We consider the problem of estimating mutual information between dependent data, an important problem in many science and engineering applications. We propose a data-driven, non-parametric estimator of mutual information in this paper. The main novelty of our solution lies in transforming the data to frequency domain to make the problem tractable. We define a novel metric--mutual information in frequency--to detect and quantify the dependence between two random processes across frequency using Cramér's spectral representation. Our solution calculates mutual information as a function of frequency to estimate the mutual information between the dependent data over time. We validate its performance on linear and nonlinear models. In addition, mutual information in frequency estimated as a part of our solution can also be used to infer cross-frequency coupling in the data.
The presence of mutual information in the research of deep learning has grown significantly. It has been proven that mutual information can be a good objective function to build a robust deep learning model. Most of the researches utilize estimation methods to approximate the true mutual information. This technical report delivers an extensive study about definitions as well as properties of mutual information. This article then delivers some reviews and current drawbacks of mutual information estimation methods afterward.
Estimators for mutual information are typically biased. However, in the case of the Kozachenko-Leonenko estimator for metric spaces, a type of nearest neighbour estimator, it is possible to calculate the bias explicitly.
Consider statistical learning (e.g. discrete distribution estimation) with local $ε$-differential privacy, which preserves each data provider's privacy locally, we aim to optimize statistical data utility under the privacy constraints. Specifically, we study maximizing mutual information between a provider's data and its private view, and give the exact mutual information bound along with an attainable mechanism: $k$-subset mechanism as results. The mutual information optimal mechanism randomly outputs a size $k$ subset of the original data domain with delicate probability assignment, where $k…
Let $X$, $Y$ be two correlated discrete random variables. We consider an estimation of $X$ from encoded data $φ(Y)$ of $Y$ by some encoder function $φ(Y)$. We derive an inequality describing a relation of the correct probability of estimation and the mutual information between $X$ and $φ(Y)$. This inequality may be useful for the secure analysis of crypto system when we use the success probability of estimating secret data as a security criterion. It also provides an intuitive meaning of the secrecy exponent in the strong secrecy criterion.
This paper compares and evaluates a set of non-parametric mutual information estimators with the goal of providing a novel toolset to progress in the analysis of the capacity of the nonlinear optical channel, which is currently an open problem. In the first part of the paper, the methods of the study are presented. The second part details their application to several optically-related channels to highlight their features.
City University of Hong Kong（香港城市大学）；Beihang University（北京航空航天大学）
We point out a limitation of the mutual information neural estimation (MINE) where the network fails to learn at the initial training phase, leading to slow convergence in the number of training iterations. To solve this problem, we propose a faster method called the mutual information neural entropic estimation (MI-NEE). Our solution first generalizes MINE to estimate the entropy using a custom reference distribution. The entropy estimate can then be used to estimate the mutual information. We argue that the seemingly redundant intermediate step of entropy estimation allows one to improve the…
We consider rank-one non-symmetric tensor estimation and derive simple formulas for the mutual information. We start by the order 2 problem, namely matrix factorization. We treat it completely in a simpler fashion than previous proofs using a new type of interpolation method developed in [1]. We then show how to harness the structure in "layers" of tensor estimation in order to obtain a formula for the mutual information for the order 3 problem from the knowledge of the formula for the order 2 problem, still using the same kind of interpolation. Our proof technique straightforwardly generalize…
Mutual information is a widely-used information theoretic measure to quantify the amount of association between variables. It is used extensively in many applications such as image registration, diagnosis of failures in electrical machines, pattern recognition, data mining and tests of independence. The main goal of this paper is to provide an efficient estimator of the mutual information based on the approach of Al Labadi et. al. (2021). The estimator is explored through various examples and is compared to its frequentist counterpart due to Berrett et al. (2019). The results show the good per…
Applied AI Institute（应用人工智能研究所）；Institute of Numerical Mathematics, RAS（俄罗斯科学院数值数学研究所）；National Research Nuclear University MEPhI（国立研究型核大学MEPhI（莫斯科工程物理学院））
