Terminal Dimension Reduction for Time Series with Applications

2026-07-10

AI Summary

This work addresses the limitation of existing terminal embedding methods in handling the ubiquitous linear interpolation structure in time series, which prevents the construction of dimension-independent coresets for clustering under the Fréchet distance. The authors propose a novel terminal embedding that preserves affine segment structure, thereby extending terminal embeddings to the time series setting for the first time. Their method leverages Johnson–Lindenstrauss (JL) embeddings to construct segment-preserving terminal embeddings and integrates coreset theory to achieve dimension-independent clustering approximations with guaranteed Fréchet distance preservation between any point and arbitrary locations in the full space. Experimental results on both synthetic and real-world datasets demonstrate that the proposed approach outperforms PCA, matches the performance of standard JL embeddings, and uniquely enables distance preservation across the entire space.

Abstract

Terminal embeddings have emerged as a powerful tool for dimension reduction. Given a set of points $P\subset \mathbb{R}^d$, a terminal embedding is a mapping $f:\mathbb{R}^d\rightarrow \mathbb{R}^t$ that preserves the pairwise distance between any pair of points $p\in P$ and $q\in \mathbb{R}^d$ up to small distortion under this mapping. Terminal embeddings have been particularly fruitful for constructing $k$-means and $k$-median coresets, where the objective is to find a typically weighted subset $Ω$ of $P$ such that for any candidate solution, the cost of the clustering objective on $Ω$ approximates the cost of the clustering objective on $P$ up to small distortion. Unfortunately, these techniques have not been extended to more complicated structures such as clustering time-series data under common straight-line interpolation between measurements. The main issue is that terminal embeddings, arguably the central technique in this line of research, cannot be linear and are thus not immediately suitable to preserve linear structures. In this work, we develop a generalization of terminal embeddings to affine line-segments that overcomes this issue. We showcase their applicability by using our lines-preserving terminal embeddings to obtain the first dimension-free coresets for clustering time-series under the Fréchet distance. The underlying dimension reduction uses Johnson-Lindenstrauss (JL) embeddings, and our experiments indicate that terminal embeddings perform similarly to JL and favorably against PCA for synthetic and real-world time-series, while only terminal embeddings extend pairwise distance preservation to the full ambient space.

Problem

• terminal embeddings
• time series clustering
• dimension reduction
• Fréchet distance
• coresets

Innovation

• terminal embedding
• time series clustering
• Fréchet distance
• dimension reduction
• coreset