[Paper Review] A Polynomial Time MCMC Method for Sampling from Continuous DPPs

Shayan Oveis Gharan, Alireza Rezaei|arXiv (Cornell University)|Oct 20, 2018

Markov Chains and Monte Carlo Methods25 references4 citations

TL;DR

This paper presents the first polynomial-time Markov Chain Monte Carlo (MCMC) method for sampling from continuous k-Determinantal Point Processes (DPPs), using a Gibbs sampler with provable rapid mixing under a warm start. It establishes that the Gibbs sampler mixes in poly(k) steps and applies this to construct an efficient algorithm for sampling from k-DPPs with a spherical Gaussian kernel on the unit sphere in d dimensions, running in time polynomial in k and d.

ABSTRACT

We study the Gibbs sampling algorithm for continuous determinantal point processes. We show that, given a warm start, the Gibbs sampler generates a random sample from a continuous $k$-DPP defined on a $d$-dimensional domain by only taking $ ext{poly}(k)$ number of steps. As an application, we design an algorithm to generate random samples from $k$-DPPs defined by a spherical Gaussian kernel on a unit sphere in $d$-dimensions, $\mathbb{S}^{d-1}$ in time polynomial in $k,d$.

Motivation & Objective

• To address the open problem of designing efficient sampling algorithms with provable guarantees for continuous k-DPPs, which lack such methods despite extensive use in machine learning and physics.
• To analyze the mixing time of the Gibbs sampler for continuous k-DPPs and establish its rapid convergence under a warm start.
• To develop a practical sampling algorithm for k-DPPs defined by a spherical Gaussian kernel on the unit sphere in d dimensions.
• To provide theoretical bounds on the conductance and mixing time of the Gibbs sampler for continuous k-DPPs using spectral and geometric arguments.
• To demonstrate that the Gibbs sampler achieves polynomial mixing time when the kernel satisfies certain spectral conditions, particularly for the spherical Gaussian kernel.

Proposed method

• The paper employs the Gibbs sampler as the MCMC method, where at each step, a point is uniformly at random removed from the current k-point set and replaced by a new point y with probability proportional to the determinant of the updated kernel matrix.
• The mixing time is analyzed via conductance bounds, leveraging the relationship between conductance and total variation distance to the target k-DPP distribution.
• The analysis relies on spectral decomposition of the continuous kernel operator, particularly the spherical Gaussian kernel, and uses Mercer's theorem to relate eigenvalues to the trace of projection operators.
• The authors derive a lower bound on the expected determinant ratio after a Gibbs step, using the trace of the orthogonal projection of feature vectors onto the complement of the current point set.
• For the spherical Gaussian kernel, the method uses known eigenvalue asymptotics on the sphere to bound the sum of tail eigenvalues, ensuring a constant lower bound on the expected determinant increase under certain conditions on σ and d.
• The algorithm is instantiated for the spherical Gaussian kernel by showing that when σ ≤ 1/(2√log k), the expected determinant ratio is bounded away from zero, enabling polynomial mixing time.

Experimental results

Research questions

• RQ1 Can the Gibbs sampler for continuous k-DPPs be shown to mix in polynomial time under a warm start?
• RQ2 What spectral or geometric conditions on the kernel ensure rapid mixing of the Gibbs sampler for continuous k-DPPs?
• RQ3 Is it possible to design a provably efficient sampling algorithm for continuous k-DPPs with a spherical Gaussian kernel?
• RQ4 How does the expected determinant ratio behave under Gibbs updates, and can it be bounded from below to ensure fast mixing?
• RQ5 What is the dependence of the mixing time on k and d for continuous k-DPPs with the spherical Gaussian kernel?

Key findings

• The Gibbs sampler for continuous k-DPPs mixes in time O(k⁴) · log(var(π/μ₀)/ε), establishing a polynomial mixing time under a warm start.
• For the spherical Gaussian kernel on the unit sphere S^{d−1}, the Gibbs sampler achieves polynomial mixing time when σ ≤ 1/(2√log k), with the mixing time bounded by poly(k, d).
• The expected determinant ratio after a Gibbs step is bounded below by a constant when σ ≤ 1/(2√log k), ensuring good conductance and rapid mixing.
• The sum of the tail eigenvalues of the spherical Gaussian kernel operator is bounded below by Ω(1) under the same condition, which is crucial for the conductance bound.
• The method provides the first MCMC-based sampling algorithm with provable polynomial-time guarantees for continuous k-DPPs, filling a key gap in the literature.
• The algorithm runs in time polynomial in k and d for the spherical Gaussian kernel, making it efficient for practical applications in diverse sampling tasks.