It is very difficult to solve the Maximum Mutual Information (MMI) or Maximum Likelihood (ML) for all possible Shannon Channels or uncertain rules of choosing hypotheses, so that we have to use iterative methods. According to the Semantic Mutual Information (SMI) and R(G) function proposed by Chenguang Lu (1993) (where R(G) is an extension of information rate distortion function R(D), and G is the lower limit of the SMI), we can obtain a new iterative algorithm of solving the MMI and ML for tests, estimations, and mixture models. The SMI is defined by the average log normalized likelihood. The…
I present several new relations between mutual information (MI) and statistical estimation error for a system that can be regarded simultaneously as a communication channel and as an estimator of an input parameter. I first derive a second-order result between MI and Fisher information (FI) that is valid for sufficiently narrow priors, but arbitrary channels. A second relation furnishes a lower bound on the MI in terms of the minimum mean-squared error (MMSE) on the Bayesian estimation of the input parameter from the channel output, one that is valid for arbitrary channels and priors. The exis…
The development of optimal and efficient machine learning-based communication systems is likely to be a key enabler of beyond 5G communication technologies. In this direction, physical layer design has been recently reformulated under a deep learning framework where the autoencoder paradigm foresees the full communication system as an end-to-end coding-decoding problem. Given the loss function, the autoencoder jointly learns the coding and decoding optimal blocks under a certain channel model. Because performance in communications typically refers to achievable rates and channel capacity, the …
IIT Delhi（印度理工学院德里分校）；University of Washington（华盛顿大学）；TIFR Mumbai（塔塔基础研究院孟买分院）
Estimation of information theoretic quantities such as mutual information and its conditional variant has drawn interest in recent times owing to their multifaceted applications. Newly proposed neural estimators for these quantities have overcome severe drawbacks of classical $k$NN-based estimators in high dimensions. In this work, we focus on conditional mutual information (CMI) estimation by utilizing its formulation as a minmax optimization problem. Such a formulation leads to a joint training procedure similar to that of generative adversarial networks. We find that our proposed estimator …
We consider the estimation of a signal from the knowledge of its noisy linear random Gaussian projections, a problem relevant in compressed sensing, sparse superposition codes or code division multiple access just to cite few. There has been a number of works considering the mutual information for this problem using the heuristic replica method from statistical physics. Here we put these considerations on a firm rigorous basis. First, we show, using a Guerra-type interpolation, that the replica formula yields an upper bound to the exact mutual information. Secondly, for many relevant practical…
We introduce the Conditional Mutual Information (CMI) for the estimation of the Markov chain order. For a Markov chain of $K$ symbols, we define CMI of order $m$, $I_c(m)$, as the mutual information of two variables in the chain being $m$ time steps apart, conditioning on the intermediate variables of the chain. We find approximate analytic significance limits based on the estimation bias of CMI and develop a randomization significance test of $I_c(m)$, where the randomized symbol sequences are formed by random permutation of the components of the original symbol sequence. The significance tes…
Deep learning based physical layer design, i.e., using dense neural networks as encoders and decoders, has received considerable interest recently. However, while such an approach is naturally training data-driven, actions of the wireless channel are mimicked using standard channel models, which only partially reflect the physical ground truth. Very recently, neural network based mutual information (MI) estimators have been proposed that directly extract channel actions from the input-output measurements and feed these outputs into the channel encoder. This is a promising direction as such a n…
Besides the different approaches suggested in the literature, accurate estimation of the order of a Markov chain from a given symbol sequence is an open issue, especially when the order is moderately large. Here, parametric significance tests of conditional mutual information (CMI) of increasing order $m$, $I_c(m)$, on a symbol sequence are conducted for increasing orders $m$ in order to estimate the true order $L$ of the underlying Markov chain. CMI of order $m$ is the mutual information of two variables in the Markov chain being $m$ time steps apart, conditioning on the intermediate variable…
Fundamental relations between information and estimation have been established in the literature for the continuous-time Gaussian and Poisson channels, in a long line of work starting from the classical representation theorems by Duncan and Kabanov respectively. In this work, we demonstrate that such relations hold for a much larger family of continuous-time channels. We introduce the family of semi-martingale channels where the channel output is a semi-martingale stochastic process, and the channel input modulates the characteristics of the semi-martingale. For these channels, which includes …
