On Geometric Asymmetry and Information in Sequential Dimension Reduction

Standard random projection techniques typically operate as a black box, mapping high-dimensional structures directly to a lower-dimensional space where the target dimension must be specified a \textit{priori}. To address scenarios where the optimal ultimate dimension is unknown, this paper investigates the retention of information through a sequential, step-by-step dimension reduction process. We examine a fixed, bounded convex body as it undergoes successive random orthogonal projections, systematically reducing the ambient dimension by one at each step. By demonstrating that this sequence of observed bodies forms a Markov chain, we quantify the information preserved through these reductions using the conditional mutual information between successive projections given the original convex body. We derive a theoretical upper bound on this conditional mutual information, parameterized by the Haar measure of the projection spaces that yield the same observed body. Leveraging the established Markov property, we extend these results to an arbitrary number of iterations, proving that the initial two-step bound characterizes information retention across the entire sequence of projections. Furthermore, by analyzing the projection space under the symmetry group of the initial body, we demonstrate that geometric asymmetry serves as a beneficial asset, resulting in higher overall information retention.