Paley-Wiener condition. Moreover, the muculant sequence and, if the random variable has finite expectation, the reconstruction of the characteristic function from its muculants converge. We furthermore develop a connection between muculants and cumulants and present the muculants of selected discrete random variables. Specifically, it is shown that the Poisson distribution is the only distribution where only the first two muculants are nonzero., Comment: 5 pages

This work investigates the information loss in a decimation system, i.e., in a downsampler preceded by an anti-aliasing filter. It is shown that, without a specific signal model in mind, the anti-aliasing filter cannot reduce information loss, while, e.g., for a simple signal-plus-noise model it can. For the Gaussian case, the optimal anti-aliasing filter is shown to coincide with the one obtained from energetic considerations. For a non-Gaussian signal corrupted by Gaussian noise, the Gaussian assumption yields an upper bound on the information loss, justifying filter design principles based on second-order statistics from an information-theoretic point-of-view., Comment: 12 pages; a shorter version of this paper was published at the 2014 International Zurich Seminar on Communications

Finite precision approximations of discrete probability distributions are considered, applicable for distribution synthesis, e.g., probabilistic shaping. Two algorithms are presented that find the optimal $M$-type approximation $Q$ of a distribution $P$ in terms of the variational distance $| Q-P|_1$ and the informational divergence $\mathbb{D}(Q| P)$. Bounds on the approximation errors are derived and shown to be asymptotically tight. Several examples illustrate that the variational distance optimal approximation can be quite different from the informational divergence optimal approximation., Comment: Submitted to the IEEE Transactions on Information Theory

In this paper, we present a method for reducing a regular, discrete-time Markov chain (DTMC) to another DTMC with a given, typically much smaller number of states. The cost of reduction is defined as the Kullback-Leibler divergence rate between a projection of the original process through a partition function and a DTMC on the correspondingly partitioned state space. Finding the reduced model with minimal cost is computationally expensive, as it requires an exhaustive search among all state space partitions, and an exact evaluation of the reduction cost for each candidate partition. Our approach deals with the latter problem by minimizing an upper bound on the reduction cost instead of minimizing the exact cost; The proposed upper bound is easy to compute and it is tight if the original chain is lumpable with respect to the partition. Then, we express the problem in the form of information bottleneck optimization, and propose using the agglomerative information bottleneck algorithm for searching a sub-optimal partition greedily, rather than exhaustively. The theory is illustrated with examples and one application scenario in the context of modeling bio-molecular interactions., Comment: 13 pages, 4 figures

In this work we present results about the rate of (relative) information loss induced by passing a real-valued, stationary stochastic process through a memoryless system. We show that for a special class of systems the information loss rate is closely related to the difference of differential entropy rates of the input and output processes. It is further shown that the rate of (relative) information loss is bounded from above by the (relative) information loss the system induces on a random variable distributed according to the process's marginal distribution. As a side result, in this work we present sufficient conditions such that for a continuous-valued Markovian input process also the output process possesses the Markov property., Comment: 9 pages, 4 figures; submitted to a conference

We present a sufficient condition for a non-injective function of a Markov chain to be a second-order Markov chain with the same entropy rate as the original chain. This permits an information-preserving state space reduction by merging states or, equivalently, lossless compression of a Markov source on a sample-by-sample basis. The cardinality of the reduced state space is bounded from below by the node degrees of the transition graph associated with the original Markov chain. We also present an algorithm listing all possible information-preserving state space reductions, for a given transition graph. We illustrate our results by applying the algorithm to a bi-gram letter model of an English text., Comment: 7 pages, 3 figures, 2 tables

In this work the information loss in deterministic, memoryless systems is investigated by evaluating the conditional entropy of the input random variable given the output random variable. It is shown that for a large class of systems the information loss is finite, even if the input is continuously distributed. Based on this finiteness, the problem of perfectly reconstructing the input is addressed and Fano-type bounds between the information loss and the reconstruction error probability are derived. For systems with infinite information loss a relative measure is defined and shown to be tightly related to R\'{e}nyi information dimension. Employing another Fano-type argument, the reconstruction error probability is bounded by the relative information loss from below. In view of developing a system theory from an information-theoretic point-of-view, the theoretical results are illustrated by a few example systems, among them a multi-channel autocorrelation receiver., Comment: 23 pages, 12 figures; submitted