We demonstrate that a popular class of nonparametric mutual information (MI) estimators based on k-nearest-neighbor graphs requires number of samples that scales exponentially with the true MI. Consequently, accurate estimation of MI between two strongly dependent variables is possible only for prohibitively large sample size. This important yet overlooked shortcoming of the existing estimators is due to their implicit reliance on local uniformity of the underlying joint distribution. We introduce a new estimator that is robust to local non-uniformity, works well with limited data, and is able…
This paper deals with arbitrarily distributed finite-power input signals observed through an additive Gaussian noise channel. It shows a new formula that connects the input-output mutual information and the minimum mean-square error (MMSE) achievable by optimal estimation of the input given the output. That is, the derivative of the mutual information (nats) with respect to the signal-to-noise ratio (SNR) is equal to half the MMSE, regardless of the input statistics. This relationship holds for both scalar and vector signals, as well as for discrete-time and continuous-time noncausal MMSE esti…
Freie Universität Berlin（柏林自由大学）；University of Siegen（锡根大学）
The Jensen inequality is a widely used tool in a multitude of fields, such as for example information theory and machine learning. It can be also used to derive other standard inequalities such as the inequality of arithmetic and geometric means or the Hölder inequality. In a probabilistic setting, the Jensen inequality describes the relationship between a convex function and the expected value. In this work, we want to look at the probabilistic setting from the reverse direction of the inequality. We show that under minimal constraints and with a proper scaling, the Jensen inequality can be r…
Mutual Information Estimation via Normalizing Flows
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Sirius University of Science and Technology（天狼星科技大学）；Skoltech（斯科尔科沃科学技术研究院）；MIPT（莫斯科物理技术学院）；Sirius（天狼星科技大学）
There has been definite progress recently in proving the variational single-letter formula given by the heuristic replica method for various estimation problems. In particular, the replica formula for the mutual information in the case of noisy linear estimation with random i.i.d. matrices, a problem with applications ranging from compressed sensing to statistics, has been proven rigorously. In this contribution we go beyond the restrictive i.i.d. matrix assumption and discuss the formula proposed by Takeda, Uda, Kabashima and later by Tulino, Verdu, Caire and Shamai who used the replica metho…
We consider a statistical model for finite-rank symmetric tensor factorization and prove a single-letter variational expression for its asymptotic mutual information when the tensor is of even order. The proof applies the adaptive interpolation method originally invented for rank-one factorization. Here we show how to extend the adaptive interpolation to finite-rank and even-order tensors. This requires new nontrivial ideas with respect to the current analysis in the literature. We also underline where the proof falls short when dealing with odd-order tensors.
On the Properties and Estimation of Pointwise Mutual Information Profiles
ETH AI Center, ETH Zurich（苏黎世联邦理工学院ETH AI中心）；Institute of Fundamental Technological Research, Polish Academy of Sciences（波兰科学院基础技术研究所）；SIB Swiss Institute of Bioinformatics（瑞士SIB生物信息学研究所）；Department of Biosystems Science and Engineering, ETH Zurich（苏黎世联邦理工学院生物系统科学与工程系）；Department of Computer Science, ETH Zurich（苏黎世联邦理工学院计算机科学系）
本文解析多元正态分布的点互信息轮廓，提出Bend and Mix模型族，用蒙特卡洛方法估计轮廓，并用于研究互信息估计器局限、神经评判器行为及异常值影响，最终实现基于模型的贝叶斯互信息估计。
Mutual Information is the metric that is used to perform link adaptation, which allows to achieve rates near capacity. The computation of adaptive transmission modes is achieved by employing the mapping between the Signal to Noise Ratio and the Mutual Information. Due to the high complexity of the computation of the Mutual Information, this process is performed off-line via Monte Carlo simulations, whose results are stored in look-up tables. However, in Index Modulations, such as Spatial Modulation or Polarized Modulation, this is not feasible since the constellation and the Mutual Information…
We consider the estimation of a n-dimensional vector x from the knowledge of noisy and possibility non-linear element-wise measurements of xxT , a very generic problem that contains, e.g. stochastic 2-block model, submatrix localization or the spike perturbation of random matrices. We use an interpolation method proposed by Guerra and later refined by Korada and Macris. We prove that the Bethe mutual information (related to the Bethe free energy and conjectured to be exact by Lesieur et al. on the basis of the non-rigorous cavity method) always yields an upper bound to the exact mutual informa…
We consider the estimation of a signal from the knowledge of its noisy linear random Gaussian projections. A few examples where this problem is relevant are compressed sensing, sparse superposition codes, and code division multiple access. There has been a number of works considering the mutual information for this problem using the replica method from statistical physics. Here we put these considerations on a firm rigorous basis. First, we show, using a Guerra-Toninelli type interpolation, that the replica formula yields an upper bound to the exact mutual information. Secondly, for many relev…
