LSMI-Sinkhorn: Semi-supervised Mutual Information Estimation with Optimal Transport

TL;DR

This paper introduces LSMI-Sinkhorn, a semi-supervised method for mutual information estimation that effectively utilizes limited paired data and unpaired samples through optimal transport, with applications in image matching and summarization.

Contribution

The paper proposes a novel semi-supervised mutual information estimation method using optimal transport and quadratic programming, along with an efficient LSMI-Sinkhorn algorithm.

Findings

Accurately estimates mutual information with few paired samples.

Demonstrates effectiveness in image matching tasks.

Shows versatility across various machine learning applications.

Abstract

Estimating mutual information is an important statistics and machine learning problem. To estimate the mutual information from data, a common practice is preparing a set of paired samples $\{(\mathbf{x}_i,\mathbf{y}_i)\}_{i=1}^n \stackrel{\mathrm{i.i.d.}}{\sim} p(\mathbf{x},\mathbf{y})$. However, in many situations, it is difficult to obtain a large number of data pairs. To address this problem, we propose the semi-supervised Squared-loss Mutual Information (SMI) estimation method using a small number of paired samples and the available unpaired ones. We first represent SMI through the density ratio function, where the expectation is approximated by the samples from marginals and its assignment parameters. The objective is formulated using the optimal transport problem and quadratic programming. Then, we introduce the Least-Squares Mutual Information with Sinkhorn (LSMI-Sinkhorn) algorithm for efficient optimization. Through experiments, we first demonstrate that the proposed method can estimate the SMI without a large number of paired samples. Then, we show the effectiveness of the proposed LSMI-Sinkhorn algorithm on various types of machine learning problems such as image matching and photo album summarization. Code can be found at https://github.com/csyanbin/LSMI-Sinkhorn.