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Integrating High-Dimensional Functions Deterministically

David Gamarnik, Devin Smedira

TL;DR

The paper develops a deterministic, quasi-polynomial-time algorithm to approximate high-dimensional integrals of separable functions defined on bounded domains by recasting the integral as a partition function of a graphical model with continuous edge potentials on a hypergraph. By enforcing positive, bounded edge potentials and applying a correlation-decay analysis to a discretized Riemann sum, the authors design recursive approximations of marginal probabilities that yield a controllable error, culminating in an overall relative error of O(1/n) for the integral. The approach crucially hinges on a contractive gradient bound for an auxiliary function g_m, which enables truncation of the recursion depth without sacrificing accuracy, and results in a runtime of n^{O(log n)} under fixed degree and parameter constants. This constitutes a deterministic alternative to randomized high-dimensional integration methods for a broad class of continuous, non-product-form graphical models. The work thus broadens the toolkit for deterministic approximation of complex integrals in high dimensions with potential applications in probabilistic inference and statistical physics.

Abstract

We design a Quasi-Polynomial time deterministic approximation algorithm for computing the integral of a multi-dimensional separable function, supported by some underlying hyper-graph structure, appropriately defined. Equivalently, our integral is the partition function of a graphical model with continuous potentials. While randomized algorithms for high-dimensional integration are widely known, deterministic counterparts generally do not exist. We use the correlation decay method applied to the Riemann sum of the function to produce our algorithm. For our method to work, we require that the domain is bounded and the hyper-edge potentials are positive and bounded on the domain. We further assume that upper and lower bounds on the potentials separated by a multiplicative factor of $1 + O(1/Δ^2)$, where $Δ$ is the maximum degree of the graph. When $Δ= 3$, our method works provided the upper and lower bounds are separated by a factor of at most $1.0479$. To the best of our knowledge, our algorithm is the first deterministic algorithm for high-dimensional integration of a continuous function, apart from the case of trivial product form distributions.

Integrating High-Dimensional Functions Deterministically

TL;DR

The paper develops a deterministic, quasi-polynomial-time algorithm to approximate high-dimensional integrals of separable functions defined on bounded domains by recasting the integral as a partition function of a graphical model with continuous edge potentials on a hypergraph. By enforcing positive, bounded edge potentials and applying a correlation-decay analysis to a discretized Riemann sum, the authors design recursive approximations of marginal probabilities that yield a controllable error, culminating in an overall relative error of O(1/n) for the integral. The approach crucially hinges on a contractive gradient bound for an auxiliary function g_m, which enables truncation of the recursion depth without sacrificing accuracy, and results in a runtime of n^{O(log n)} under fixed degree and parameter constants. This constitutes a deterministic alternative to randomized high-dimensional integration methods for a broad class of continuous, non-product-form graphical models. The work thus broadens the toolkit for deterministic approximation of complex integrals in high dimensions with potential applications in probabilistic inference and statistical physics.

Abstract

We design a Quasi-Polynomial time deterministic approximation algorithm for computing the integral of a multi-dimensional separable function, supported by some underlying hyper-graph structure, appropriately defined. Equivalently, our integral is the partition function of a graphical model with continuous potentials. While randomized algorithms for high-dimensional integration are widely known, deterministic counterparts generally do not exist. We use the correlation decay method applied to the Riemann sum of the function to produce our algorithm. For our method to work, we require that the domain is bounded and the hyper-edge potentials are positive and bounded on the domain. We further assume that upper and lower bounds on the potentials separated by a multiplicative factor of , where is the maximum degree of the graph. When , our method works provided the upper and lower bounds are separated by a factor of at most . To the best of our knowledge, our algorithm is the first deterministic algorithm for high-dimensional integration of a continuous function, apart from the case of trivial product form distributions.
Paper Structure (8 sections, 7 theorems, 48 equations, 2 algorithms)

This paper contains 8 sections, 7 theorems, 48 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model and the results
  3. Preliminary Results
  4. Correlation Decay
  5. Computation of Derivatives
  6. Bounding the Gradient
  7. The Approximation Algorithm
  8. From Discrete to Continuous

Key Result

Theorem 1

Suppose we have a function $f$ taking the form of equation eq:f def with respect to some hypergraph $G$. Further, suppose each function $\Phi_e$ is differentiable, satisfies the condition in equation eq:Phi assum, and there exists some $k > 0$ such that every $\Phi_e$ satisfies $\|\nabla \Phi_e(\tex Taking $k, \Delta,$ and $\eta$ to be constants, this algorithm will run in $n^{O(\log(n))}$ time.

Theorems & Definitions (13)

  • Theorem 1: Main Result
  • Proposition 2
  • proof
  • Theorem 3: Bounded Gradients
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • ...and 3 more