A GPU-Oriented Algorithm Design for Secant-Based Dimensionality Reduction

TL;DR

This paper introduces a GPU-accelerated algorithm for secant-based dimensionality reduction, enabling efficient computation of low-dimensional embeddings that preserve manifold geometry in high-dimensional data sets.

Contribution

The paper presents a novel polynomial-time GPU-based algorithm for secant set computation and data reduction, inspired by Whitney's embedding theorem, improving efficiency over traditional methods.

Findings

GPU implementation significantly reduces secant computation time

Algorithm produces meaningful low-dimensional data representations

Applicable to high-dimensional data sets with manifold structures

Abstract

Dimensionality-reduction techniques are a fundamental tool for extracting useful information from high-dimensional data sets. Because secant sets encode manifold geometry, they are a useful tool for designing meaningful data-reduction algorithms. In one such approach, the goal is to construct a projection that maximally avoids secant directions and hence ensures that distinct data points are not mapped too close together in the reduced space. This type of algorithm is based on a mathematical framework inspired by the constructive proof of Whitney's embedding theorem from differential topology. Computing all (unit) secants for a set of points is by nature computationally expensive, thus opening the door for exploitation of GPU architecture for achieving fast versions of these algorithms. We present a polynomial-time data-reduction algorithm that produces a meaningful low-dimensional representation of a data set by iteratively constructing improved projections within the framework described above. Key to our algorithm design and implementation is the use of GPUs which, among other things, minimizes the computational time required for the calculation of all secant lines. One goal of this report is to share ideas with GPU experts and to discuss a class of mathematical algorithms that may be of interest to the broader GPU community.