# Spectral Approximate Inference

## TL;DR

This paper introduces a spectral-based approximation method for computing the partition function in graphical models, overcoming limitations of local iterative algorithms by leveraging global spectral features for improved robustness and accuracy.

## Contribution

It presents a polynomial-time approximation scheme for low-rank GMs and a spectral mean-field scheme for high-rank GMs, enhancing robustness over prior methods.

## Key findings

- The spectral approach outperforms prior algorithms in accuracy.
- The method is robust and does not suffer from convergence issues.
- Experiments demonstrate improved efficiency and reliability.

## Abstract

Given a graphical model (GM), computing its partition function is the most essential inference task, but it is computationally intractable in general. To address the issue, iterative approximation algorithms exploring certain local structure/consistency of GM have been investigated as popular choices in practice. However, due to their local/iterative nature, they often output poor approximations or even do not converge, e.g., in low-temperature regimes (hard instances of large parameters). To overcome the limitation, we propose a novel approach utilizing the global spectral feature of GM. Our contribution is two-fold: (a) we first propose a fully polynomial-time approximation scheme (FPTAS) for approximating the partition function of GM associating with a low-rank coupling matrix; (b) for general high-rank GMs, we design a spectral mean-field scheme utilizing (a) as a subroutine, where it approximates a high-rank GM into a product of rank-1 GMs for an efficient approximation of the partition function. The proposed algorithm is more robust in its running time and accuracy than prior methods, i.e., neither suffers from the convergence issue nor depends on hard local structures, as demonstrated in our experiments.