Random Projections and Sampling Algorithms for Clustering of High-Dimensional Polygonal Curves

TL;DR

This paper introduces a Johnson-Lindenstrauss projection for high-dimensional polygonal curves to enable efficient clustering using the Fréchet distance, with theoretical error analysis and empirical validation.

Contribution

It proposes a novel projection method for curves, analyzes the approximation limits of Fréchet distance, and provides a CUDA-accelerated clustering algorithm.

Findings

Sublinear complexity in clustering algorithms

Fréchet distance cannot be approximated within factor less than √2

Empirical validation of the proposed methods

Abstract

We study the $k$-median clustering problem for high-dimensional polygonal curves with finite but unbounded number of vertices. We tackle the computational issue that arises from the high number of dimensions by defining a Johnson-Lindenstrauss projection for polygonal curves. We analyze the resulting error in terms of the Fr\'echet distance, which is a tractable and natural dissimilarity measure for curves. Our clustering algorithms achieve sublinear dependency on the number of input curves via subsampling. Also, we show that the Fr\'echet distance can not be approximated within any factor of less than $\sqrt{2}$ by probabilistically reducing the dependency on the number of vertices of the curves. As a consequence we provide a fast, CUDA-parallelized version of the Alt and Godau algorithm for computing the Fr\'echet distance and use it to evaluate our results empirically.