On the Effectiveness of Hybrid Mutual Information Estimation (2306.00608v2)

Abstract: Estimating the mutual information from samples from a joint distribution is a challenging problem in both science and engineering. In this work, we realize a variational bound that generalizes both discriminative and generative approaches. Using this bound, we propose a hybrid method to mitigate their respective shortcomings. Further, we propose Predictive Quantization (PQ): a simple generative method that can be easily combined with discriminative estimators for minimal computational overhead. Our propositions yield a tighter bound on the information thanks to the reduced variance of the estimator. We test our methods on a challenging task of correlated high-dimensional Gaussian distributions and a stochastic process involving a system of free particles subjected to a fixed energy landscape. Empirical results show that hybrid methods consistently improved mutual information estimates when compared to the corresponding discriminative counterpart.

• The paper demonstrates that combining generative (using Predictive Quantization) and discriminative methods reduces estimator variance in mutual information estimation.
• The proposed variational bound unifies the strengths of both approaches to yield tighter and more accurate MI estimates.
• Empirical tests on high-dimensional Gaussian mixtures and free-particle systems confirm the method's superior accuracy and practical versatility.

On the Effectiveness of Hybrid Mutual Information Estimation

The paper explores a novel method for estimating mutual information (MI) by introducing a hybrid approach that generalizes existing generative and discriminative techniques. Researchers Marco Federici, David Ruhe, and Patrick Forré aim to address the limitations inherent in traditional MI estimation methods by leveraging the complementary strengths of generative and discriminative frameworks.

Mutual Information, a critical concept in information theory, quantifies the dependency between two random variables. Estimating MI is challenging when dealing with unknown or intractable probability distributions. Traditional estimators falter as data becomes higher-dimensional, prompting the need for more sophisticated techniques.

Central to this paper is the development of a variational bound that encapsulates both generative and discriminative methods. By doing so, the hybrid method mitigates the shortcomings associated with each approach. Generative models often lack flexibility, while discriminative models suffer from high bias-variance trade-offs under large information estimates. The proposed hybrid approach integrates these models, potentially improving the accuracy of MI estimation by decreasing variance.

The researchers introduce Predictive Quantization (PQ) as a novel, generative method. PQ utilizes a quantized or discrete representation of data to approximate usually intractable quantities, such as marginal entropy. With minimal computational overhead, PQ can be combined with existing discriminative estimators. This hybrid approach yields tighter bounds on MI by minimizing the estimator variance.

Empirically, the authors demonstrate the efficacy of their methods using two scenarios: a mixture of high-dimensional Gaussian distributions and a system of free particles under a fixed energy landscape. Results unequivocally show that hybrid methods surpass purely discriminative approaches in providing more accurate MI estimates.

For theoretical insights, the paper examines variational MI estimation, where a lower bound on MI is utilized for optimization. The unification of methods involves parameterizing joint distributions as normalized exponentials and computing MI through a Monte Carlo approach. The decomposition of MI into components of generative and discriminative origins underscores the potential for improved accuracy with hybrid methods.

From a practical standpoint, this research presents significant advancements in areas relying on MI estimation, such as self-supervised learning and Bayesian experimental design. The hybrid approach fosters versatility and scalability, making it viable for diverse applications in machine learning and beyond.

Looking forward, the integration of simple or non-parametric proposals with discriminative models may further enhance representation learning and information maximization techniques. While the current paper focuses on statistical robustness, the potential for applying these methods to larger, more complex datasets remains an exciting avenue for future exploration.