A lumping of a Markov chain is a coordinate-wise projection of the chain. We characterise the entropy rate preservation of a lumping of an aperiodic and irreducible Markov chain on a finite state space by the random growth rate of the cardinality of the realisable preimage of a finite-length trajectory of the lumped chain and by the information needed to reconstruct original trajectories from their lumped images. Both are purely combinatorial criteria, depending only on the transition graph of the Markov chain and the lumping function. A lumping is strongly k-lumpable, iff the lumped process is a k-th order Markov chain for each starting distribution of the original Markov chain. We characterise strong k-lumpability via tightness of stationary entropic bounds. In the sparse setting, we give sufficient conditions on the lumping to both preserve the entropy rate and be strongly k-lumpable.

We introduce the notion of relevant information loss for the purpose of casting the signal enhancement problem in information-theoretic terms. We show that many algorithms from machine learning can be reformulated using relevant information loss, which allows their application to the aforementioned problem. As a particular example we analyze principle component analysis for dimensionality reduction, discuss its optimality, and show that the relevant information loss can indeed vanish if the relevant information is concentrated on a lower-dimensional subspace of the input space., Comment: 9 pages; 4 figures; accepted for presentation at a conference

In this work we analyze principle component analysis (PCA) as a deterministic input-output system. We show that the relative information loss induced by reducing the dimensionality of the data after performing the PCA is the same as in dimensionality reduction without PCA. Finally, we analyze the case where the PCA uses the sample covariance matrix to compute the rotation. If the rotation matrix is not available at the output, we show that an infinite amount of information is lost. The relative information loss is shown to decrease with increasing sample size., Comment: 9 pages, 4 figure; extended version of a paper accepted for publication

We introduce a relative variant of information loss to characterize the behavior of deterministic input-output systems. We show that the relative loss is closely related to Renyi's information dimension. We provide an upper bound for continuous input random variables and an exact result for a class of functions (comprising quantizers) with infinite absolute information loss. A connection between relative information loss and reconstruction error is investigated., Comment: 6 pages; submitted to a conference

In this work we give a concise definition of information loss from a system-theoretic point of view. Based on this definition, we analyze the information loss in static input-output systems subject to a continuous-valued input. For a certain class of multiple-input, multiple-output systems the information loss is quantified. An interpretation of this loss is accompanied by upper bounds which are simple to evaluate. Finally, a class of systems is identified for which the information loss is necessarily infinite. Quantizers and limiters are shown to belong to this class., Comment: 7 pages, 6 figures; submitted to a conference

Acquisition is a search in two continuous dimensions, where the digital algorithms require a partitioning of the search space into cells. Depending on the partitioning of the Doppler frequency domain, more than one cell might contain significant signal energy. We present an expression for the expected values of the cells' energies to analyze the impact of the Doppler bin width on detection and false alarm probabilities., Comment: 24 pages, 12 figures; submitted

In this work we investigate the information loss in (nonlinear) dynamical input-output systems and provide some general results. In particular, we present an upper bound on the information loss rate, defined as the (non-negative) difference between the entropy rates of the jointly stationary stochastic processes at the input and output of the system. We further introduce a family of systems with vanishing information loss rate. It is shown that not only linear filters belong to that family, but - under certain circumstances - also finite-precision implementations of the latter, which typically consist of nonlinear elements., Comment: 6 pages, 2 figures, submitted to a conference

In this work, conditional entropy is used to quantify the information loss induced by passing a continuous random variable through a memoryless nonlinear input-output system. We derive an expression for the information loss depending on the input density and the nonlinearity and show that the result is strongly related to the non-injectivity of the considered system. Tight upper bounds are presented, which can be evaluated with less difficulty than a direct evaluation of the information loss, which involves the logarithm of a sum. Application of our results is illustrated on a set of examples., Comment: 9 pages, 6 figures; A short version of this paper is submitted to an IEEE conference