Information Retention in Iterative Random Projection of Convex Bodies to Lower Dimensions

In this paper, we consider a bounded convex body $K_0 \subset \mathbb{R}^{n}$ subjected to two successive random orthogonal projections onto $\mathbb{R}^{n-1}$ and $\mathbb{R}^{n-2}$, respectively. First, we project $K_0$ orthogonally onto $U_{1}^{\perp}$, the orthogonal complement of $\mbox{\boldmath $U$}_1$, where $\mbox{\boldmath $U$}_1$ is uniformly distributed on the unit sphere $S^{n-1}$. This yields a random convex body $K_1 = \mathrm{Proj}_{{U_1}^{\perp}}(K_0) \subset \mathbb{R}^{n-1}$. We then repeat the process, projecting $K_1$ orthogonally onto ${U}_{2}^{\perp}$, the orthogonal complement of $\mbox{\boldmath $U$}_2$ chosen uniformly from the unit sphere in $\mbox{\boldmath $U$}_{1}^{\perp}$ or $S^{n-2}$, resulting in a second random convex body $K_2 = \mathrm{Proj}_{{U_2}^{\perp}}(K_1) \subset \mathbb{R}^{n-2}$. To quantify information retention through these sequential dimension reductions, we derive an upper bound for the conditional mutual information $I(K_1;K_2 \mid K_0)$. Furthermore, we extend this process to $m$ iterations and generalize the upper bound on $I(K_1;K_2 \mid K_0)$ to establish an analogous upper bound for $I(K_1;K_m \mid K_0)$. Finally, we examine the influence of $K_0$'s symmetry on the achievability of this upper bound for $I(K_1;K_m \mid K_0)$.