arXiv subjects

Ching-Peng Huang

Publications and source records attributed to Ching-Peng Huang.

4 recordsLinked to original sources

arXiv2024

How do correlations shape the landscape of information?

We explore a few common models on how correlations affect information. The main model considered is the Shannon mutual information $I(S:R_1,\cdots, R_i)$ over distributions with marginals $P_{S,R_i}$ fixed for each $i$, with the analogy in which $S$ is the stimulus and $R_i$'s are neurons. We work out basic models in details, using algebro-geometric tools to write down discriminants that separate distributions with distinct qualitative behaviours in the probability simplex into toric chambers and evaluate the volumes of them algebraically. As a byproduct, we provide direct translation between a decomposition of mutual information inspired by a series expansion and one from partial information decomposition (PID) problems, characterising the synergistic terms of the former. We hope this paper serves for communication between communities especially mathematics and theoretical neuroscience on the topic. KEYWORDS: information theory, algebraic statistics, mathematical neuroscience, partial information decomposition

cs.IT↗

arXiv2023

Eigenvalue processes in light of Riemannian submersion and gradient flow of isospectral orbits

We prove eigenvalue processes from dynamical random matrix theory including Dyson Brownian motion, Wishart process, and Dynkin's Brownian motion of ellipsoids are results of projecting Brownian motion through Riemannian submersions induced by isometric action of compact Lie groups, whose orbits have nonzero mean curvature, which contributes to drift terms and is the log gradient of orbit volume function, showing in another way that eigenvalues collide whenever the fibre is degenerate. We thus provide a unified treatment and better connection between eigenvalue processes in different settings with the language of Riemannian geometry. Under such interpretation, we see how we can naturally recover eigenvector processes and derive $β$ process such as $β$-Dyson Brownian motion for general $β>0$. \textbf{KEYWORDS}: eigenvalue process, symmetric space, mean curvature flow, gradient flow, isospectral manifold, random matrix ensemble

math.PR↗

arXiv2023

Dyson Brownian Motion and motion by mean curvature

We construct Dyson Brownian motion for $β\in (0,\infty]$ by adapting the extrinsic construction of Brownian motion on Riemannian manifolds to the geometry of group orbits within the space of Hermitian matrices. When $β$ is infinite, the eigenvalues evolve by Coulombic repulsion and the group orbits evolve by motion by (minus one half times) mean curvature.

math.PR↗

arXiv2022

A model of invariant control system using mean curvature drift from Brownian motion under submersions

Given a submersion $ϕ: M \to N$, where $M$ is Riemannian, we construct a stochastic process $X$ on $M$ such that the image $Y:=ϕ(X)$ is a (reversed, scaled) mean curvature flow of the fibers of the submersion. The model example is the mapping $π: GL(n) \to GL(n)/O(n)$, whose image is equivalent to the space of $n$-by-$n$ positive definite matrices, $\pdef$, and the said MCF has deterministic image. We are able to compute explicitly the mean curvature (and hence the drift term) of the fibers w.r.t. this map, (i) under diagonalization and (ii) in matrix entries, writing mean curvature as the gradient of log volume of orbits. As a consequence, we are able to write down Brownian motions explicitly on several common homogeneous spaces, such as Poincaré's upper half plane and the Bures-Wasserstein geometry on $\pdef$, on which we can see the eigenvalue processes of Brownian motion reminiscent of Dyson's Brownian motion. By choosing the background metric via natural $GL(n)$ action, we arrive at an invariant control system on the $GL(n)$-homogenous space $GL(n)/O(n)$. We investigate feasibility of developing stochastic algorithms using the mean curvature flow. KEY WORDS: mean curvature flow, gradient flow, Brownian motion, Riemannian submersion, random matrix, eigenvalue processes, geometry of positive definite matrices, stochastic algorithm, control theory on homogeneous space

math.PR↗