Dimension Reduction for Curves: Simplified and Generalized

2026-07-03

AI Summary

This work addresses the challenge of efficiently preserving continuous curve distances—such as the Fréchet distance—under dimensionality reduction for high-dimensional polygonal curves. The authors propose a randomized projection method based on sparse oblivious subspace embeddings that simultaneously approximates multiple curve dissimilarity measures, including Fréchet, q-DTW, and Hausdorff distances, within a relative error of (1±ε) using a target dimension of O(ε⁻² log(nm)). By constructing a unified framework for generalized curve distance metrics, the approach extends dimensionality reduction theory to piecewise linear surfaces, substantially simplifying existing analyses and broadening applicability across diverse curve comparison tasks.

Abstract

We revisit random projections for reducing the dimension of high-dimensional polygonal curves. Drawing from the toolbox of randomized linear algebra, we give a considerably simplified proof of the known $O(\varepsilon^{-2}\log(nm))$ bound on the target dimension of a random projection that preserves the continuous Fréchet distance of polygonal curves up to a factor $(1\pm\varepsilon)$. Our proof is based on the concept of sparse oblivious subspace embeddings. While previous techniques were limited to the case of the Fréchet distance, our techniques are fairly general and extend to all possible distance measures that involve the maximum, a sum or an integral over Euclidean distances between pairs of points on both input curves. We define a generalized dissimilarity measure for curves that includes several popular measures such as Fréchet, $q$-DTW, Hausdorff, etc. as special cases and show that the same dimension reduction technique works for this generalized dissimilarity measure. Finally, we apply the same framework for dimension reduction to piecewise linear surfaces, after extending the distance measure suitably to such surfaces.

Problem

• dimension reduction
• polygonal curves
• Fréchet distance
• random projections
• curve dissimilarity

Innovation

• dimension reduction
• random projections
• Fréchet distance
• oblivious subspace embeddings
• generalized dissimilarity measures