On the Kullback-Leibler divergence between discrete normal distributions

Abstract: Discrete normal distributions are defined as the distributions with prescribed means and covariance matrices which maximize entropy on the integer lattice support. The set of discrete normal distributions form an exponential family with cumulant function related to the Riemann theta function. In this paper, we present several formula for common statistical divergences between discrete normal distributions including the Kullback-Leibler divergence. In particular, we describe an efficient approximation technique for calculating the Kullback-Leibler divergence between discrete normal distributions via the R\'enyi $\alpha$-divergences or the projective $\gamma$-divergences.

• The paper analyzes Kullback-Leibler divergence for discrete normal distributions, providing formulations and efficient approximation methods to address computational challenges.
• Discrete normal distributions belong to a unique exponential family whose structure and parameterization are linked to the complex Riemann theta function, posing significant computational difficulties.
• These theoretical advancements and computational techniques for discrete normal distributions have practical implications for fields requiring robust data handling, such as differential privacy and cryptography.

Analyzing the Kullback-Leibler Divergence between Discrete Normal Distributions

The paper under discussion presents an intricate study of the Kullback-Leibler divergence (KLD) between discrete normal distributions. Discrete normal distributions, defined on the integer lattice, maximize entropy while maintaining prescribed mean and covariance matrix, akin to their continuous counterparts. As a result, they form a unique exponential family whose cumulant functions relate to the Riemann theta function. This study provides significant insights into the mathematical properties and computational approaches for dealing with statistical divergences specifically tailored for discrete normal distributions.

Summary of Key Results

1. Exponential Family Structure:

Discrete normal distributions are characterized by their placement in an exponential family context. This paper explores their parameterization via natural parameters correlated with the Riemann theta function. Unlike continuous normal distributions, the discrete variants require novel approaches due to their discrete lattice support.

2. Statistical Divergences:

The study explores various statistical divergences, providing explicit formulations for the R\'enyi $\alpha$-divergence, cross-entropy, and the KLD between discrete normal distributions. A notable contribution is an efficient approximation technique for calculating the KLD, which is essential for practical applications given the intensive computation required for theta functions.

3. Computing Challenges and Approximations:

Sophisticated methods are developed for approximating the complex Riemann theta function, crucial for computing divergences. The paper covers the intricacies of transforming between natural and moment parameters numerically, confronting the lack of closed-form expressions with robust computational techniques.

4. Applications and Implications:

The theoretical advancements have implications for fields such as differential privacy and cryptography, where discrete normal distributions facilitate robust data handling. This research underscores the significance of exploring discrete counterparts to continuous families in algorithmic implementations necessitating high precision and efficiency.

Implications and Future Directions

This work not only advances the understanding of discrete normal distributions but also offers a practical roadmap for their application in computationally demanding tasks. The efficacy of the proposed estimation techniques and divergence measures open avenues for further research into refined algorithms for theta function approximation and its integration in machine learning algorithms like Riemann-Theta Boltzmann machines.

Given the expanding scope of artificial intelligence and data privacy demands, these findings will likely influence the development of algorithms in privacy-preserving data analytics and secure communications. Future work could explore the integration of these mathematical innovations in real-time applications, potential extensions to other types of exponential families, and continued optimization of numerical methods for higher-dimensional lattices.

Conclusion

The paper provides an authoritative mathematical treatment of Kullback-Leibler divergence in the context of discrete normal distributions, showcasing how classical information geometry intersects with modern computational needs. It exemplifies how abstract mathematical constructs like theta functions can have direct, impactful applications in fields reliant on discrete statistics and secure data handling.