Total correlation (TC) is a fundamental concept in information theory that measures statistical dependency among multiple random variables. Recently, TC has shown noticeable effectiveness as a regularizer in many learning tasks, where the correlation among multiple latent embeddings requires to be jointly minimized or maximized. However, calculating precise TC values is challenging, especially when the closed-form distributions of embedding variables are unknown. In this paper, we introduce a unified framework to estimate total correlation values with sample-based mutual information (MI) estimators. More specifically, we discover a relation between TC and MI and propose two types of calculation paths (tree-like and line-like) to decompose TC into MI terms. With each MI term being bounded, the TC values can be successfully estimated. Further, we provide theoretical analyses concerning the statistical consistency of the proposed TC estimators. Experiments are presented on both synthetic and real-world scenarios, where our estimators demonstrate effectiveness in all TC estimation, minimization, and maximization tasks. The code is available at https://github.com/Linear95/TC-estimation.
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.
Inspired by Hilberg's hypothesis, which states that mutual information between blocks for natural language grows like a power law, we seek for links between power-law growth rate of algorithmic mutual information and of some estimator of the unifilar order, i.e., the number of hidden states in the generating stationary ergodic source in its minimal unifilar hidden Markov representation. We consider an order estimator which returns the smallest order for which the maximum likelihood is larger than a weakly penalized universal probability. This order estimator is intractable and follows the idea…
In the era of big data and the Internet of Things (IoT), data owners need to share a large amount of data with the intended receivers in an insecure environment, posing a trade-off issue between user privacy and data utility. The privacy utility trade-off was facilitated through a privacy funnel based on mutual information. Nevertheless, it is challenging to characterize the mutual information accurately with small sample size or unknown distribution functions. In this article, we propose a privacy funnel based on mutual information neural estimator (MINE) to optimize the privacy utility trade…
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…
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 …
Estimating mutual information (MI) from samples is a fundamental problem in statistics, machine learning, and data analysis. Recently it was shown that a popular class of non-parametric MI estimators perform very poorly for strongly dependent variables and have sample complexity that scales exponentially with the true MI. This undesired behavior was attributed to the reliance of those estimators on local uniformity of the underlying (and unknown) probability density function. Here we present a novel semi-parametric estimator of mutual information, where at each sample point, densities are {\em…
Beyond Normal: On the Evaluation of Mutual Information Estimators
ETH AI Center, ETH Zurich（苏黎世联邦理工学院ETH人工智能中心）；Institute of Fundamental Technological Research, Polish Academy of Sciences（波兰科学院基础技术研究所）；Department of Computer Science, ETH Zurich（苏黎世联邦理工学院计算机科学系）；SIB Swiss Institute of Bioinformatics（SIB瑞士生物信息学研究所）；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）
The conditional mutual information I(X;Y|Z) measures the average information that X and Y contain about each other given Z. This is an important primitive in many learning problems including conditional independence testing, graphical model inference, causal strength estimation and time-series problems. In several applications, it is desirable to have a functional purely of the conditional distribution p_{Y|X,Z} rather than of the joint distribution p_{X,Y,Z}. We define the potential conditional mutual information as the conditional mutual information calculated with a modified joint distribut…
Several recent works in communication systems have proposed to leverage the power of neural networks in the design of encoders and decoders. In this approach, these blocks can be tailored to maximize the transmission rate based on aggregated samples from the channel. Motivated by the fact that, in many communication schemes, the achievable transmission rate is determined by a conditional mutual information term, this paper focuses on neural-based estimators for this information-theoretic quantity. Our results are based on variational bounds for the KL-divergence and, in contrast to some previo…
Variational approaches based on neural networks are showing promise for estimating mutual information (MI) between high dimensional variables. However, they can be difficult to use in practice due to poorly understood bias/variance tradeoffs. We theoretically show that, under some conditions, estimators such as MINE exhibit variance that could grow exponentially with the true amount of underlying MI. We also empirically demonstrate that existing estimators fail to satisfy basic self-consistency properties of MI, such as data processing and additivity under independence. Based on a unified pers…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Washington（华盛顿大学）
