Towards Diverse and Comprehensive Benchmarks for Mutual Information Estimation

Mutual information (MI) estimation is a central problem in machine learning and statistics; however, existing benchmarks typically evaluate estimators on simplified, low-dimensional distributions, leaving their performance on complex, realistic data largely unexplored. We address this gap with a comprehensive benchmarking framework grounded in a unified copula-theoretic perspective that subsumes existing benchmarks as special cases. Within this framework, we propose two complementary families of tests: a copula-first family that systematically varies ground-truth MI, dimensionality, and marginal complexity using synthetic and flow-based transformations; and a marginals-first family that couples real-world image data with controlled dependency structures, extending the classic same-class-pairing paradigm. We use this suite to extensively evaluate three classes of estimators: non-parametric, discriminative, and generative. Contrary to prevailing assumptions, our results indicate that there is no universal winner: each category can systematically outperform all other estimators under specific setups. By analyzing these cases, we identify fundamental estimation barriers and propose new tests that more effectively stress these specific limitations. We share the open source code at https://github.com/VanessB/mutinfo.

Recurrent Transformer-Based Near- and Far-Field THz Wideband Channel Estimation for UM-MIMO

The integration of terahertz communications and ultra-massive multiple-input multiple-output (UM-MIMO) systems in 6G networks is motivated by their ability to enable unprecedented data rates, mitigate spectrum congestion, and enhance overall network performance. However, the enlarged antenna apertures and higher carrier frequencies in these systems increase the Rayleigh distance, causing users to span both the near-field and conventional far-field regions. Accurate spatial precoding thus requires exact channel estimation at the base station - a task made more challenging by the hybrid coexistence of near- and far-field effects and the limited number of digital chains available in hybrid beamforming architectures. In this paper, we propose a block recurrent transformer model to address this challenge. We demonstrate that a single transformer block equipped with state memory can be trained once and then iteratively applied for hybrid-field channel estimation. Furthermore, we train the model such that it generalizes to wireless channels with varying scatterer distances, different numbers of propagation paths, and wideband operation. Simulation results show that the proposed method achieves performance gains of approximately 5 dB and 7.5 dB in normalized mean squared error (NMSE) over state-of-the-art solutions in narrowband and wideband scenarios, respectively.

FMMI: Flow Matching Mutual Information Estimation

We introduce a novel Mutual Information (MI) estimator that fundamentally reframes the discriminative approach. Instead of training a classifier to discriminate between joint and marginal distributions, we learn a normalizing flow that transforms one into the other. This technique produces a computationally efficient and precise MI estimate that scales well to high dimensions and across a wide range of ground-truth MI values.

Curse of Slicing: Why Sliced Mutual Information is a Deceptive Measure of Statistical Dependence

Sliced Mutual Information (SMI) is widely used as a scalable alternative to mutual information for measuring non-linear statistical dependence. Despite its advantages, such as faster convergence, robustness to high dimensionality, and nullification only under statistical independence, we demonstrate that SMI is highly susceptible to data manipulation and exhibits counterintuitive behavior. Through extensive benchmarking and theoretical analysis, we show that SMI saturates easily, fails to detect increases in statistical dependence (even under linear transformations designed to enhance the extraction of information), prioritizes redundancy over informative content, and in some cases, performs worse than simpler dependence measures like the correlation coefficient.

Efficient Distribution Matching of Representations via Noise-Injected Deep InfoMax

Deep InfoMax (DIM) is a well-established method for self-supervised representation learning (SSRL) based on maximization of the mutual information between the input and the output of a deep neural network encoder. Despite the DIM and contrastive SSRL in general being well-explored, the task of learning representations conforming to a specific distribution (i.e., distribution matching, DM) is still under-addressed. Motivated by the importance of DM to several downstream tasks (including generative modeling, disentanglement, outliers detection and other), we enhance DIM to enable automatic matching of learned representations to a selected prior distribution. To achieve this, we propose injecting an independent noise into the normalized outputs of the encoder, while keeping the same InfoMax training objective. We show that such modification allows for learning uniformly and normally distributed representations, as well as representations of other absolutely continuous distributions. Our approach is tested on various downstream tasks. The results indicate a moderate trade-off between the performance on the downstream tasks and quality of DM.

Mutual Information Estimation via Normalizing Flows

We propose a novel approach to the problem of mutual information (MI) estimation via introducing normalizing flows-based estimator. The estimator maps original data to the target distribution with known closed-form expression for MI. We demonstrate that our approach yields MI estimates for the original data. Experiments with high-dimensional data are provided to show the advantages of the proposed estimator.

Information Bottleneck Analysis of Deep Neural Networks via Lossy Compression

The Information Bottleneck (IB) principle offers an information-theoretic framework for analyzing the training process of deep neural networks (DNNs). Its essence lies in tracking the dynamics of two mutual information (MI) values: one between the hidden layer and the class label, and the other between the hidden layer and the DNN input. According to the hypothesis put forth by Shwartz-Ziv and Tishby (2017), the training process consists of two distinct phases: fitting and compression. The latter phase is believed to account for the good generalization performance exhibited by DNNs. Due to the challenging nature of estimating MI between high-dimensional random vectors, this hypothesis has only been verified for toy NNs or specific types of NNs, such as quantized NNs and dropout NNs. In this paper, we introduce a comprehensive framework for conducting IB analysis of general NNs. Our approach leverages the stochastic NN method proposed by Goldfeld et al. (2019) and incorporates a compression step to overcome the obstacles associated with high dimensionality. In other words, we estimate the MI between the compressed representations of high-dimensional random vectors. The proposed method is supported by both theoretical and practical justifications. Notably, we demonstrate the accuracy of our estimator through synthetic experiments featuring predefined MI values. Finally, we perform IB analysis on a close-to-real-scale convolutional DNN, which reveals new features of the MI dynamics.

Soft-Output Deep Neural Network-Based Decoding

Deep neural network (DNN)-based channel decoding is widely considered in the literature. The existing solutions are investigated for the case of hard output, i.e. when the decoder returns the estimated information word. At the same time, soft-output decoding is of critical importance for iterative receivers and decoders. In this paper, we focus on the soft-output DNN-based decoding problem. We start with the syndrome-based approach proposed by Bennatan et al. (2018) and modify it to provide soft output in the AWGN channel. The new decoder can be considered as an approximation of the MAP decoder with smaller computation complexity. We discuss various regularization functions for joint DNN-MAP training and compare the resulting distributions for [64, 45] BCH code. Finally, to demonstrate the soft-output quality we consider the turbo-product code with [64, 45] BCH codes as row and column codes. We show that the resulting DNN-based scheme is very close to the MAP-based performance and significantly outperforms the solution based on the Chase decoder. We come to the conclusion that the new method is prospective for the challenging problem of DNN-based decoding of long codes consisting of short component codes.