Freie Universität Berlin（柏林自由大学）；Technische Universität Berlin（柏林工业大学）
End-to-end deep learning for communication systems, i.e., systems whose encoder and decoder are learned, has attracted significant interest recently, due to its performance which comes close to well-developed classical encoder-decoder designs. However, one of the drawbacks of current learning approaches is that a differentiable channel model is needed for the training of the underlying neural networks. In real-world scenarios, such a channel model is hardly available and often the channel density is not even known at all. Some works, therefore, focus on a generative approach, i.e., generating …
Exploiting Mutual Coupling Structure for Channel Estimation of Active RIS-Assisted Links
Technische Universität Berlin（柏林工业大学）；University of British Columbia, Kelowna（不列颠哥伦比亚大学（基洛纳校区））；American University of Beirut (AUB)（贝鲁特美国大学）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）
This paper considers the model of an arbitrary distributed signal x observed through an added independent white Gaussian noise w, y=x+w. New relations between the minimal mean square error of the non-causal estimator and the likelihood ratio between y and ωare derived. This is followed by an extended version of a recently derived relation between the mutual information I(x;y) and the minimal mean square error. These results are applied to derive infinite dimensional versions of the Fisher information and the de Bruijn identity. The derivation of the results is based on the Malliavin calculus.
Department of EECS, MIT（麻省理工学院电子工程与计算机科学系）；Department of Computer Science, University of Rochester（罗切斯特大学计算机科学系）
The InfoNCE objective, originally introduced for contrastive representation learning, has become a popular choice for mutual information (MI) estimation, despite its indirect connection to MI. In this paper, we demonstrate why InfoNCE should not be regarded as a valid MI estimator, and we introduce a simple modification, which we refer to as InfoNCE-anchor, for accurate MI estimation. Our modification introduces an auxiliary anchor class, enabling consistent density ratio estimation and yielding a plug-in MI estimator with significantly reduced bias. Beyond this, we generalize our framework us…
We investigate the sets of joint probability distributions that maximize the average multi-information over a collection of margins. These functionals serve as proxies for maximizing the multi-information of a set of variables or the mutual information of two subsets of variables, at a lower computation and estimation complexity. We describe the maximizers and their relations to the maximizers of the multi-information and the mutual information.
Northeastern University（东北大学）；Graz University of Technology（格拉茨技术大学）；Silicon Austria Labs（硅谷奥地利实验室）
Feature ranking and selection is a widely used approach in various applications of supervised dimensionality reduction in discriminative machine learning. Nevertheless there exists significant evidence on feature ranking and selection algorithms based on any criterion leading to potentially sub-optimal solutions for class separability. In that regard, we introduce emerging information theoretic feature transformation protocols as an end-to-end neural network training approach. We present a dimensionality reduction network (MMINet) training procedure based on the stochastic estimate of the mutu…
The problem of estimating the dynamic direction of arrival of far field signals impinging on a uniform linear array, with mutual coupling effects, is addressed. This work proposes two novel approaches able to provide accurate solutions, including at the endfire regions of the array. Firstly, a Bayesian compressive sensing Kalman filter is developed, which accounts for the predicted estimated signals rather than using the traditional sparse prior. The posterior probability density function of the received source signals and the expression for the related marginal likelihood function are derived…
Conditional Mutual Information (CMI) is a measure of conditional dependence between random variables X and Y, given another random variable Z. It can be used to quantify conditional dependence among variables in many data-driven inference problems such as graphical models, causal learning, feature selection and time-series analysis. While k-nearest neighbor (kNN) based estimators as well as kernel-based methods have been widely used for CMI estimation, they suffer severely from the curse of dimensionality. In this paper, we leverage advances in classifiers and generative models to design metho…
Virginia Tech（弗吉尼亚理工大学）；Duke University（杜克大学）；KAUST（阿卜杜拉国王科技大学）；Amazon（亚马逊）
Successful applications of InfoNCE and its variants have popularized the use of contrastive variational mutual information (MI) estimators in machine learning. While featuring superior stability, these estimators crucially depend on costly large-batch training, and they sacrifice bound tightness for variance reduction. To overcome these limitations, we revisit the mathematics of popular variational MI bounds from the lens of unnormalized statistical modeling and convex optimization. Our investigation not only yields a new unified theoretical framework encompassing popular variational MI bounds…