Estimating mutual information from observed samples is a basic primitive, useful in several machine learning tasks including correlation mining, information bottleneck clustering, learning a Chow-Liu tree, and conditional independence testing in (causal) graphical models. While mutual information is a well-defined quantity in general probability spaces, existing estimators can only handle two special cases of purely discrete or purely continuous pairs of random variables. The main challenge is that these methods first estimate the (differential) entropies of X, Y and the pair (X;Y) and add the…
Estimating conditional mutual information (CMI) is an essential yet challenging step in many machine learning and data mining tasks. Estimating CMI from data that contains both discrete and continuous variables, or even discrete-continuous mixture variables, is a particularly hard problem. In this paper, we show that CMI for such mixture variables, defined based on the Radon-Nikodym derivate, can be written as a sum of entropies, just like CMI for purely discrete or continuous data. Further, we show that CMI can be consistently estimated for discrete-continuous mixture variables by learning an…
Multivariate pattern analyses approaches in neuroimaging are fundamentally concerned with investigating the quantity and type of information processed by various regions of the human brain; typically, estimates of classification accuracy are used to quantify information. While a extensive and powerful library of methods can be applied to train and assess classifiers, it is not always clear how to use the resulting measures of classification performance to draw scientific conclusions: e.g. for the purpose of evaluating redundancy between brain regions. An additional confound for interpreting cl…
The estimation of mutual information (MI) or conditional mutual information (CMI) from a set of samples is a long-standing problem. A recent line of work in this area has leveraged the approximation power of artificial neural networks and has shown improvements over conventional methods. One important challenge in this new approach is the need to obtain, given the original dataset, a different set where the samples are distributed according to a specific product density function. This is particularly challenging when estimating CMI. In this paper, we introduce a new technique, based on k neare…
Mutual information is a measure of the dependence between random variables that has been used successfully in myriad applications in many fields. Generalized mutual information measures that go beyond classical Shannon mutual information have also received much interest in these applications. We derive the mean squared error convergence rates of kernel density-based plug-in estimators of general mutual information measures between two multidimensional random variables $\mathbf{X}$ and $\mathbf{Y}$ for two cases: 1) $\mathbf{X}$ and $\mathbf{Y}$ are continuous; 2) $\mathbf{X}$ and $\mathbf{Y}$ …
This paper proposes an estimation framework to assess the performance of sorting over perturbed/noisy data. In particular, the recovering accuracy is measured in terms of Minimum Mean Square Error (MMSE) between the values of the sorting function computed on data without perturbation and the estimator that operates on the sorted noisy data. It is first shown that, under certain symmetry conditions, satisfied for example by the practically relevant Gaussian noise perturbation, the optimal estimator can be expressed as a linear combination of estimators on the unsorted data. Then, two suboptimal…
Channel capacity plays a crucial role in the development of modern communication systems as it represents the maximum rate at which information can be reliably transmitted over a communication channel. Nevertheless, for the majority of channels, finding a closed-form capacity expression remains an open challenge. This is because it requires to carry out two formidable tasks a) the computation of the mutual information between the channel input and output, and b) its maximization with respect to the signal distribution at the channel input. In this paper, we address both tasks. Inspired by impl…
Estimating the number of signals embedded in noise is a fundamental problem in signal processing. As a classic estimator based on random matrix theory (RMT), the RMT estimator estimates the number of signals via sequentially testing the likelihood of an eigenvalue as arising from a signal or noise for a given over-detection probability. However, it tends to under-estimate the number of signals as weak signal eigenvalues may be immersed in the non-negligible bias term among eigenvalues for finite sample size. In order to solve this problem, we propose an RMT estimator with adaptive decision cri…
