Information theoretic quantities play an important role in various settings in machine learning, including causality testing, structure inference in graphical models, time-series problems, feature selection as well as in providing privacy guarantees. A key quantity of interest is the mutual information and generalizations thereof, including conditional mutual information, multivariate mutual information, total correlation and directed information. While the aforementioned information quantities are well defined in arbitrary probability spaces, existing estimators add or subtract entropies (we term them ΣH methods). These methods work only in purely discrete space or purely continuous case since entropy (or differential entropy) is well defined only in that regime. In this paper, we define a general graph divergence measure (GDM),as a measure of incompatibility between the observed distribution and a given graphical model structure. This generalizes the aforementioned information measures and we construct a novel estimator via a coupling trick that directly estimates these multivariate information measures using the Radon-Nikodym derivative. These estimators are proven to be consistent in a general setting which includes several cases where the existing estimators fail, thus providing the only known estimators for the following settings: (1) the data has some discrete and some continuous valued components (2) some (or all) of the components themselves are discrete-continuous mixtures (3) the data is real-valued but does not have a joint density on the entire space, rather is supported on a low-dimensional manifold. We show that our proposed estimators significantly outperform known estimators on synthetic and real datasets.

Neural Information Processing Systems http://nips.cc/

Designing codes that combat the noise in a communication medium has remained a significant area of research in information theory as well as wireless communications. Asymptotically optimal channel codes have been developed by mathematicians for communicating under canonical models after over 60 years of research. On the other hand, in many non-canonical channel settings, optimal codes do not exist and the codes designed for canonical models are adapted via heuristics to these channels and are thus not guaranteed to be optimal. In this work, we make significant progress on this problem by designing a fully end-to-end jointly trained neural encoder and decoder, namely, Turbo Autoencoder (TurboAE), with the following contributions: (a) under moderate block lengths, TurboAE approaches state-of-the-art performance under canonical channels; (b) moreover, TurboAE outperforms the state-of-the-art codes under non-canonical settings in terms of reliability. TurboAE shows that the development of channel coding design can be automated via deep learning, with near-optimal performance.

Designing codes that combat the noise in a communication medium has remained a significant area of research in information theory as well as wireless communications. Asymptotically optimal channel codes have been developed by mathematicians for communicating under canonical models after over 60 years of research. On the other hand, in many non-canonical channel settings, optimal codes do not exist and the codes designed for canonical models are adapted via heuristics to these channels and are thus not guaranteed to be optimal. In this work, we make significant progress on this problem by designing a fully end-to-end jointly trained neural encoder and decoder, namely, Turbo Autoencoder (TurboAE), with the following contributions: (a) under moderate block lengths, TurboAE approaches state-of-the-art performance under canonical channels; (b) moreover, TurboAE outperforms the state-of-the-art codes under non-canonical settings in terms of reliability. TurboAE shows that the development of channel coding design can be automated via deep learning, with near-optimal performance.

Estimation of mutual information from observed samples is a basic primitive in machine learning, useful in several learning tasks including correlation mining, information bottleneck, Chow-Liu tree, and conditional independence testing in (causal) graphical models. While mutual information is a quantity well-defined for general probability spaces, estimators have been developed only in the special case of discrete or continuous pairs of random variables. Most of these estimators operate using the 3H-principle, i.e., by calculating the three (differential) entropies of X, Y and the pair (X, Y). However, in general mixture spaces, such individual entropies are not well defined, even though mutual information is. In this paper, we develop a novel estimator for estimating mutual information in discrete-continuous mixtures. We prove the consistency of this estimator theoretically as well as demonstrate its excellent empirical performance. This problem is relevant in a wide-array of applications, where some variables are discrete, some continuous, and others are a mixture between continuous and discrete components.

Discovering a correlation from one variable to another variable is of fundamental scientific and practical interest. While existing correlation measures are suitable for discovering average correlation, they fail to discover hidden or potential correlations. To bridge this gap, (i) we postulate a set of natural axioms that we expect a measure of potential correlation to satisfy; (ii) we show that the rate of information bottleneck, i.e., the hypercontractivity coefficient, satisfies all the proposed axioms; (iii) we provide a novel estimator to estimate the hypercontractivity coefficient from samples; and (iv) we provide numerical experiments demonstrating that this proposed estimator discovers potential correlations among various indicators of WHO datasets, is robust in discovering gene interactions from gene expression time series data, and is statistically more powerful than the estimators for other correlation measures in binary hypothesis testing of canonical examples of potential correlations.

Information theoretic quantities play an important role in various settings in machine learning, including causality testing, structure inference in graphical models, time-series problems, feature selection as well as in providing privacy guarantees. A key quantity of interest is the mutual information and generalizations thereof, including conditional mutual information, multivariate mutual information, total correlation and directed information. While the aforementioned information quantities are well defined in arbitrary probability spaces, existing estimators add or subtract entropies (we term them ΣH methods). These methods work only in purely discrete space or purely continuous case since entropy (or differential entropy) is well defined only in that regime. In this paper, we define a general graph divergence measure (GDM),as a measure of incompatibility between the observed distribution and a given graphical model structure. This generalizes the aforementioned information measures and we construct a novel estimator via a coupling trick that directly estimates these multivariate information measures using the Radon-Nikodym derivative. These estimators are proven to be consistent in a general setting which includes several cases where the existing estimators fail, thus providing the only known estimators for the following settings: (1) the data has some discrete and some continuous valued components (2) some (or all) of the components themselves are discrete-continuous mixtures (3) the data is real-valued but does not have a joint density on the entire space, rather is supported on a low-dimensional manifold. We show that our proposed estimators significantly outperform known estimators on synthetic and real datasets.

The design of codes for communicating reliably over a statistically well defined channel is an important endeavor involving deep mathematical research and wideranging practical applications. In this work, we present the first family of codes obtained via deep learning, which significantly beats state-of-the-art codes designed over several decades of research. The communication channel under consideration is the Gaussian noise channel with feedback, whose study was initiated by Shannon; feedback is known theoretically to improve reliability of communication, but no practical codes that do so have ever been successfully constructed. We break this logjam by integrating information theoretic insights harmoniously with recurrent-neural-network based encoders and decoders to create novel codes that outperform known codes by 3 orders of magnitude in reliability. We also demonstrate several desirable properties in the codes: (a) generalization to larger block lengths; (b) composability with known codes; (c) adaptation to practical constraints. This result also presents broader ramifications to coding theory: even when the channel has a clear mathematical model, deep learning methodologies, when combined with channel-specific information-theoretic insights, can potentially beat state-of-the-art codes, constructed over decades of mathematical research.

Neural Information Processing Systems http://nips.cc/

Discovering a correlation from one variable to another variable is of fundamental scientific and practical interest. While existing correlation measures are suitable for discovering average correlation, they fail to discover hidden or potential correlations. To bridge this gap, (i) we postulate a set of natural axioms that we expect a measure of potential correlation to satisfy; (ii) we show that the rate of information bottleneck, i.e., the hypercontractivity coefficient, satisfies all the proposed axioms; (iii) we provide a novel estimator to estimate the hypercontractivity coefficient from samples; and (iv) we provide numerical experiments demonstrating that this proposed estimator discovers potential correlations among various indicators of WHO datasets, is robust in discovering gene interactions from gene expression time series data, and is statistically more powerful than the estimators for other correlation measures in binary hypothesis testing of canonical examples of potential correlations.