Improving Numerical Stability of Normalized Mutual Information Estimator on High Dimensions

TL;DR

This paper introduces a logarithmic transformation technique to improve the numerical stability of normalized mutual information estimation in high-dimensional data, preventing overflow and maintaining accuracy.

Contribution

It proposes a novel logarithmic transformation method that enhances the stability of k-NN based mutual information estimators in high-dimensional spaces.

Findings

Transformation prevents numerical overflow in high dimensions

Maintains estimator precision and variance

No significant computational overhead introduced

Abstract

Mutual information provides a powerful, general-purpose metric for quantifying the amount of shared information between variables. Estimating normalized mutual information using a k-Nearest Neighbor (k-NN) based approach involves the calculation of the scaling-invariant k-NN radius. Calculation of the radius suffers from numerical overflow when the joint dimensionality of the data becomes high, typically in the range of several hundred dimensions. To address this issue, we propose a logarithmic transformation technique that improves the numerical stability of the radius calculation in high-dimensional spaces. By applying the proposed transformation during the calculation of the radius, numerical overflow is avoided, and precision is maintained. Proposed transformation is validated through both theoretical analysis and empirical evaluation, demonstrating its ability to stabilize the calculation without compromising precision, increasing bias, or adding significant computational overhead, while also helping to maintain estimator variance.