Recently, several methods have been proposed for estimating the mutual information from sample data using deep neural networks and without the knowing closed form distribution of the data. This class of estimators is referred to as neural mutual information estimators. Although very promising, such techniques have yet to be rigorously bench-marked so as to establish their efficacy, ease of implementation, and stability for capacity estimation which is joint maximization frame-work. In this paper, we compare the different techniques proposed in the literature for estimating capacity and provide…
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 …
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…
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…
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…
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…
KSG mutual information estimator, which is based on the distances of each sample to its k-th nearest neighbor, is widely used to estimate mutual information between two continuous random variables. Existing work has analyzed the convergence rate of this estimator for random variables whose densities are bounded away from zero in its support. In practice, however, KSG estimator also performs well for a much broader class of distributions, including not only those with bounded support and densities bounded away from zero, but also those with bounded support but densities approaching zero, and th…
We propose a new estimator to measure directed dependencies in time series. The dimensionality of data is first reduced using a new non-uniform embedding technique, where the variables are ranked according to a weighted sum of the amount of new information and improvement of the prediction accuracy provided by the variables. Then, using a greedy approach, the most informative subsets are selected in an iterative way. The algorithm terminates, when the highest ranked variable is not able to significantly improve the accuracy of the prediction as compared to that obtained using the existing sele…
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…
Fritz Haber Institute of the Max Planck Society（马克斯·普朗克学会弗里茨·哈伯研究所）；Humboldt-Universität zu Berlin（柏林洪堡大学）
The identification of relevant features, i.e., the driving variables that determine a process or the properties of a system, is an essential part of the analysis of data sets with a large number of variables. A mathematical rigorous approach to quantifying the relevance of these features is mutual information. Mutual information determines the relevance of features in terms of their joint mutual dependence to the property of interest. However, mutual information requires as input probability distributions, which cannot be reliably estimated from continuous distributions such as physical quanti…
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 entropy and mutual information consistently is important for many machine learning applications. The Kozachenko-Leonenko (KL) estimator (Kozachenko & Leonenko, 1987) is a widely used nonparametric estimator for the entropy of multivariate continuous random variables, as well as the basis of the mutual information estimator of Kraskov et al. (2004), perhaps the most widely used estimator of mutual information in this setting. Despite the practical importance of these estimators, major theoretical questions regarding their finite-sample behavior remain open. This paper proves finite-s…
Conditional independence testing is a fundamental problem underlying causal discovery and a particularly challenging task in the presence of nonlinear and high-dimensional dependencies. Here a fully non-parametric test for continuous data based on conditional mutual information combined with a local permutation scheme is presented. Through a nearest neighbor approach, the test efficiently adapts also to non-smooth distributions due to strongly nonlinear dependencies. Numerical experiments demonstrate that the test reliably simulates the null distribution even for small sample sizes and with hi…
We study the problem of using i.i.d. samples from an unknown multivariate probability distribution $p$ to estimate the mutual information of $p$. This problem has recently received attention in two settings: (1) where $p$ is assumed to be Gaussian and (2) where $p$ is assumed only to lie in a large nonparametric smoothness class. Estimators proposed for the Gaussian case converge in high dimensions when the Gaussian assumption holds, but are brittle, failing dramatically when $p$ is not Gaussian. Estimators proposed for the nonparametric case fail to converge with realistic sample sizes except…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Columbia University（哥伦比亚大学）
We consider the problem of estimating the common mean of independently sampled data, where samples are drawn in a possibly non-identical manner from symmetric, unimodal distributions with a common mean. This generalizes the setting of Gaussian mixture modeling, since the number of distinct mixture components may diverge with the number of observations. We propose an estimator that adapts to the level of heterogeneity in the data, achieving near-optimality in both the i.i.d. setting and some heterogeneous settings, where the fraction of ``low-noise'' points is as small as $\frac{\log n}{n}$. Ou…
