[Paper Review] The accessibility of convex bodies and derandomization of the hit and run algorithm

Benoı̂t Collins, Termeh Kousha+3 more|arXiv (Cornell University)|Dec 26, 2013

Markov Chains and Monte Carlo Methods19 references3 citations

TL;DR

This paper introduces the concept of accessibility for convex bodies in R^d, proving that any such body is k-accessible with constants depending only on dimension d. It proposes a derandomized hit-and-run algorithm using a fixed set of directions, which converges exponentially fast to the uniform distribution on the body, with convergence rate quantified in terms of the inradius-to-diameter ratio and accessibility parameter k.

ABSTRACT

We introduce the concept of accessibility and prove that any convex body $X$ in $\mathbb R^d$ is accessible with relevant constants depending on $d$ only. This property leads to a new algorithm which may be considered as a natural derandomization of the hit and run algorithm applied to generate a sequence of random points covering $X$ uniformly. We prove stability of the Markov chain generated by the proposed algorithm and provide its rate of convergence.

Motivation & Objective

• To formalize and prove that any convex body in R^d is k-accessible with k and l depending only on d.
• To develop a deterministic alternative to the stochastic hit-and-run algorithm for uniform sampling over convex bodies.
• To establish exponential convergence of the resulting Markov chain to the uniform (Lebesgue) measure on the convex body.
• To provide explicit bounds on the convergence rate in terms of geometric parameters like inradius, diameter, and accessibility k.
• To apply the algorithm to quantum information and statistical physics settings, such as quantum states and bistochastic matrices.

Proposed method

• Define k-accessibility: from any point in a convex body X, one can reach a fixed interior point x* in at most k steps along a fixed set of l vectors, staying within X.
• Prove that any convex body with an inscribed r-ball and diameter R is (d+1)-accessible using a finite cover of the sphere S^{d-1} with balls of radius r/2.
• Use John's theorem to show that any convex body contains a maximal volume ellipsoid, which allows reduction to the ellipsoidal case.
• Construct a Markov chain over X using deterministic steps along a finite set of directions {e_1, ..., e_l}, ensuring the chain is irreducible and aperiodic.
• Establish exponential convergence of the Markov chain to the uniform measure by analyzing the spectral gap via the accessibility parameter k and geometric ratios.
• Apply the algorithm to specific convex sets: quantum states, stochastic matrices, and bistochastic matrices, showing k-accessibility with explicit bounds.

Experimental results

Research questions

• RQ1 Can every convex body in R^d be reached from any point in finitely many steps using only a fixed finite set of directions, with bounds depending only on d?
• RQ2 Does a derandomized version of the hit-and-run algorithm exist that still converges exponentially fast to the uniform distribution on a convex body?
• RQ3 What is the rate of convergence of the proposed deterministic Markov chain, and how does it depend on geometric parameters like r/R and k?
• RQ4 Can the algorithm be effectively applied to structured convex sets arising in quantum information, such as the set of quantum states or bistochastic matrices?
• RQ5 How does the choice of direction set affect the convergence speed, and what is the minimal k for which exponential convergence is guaranteed?

Key findings

• Any convex body in R^d is (d+1)-accessible with respect to a finite set of vectors l ≤ (1 + 2R/r)^d + d, where R is the diameter and r is the inradius.
• The proposed derandomized algorithm converges exponentially fast to the uniform distribution on the convex body, with convergence rate depending on the ratio r/R and the accessibility parameter k.
• For the set of bistochastic matrices, the algorithm is (N-1)^3-accessible with k = (N-1)^3 and l = N^2(N-1)^2, ensuring exponential convergence.
• The convergence rate is quantified via the parameter α = (1 - θ)^{1/M}, where M = N(N-1)^2 and θ = b_{(N-1)^2} (N-1)^{-4N(N-1)^2}, showing explicit decay bounds.
• The algorithm fails to converge exponentially if the set is not k-accessible; for example, a triangle not accessible with respect to a given basis only covers an open dense subset, not the full measure.
• The method provides a practical alternative to standard hit-and-run sampling in high-dimensional settings where the volume of relevant subsets (e.g., PPT states) is exponentially small.