- The paper introduces a geometric extension of Jensen-Shannon divergence for Gaussian measures in infinite-dimensional Hilbert spaces.
- It derives closed-form expressions using Log-Determinant divergence and a regularized variant to handle non-equivalent measures.
- The findings enhance practical applications in fields like statistical physics and machine learning by improving high-dimensional data analysis.
Geometric Jensen-Shannon Divergence Between Gaussian Measures on Hilbert Space
The paper presents an exploration of the Geometric Jensen-Shannon (GJS) divergence in the domain of Gaussian measures defined on an infinite-dimensional Hilbert space. This paper fundamentally extends the classical Jensen-Shannon divergence, notable in information theory, into a geometric form accommodating Gaussian measures, which are crucial in probabilistic and statistical applications.
Mathematical and Theoretical Foundations
The authors provide a comprehensive formulation of the GJS divergence using Gaussian measures equivalent to a predetermined Gaussian measure, maintained within a Hilbert space. This choice of space imposes considerable mathematical challenges due to the absence of a natural reference measure, such as the Lebesgue measure in finite-dimensional settings, complicating the definition of Gaussian densities.
Construction and Extension
The paper leverages the closed-form expression of the GJS divergence, derived from the Log-Determinant divergence between positive definite trace class operators, introducing a Regularized Geometric Jensen-Shannon divergence. This regularized variant is applicable to any pair of Gaussian measures, transcending previous limitations which required equivalent measures.
Key formulaic constructions include the use of abstract means, particularly the geometric mean, which allows the derivation of weighted mixtures of Gaussian measures. This facilitates the development of new GJS divergence expressions expanding beyond traditional settings. The regularized expression recovers the exact GJS divergence when the regularization parameter approaches zero, enabling practical implementations that necessitate adjusting for non-equivalent measures.
Implications and Future Directions
The methodologies and results contribute to a methodological arsenal for handling Gaussian distributions in infinite-dimensional contexts, reasonably extending into areas such as statistical physics, quantum mechanics, and functional data analysis. As the analysis of infinite-dimensional spaces is computationally intensive, the closed form of the regularized divergence may provide substantial improvements in computational methodologies involving Gaussian measures.
The paper anticipates future work extending practical applications, especially within Artificial Intelligence and Machine Learning domains where handling infinite-dimensional data structures and distributions has become critical, particularly in high-dimensional Bayesian statistics and functional data analysis. Potential enhancements regarding computational efficiency and integration within existing machine learning frameworks are expected to yield impactful advancements in the handling of complex data structures.
Thus, the paper stands as a significant contribution to the mathematical theory of divergence measures and their applications in high-dimensional statistical learning and data science, providing foundational techniques ensuring robust analysis in complex probability spaces.