Approximating the Permanent by Sampling from Adaptive Partitions

TL;DR

This paper introduces AdaPart, an efficient sampling method for approximating the permanent of non-negative matrices, enabling faster computations and improved multi-target tracking performance.

Contribution

AdaPart provides a simple, efficient way to sample exactly from complex distributions, enabling tight bounds on the permanent with guaranteed polynomial runtime.

Findings

AdaPart achieves over 25x speedup compared to previous methods.

Exact sampling improves multi-target tracking with fewer samples.

The method guarantees polynomial runtime for dense matrices.

Abstract

Computing the permanent of a non-negative matrix is a core problem with practical applications ranging from target tracking to statistical thermodynamics. However, this problem is also #P-complete, which leaves little hope for finding an exact solution that can be computed efficiently. While the problem admits a fully polynomial randomized approximation scheme, this method has seen little use because it is both inefficient in practice and difficult to implement. We present AdaPart, a simple and efficient method for drawing exact samples from an unnormalized distribution. Using AdaPart, we show how to construct tight bounds on the permanent which hold with high probability, with guaranteed polynomial runtime for dense matrices. We find that AdaPart can provide empirical speedups exceeding 25x over prior sampling methods on matrices that are challenging for variational based approaches. Finally, in the context of multi-target tracking, exact sampling from the distribution defined by the matrix permanent allows us to use the optimal proposal distribution during particle filtering. Using AdaPart, we show that this leads to improved tracking performance using an order of magnitude fewer samples.