ProHD: Projection-Based Hausdorff Distance Approximation

TL;DR

ProHD is a projection-based approximation method that significantly speeds up Hausdorff distance computation on large, high-dimensional datasets while maintaining high accuracy, enabling practical applications in large-scale data analysis.

Contribution

ProHD introduces a novel projection-guided approximation algorithm for Hausdorff distance that guarantees bounded error and outperforms existing sampling methods in speed and accuracy.

Findings

Runs 10-100x faster than exact algorithms on large datasets.

Achieves 5-20x lower error than random sampling approximations.

Effective on datasets with up to two million points in 256 dimensions.

Abstract

The Hausdorff distance (HD) is a robust measure of set dissimilarity, but computing it exactly on large, high-dimensional datasets is prohibitively expensive. We propose \textbf{ProHD}, a projection-guided approximation algorithm that dramatically accelerates HD computation while maintaining high accuracy. ProHD identifies a small subset of candidate "extreme" points by projecting the data onto a few informative directions (such as the centroid axis and top principal components) and computing the HD on this subset. This approach guarantees an underestimate of the true HD with a bounded additive error and typically achieves results within a few percent of the exact value. In extensive experiments on image, physics, and synthetic datasets (up to two million points in $D=256$), ProHD runs 10--100$\times$ faster than exact algorithms while attaining 5--20$\times$ lower error than random sampling-based approximations. Our method enables practical HD calculations in scenarios like large vector databases and streaming data, where quick and reliable set distance estimation is needed.