Mutual information (MI) is one of the most general ways to measure relationships between random variables, but estimating this quantity for complex systems is challenging. Denoising diffusion models have recently set a new bar for density estimation, so it is natural to consider whether these methods could also be used to improve MI estimation. Using the recently introduced information-theoretic formulation of denoising diffusion models, we show the diffusion models can be used in a straightforward way to estimate MI. In particular, the MI corresponds to half the gap in the Minimum Mean Square Error (MMSE) between conditional and unconditional diffusion, integrated over all Signal-to-Noise-Ratios (SNRs) in the noising process. Our approach not only passes self-consistency tests but also outperforms traditional and score-based diffusion MI estimators. Furthermore, our method leverages adaptive importance sampling to achieve scalable MI estimation, while maintaining strong performance even when the MI is high.

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.

Representation learning methods utilizing the InfoNCE loss have demonstrated considerable capacity in reducing human annotation effort by training invariant neural feature extractors. Although different variants of the training objective adhere to the information maximization principle between the data and learned features, data selection and augmentation still rely on human hypotheses or engineering, which may be suboptimal. For instance, data augmentation in contrastive learning primarily focuses on color jittering, aiming to emulate real-world illumination changes. In this work, we investigate the potential of selecting training data based on their mutual information computed from real-world distributions, which, in principle, should endow the learned features with better generalization when applied in open environments. Specifically, we consider patches attached to scenes that exhibit high mutual information under natural perturbations, such as color changes and motion, as positive samples for learning with contrastive loss. We evaluate the proposed mutual-information-informed data augmentation method on several benchmarks across multiple state-of-the-art representation learning frameworks, demonstrating its effectiveness and establishing it as a promising direction for future research.

We show that optimizing this lower bound is equivalent to maximizing the likelihood of a one-step improved policy on the offline dataset. Hence, we constrain the policy improvement direction to lie in the data manifold.

V ariational mutual information (MI) estimators are widely used in unsupervised representation learning methods such as contrastive predictive coding (CPC).

Evaluation of statistical dependencies between two data samples is a basic problem of data science/machine learning, and HSIC (Hilbert-Schmidt Information Criterion)~\cite{HSIC} is considered the state-of-art method. However, for size $n$ data sample it requires multiplication of $n\times n$ matrices, what currently needs $\sim O(n^{2.37})$ computational complexity~\cite{mult}, making it impractical for large data samples. We discuss HCR (Hierarchical Correlation Reconstruction) as its linear cost practical alternative, in tests of even higher sensitivity to dependencies, and additionally providing actual joint distribution model for chosen significance level, by description of dependencies through features being mixed moments, starting with correlation and homoscedasticity. Also allowing to approximate mutual information as just sum of squares of such nontrivial mixed moments between two data samples. Such single dependence describing feature is calculated in $O(n)$ linear time. Their number to test varies with dimension $d$ - requiring $O(d^2)$ for pairwise dependencies, $O(d^3)$ if wanting to also consider more subtle triplewise, and so on.

Estimating mutual information accurately is pivotal across diverse applications, from machine learning to communications and biology, enabling us to gain insights into the inner mechanisms of complex systems. Yet, dealing with high-dimensional data presents a formidable challenge, due to its size and the presence of intricate relationships. Recently proposed neural methods employing variational lower bounds on the mutual information have gained prominence. However, these approaches suffer from either high bias or high variance, as the sample size and the structure of the loss function directly influence the training process. In this paper, we propose a novel class of discriminative mutual information estimators based on the variational representation of the f -divergence. We investigate the impact of the permutation function used to obtain the marginal training samples and present a novel architectural solution based on derangements.

Diffusion bridge models have recently become a powerful tool in the field of generative modeling. In this work, we leverage their power to address another important problem in machine learning and information theory - the estimation of the mutual information (MI) between two random variables. We show that by using the theory of diffusion bridges, one can construct an unbiased estimator for data posing difficulties for conventional MI estimators. We showcase the performance of our estimator on a series of standard MI estimation benchmarks.