Manifold-based transformation of probability distributions: application to the inverse problem of reconstructing distributions from experimental data
Information geometry is a mathematical framework that elucidates the manifold structure of the probability distribution space (p-space), providing a systematic approach to transforming probability distributions (PDs). In this study, we utilized information geometry to address the inverse problems associated with reconstructing PDs from experimental data. Our initial finding is that the Kullback-Leibler divergence, often considered non-metric owing to its asymmetry, can serve as a valid metric under specific geometric conditions on the manifold. Based on this finding, we formulated the manifold-based gradient descent (MBGD) method, which was employed to visualize the internal structures -- represented as PDs -- of two types of systems: those with static constituent elements and those with dynamic state transitions. Through the application of MBGD, we successfully reconstructed the underlying PDs for both types of systems, outperforming the standard gradient descent methods that neglect the manifold structure of p-space. Therefore, the present results demonstrate the essentiality of accounting for the manifold structure of p-space in the inverse problems of reconstructing PDs. The ability of MBGD to accurately reconstruct PDs for systems with dynamic state transitions underscores its potential to provide valuable physical insights by visualizing internal structures.