Nonparametric estimation of mutual information is used in a wide range of scientific problems to quantify dependence between variables. The k-nearest neighbor (knn) methods are consistent, and therefore expected to work well for large sample size. These methods use geometrically regular local volume elements. This practice allows maximum localization of the volume elements, but can also induce a bias due to a poor description of the local geometry of the underlying probability measure. We introduce a new class of knn estimators that we call geometric knn estimators (g-knn), which use more comp…
Estimating mutual information (MI) between two continuous random variables $X$ and $Y$ allows to capture non-linear dependencies between them, non-parametrically. As such, MI estimation lies at the core of many data science applications. Yet, robustly estimating MI for high-dimensional $X$ and $Y$ is still an open research question. In this paper, we formulate this problem through the lens of manifold learning. That is, we leverage the common assumption that the information of $X$ and $Y$ is captured by a low-dimensional manifold embedded in the observed high-dimensional space and transfer it …
The Mutual Information (MI) is an often used measure of dependency between two random variables utilized in information theory, statistics and machine learning. Recently several MI estimators have been proposed that can achieve parametric MSE convergence rate. However, most of the previously proposed estimators have the high computational complexity of at least $O(N^2)$. We propose a unified method for empirical non-parametric estimation of general MI function between random vectors in $\mathbb{R}^d$ based on $N$ i.i.d. samples. The reduced complexity MI estimator, called the ensemble dependen…
This work addresses testing the independence of two continuous and finite-dimensional random variables from the design of a data-driven partition. The empirical log-likelihood statistic is adopted to approximate the sufficient statistics of an oracle test against independence (that knows the two hypotheses). It is shown that approximating the sufficient statistics of the oracle test offers a learning criterion for designing a data-driven partition that connects with the problem of mutual information estimation. Applying these ideas in the context of a data-dependent tree-structured partition (…
Kalman filter is a best linear unbiased state estimator. It is also comprehensible from the point view of the Bayesian estimation. However, this note gives a detailed derivation of Kalman filter from the mutual information perspective for the first time. Then we extend this result to the Rényi mutual information. Finally we draw the conclusion that the measurement update of the Kalman filter is the key step to minimize the uncertainty of the state of the dynamical system.
Let $X$ be a non-negative random variable and let the conditional distribution of a random variable $Y$, given $X$, be ${Poisson}(γ\cdot X)$, for a parameter $γ\geq 0$. We identify a natural loss function such that: 1) The derivative of the mutual information between $X$ and $Y$ with respect to $γ$ is equal to the \emph{minimum} mean loss in estimating $X$ based on $Y$, regardless of the distribution of $X$. 2) When $X \sim P$ is estimated based on $Y$ by a mismatched estimator that would have minimized the expected loss had $X \sim Q$, the integral over all values of $γ$ of the excess mean lo…
We consider the processing of statistical samples $X\sim P_θ$ by a channel $p(y|x)$, and characterize how the statistical information from the samples for estimating the parameter $θ\in\mathbb{R}^d$ can scale with the mutual information or capacity of the channel. We show that if the statistical model has a sub-Gaussian score function, then the trace of the Fisher information matrix for estimating $θ$ from $Y$ can scale at most linearly with the mutual information between $X$ and $Y$. We apply this result to obtain minimax lower bounds in distributed statistical estimation problems, and obtain…
CRYPTO-MINE: Cryptanalysis via Mutual Information Neural Estimation
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Massachusetts Institute of Technology（麻省理工学院）；Technion（以色列理工学院）；Clemson University（克莱姆森大学）；MIT Lincoln Laboratory（麻省理工学院林肯实验室）
In this paper a numerical method is presented, which finds a lower bound for the mutual information between a binary and an arbitrary finite random variable with joint distributions that have a variational distance not greater than a known value to a known joint distribution. This lower bound can be applied to mutual information estimation with confidence intervals.
Estimating the entropy of a discrete random variable is a fundamental problem in information theory and related fields. This problem has many applications in various domains, including machine learning, statistics and data compression. Over the years, a variety of estimation schemes have been suggested. However, despite significant progress, most methods still struggle when the sample is small, compared to the variable's alphabet size. In this work, we introduce a practical solution to this problem, which extends the work of McAllester and Statos (2020). The proposed scheme uses the generaliza…
