A Sketching Method for Finding the Closest Point on a Convex Hull

TL;DR

This paper introduces a sketching algorithm that efficiently finds the closest point on a convex hull to an external query point, especially suitable for large, high-dimensional datasets in machine learning.

Contribution

The paper presents a novel sketching-based approach that accelerates the computation of the closest point on a convex hull by exploiting data structure and iterative optimization.

Findings

Faster convergence to the optimal solution compared to standard algorithms

Effective handling of large, high-dimensional datasets

Reduces computational complexity of convex hull proximity queries

Abstract

We develop a sketching algorithm to find the point on the convex hull of a dataset, closest to a query point outside it. Studying the convex hull of datasets can provide useful information about their geometric structure and their distribution. Many machine learning datasets have large number of samples with large number of features, but exact algorithms in computational geometry are usually not designed for such setting. Alternatively, the problem can be formulated as a linear least-squares problem with linear constraints. However, solving the problem using standard optimization algorithms can be very expensive for large datasets. Our algorithm uses a sketching procedure to exploit the structure of the data and unburden the optimization process from irrelevant points. This involves breaking the data into pieces and gradually putting the pieces back together, while improving the optimal solution using a gradient project method that can rapidly change its active set of constraints. Our method eventually leads to the optimal solution of our convex problem faster than off-the-shelf algorithms.