Fast Exact Inference for Recursive Cardinality Models

TL;DR

This paper introduces Recursive Cardinality models, enabling efficient exact inference, marginalization, and sampling in high-order potential models, significantly improving computational efficiency and expressive power in probabilistic modeling.

Contribution

The paper presents a novel class of Recursive Cardinality models and an efficient O(D log^2 D) algorithm for exact marginalization and sampling, extending inference capabilities.

Findings

Efficient exact inference algorithms for Recursive Cardinality models.

Demonstrated the models' expressive power through empirical experiments.

Achieved faster inference in models with high-order potentials.

Abstract

Cardinality potentials are a generally useful class of high order potential that affect probabilities based on how many of D binary variables are active. Maximum a posteriori (MAP) inference for cardinality potential models is well-understood, with efficient computations taking O(DlogD) time. Yet efficient marginalization and sampling have not been addressed as thoroughly in the machine learning community. We show that there exists a simple algorithm for computing marginal probabilities and drawing exact joint samples that runs in O(Dlog2 D) time, and we show how to frame the algorithm as efficient belief propagation in a low order tree-structured model that includes additional auxiliary variables. We then develop a new, more general class of models, termed Recursive Cardinality models, which take advantage of this efficiency. Finally, we show how to do efficient exact inference in models composed of a tree structure and a cardinality potential. We explore the expressive power of Recursive Cardinality models and empirically demonstrate their utility.