发表机构
TU Dortmund; EPFL; CNRS & Université Paris Cité, IRIF; Aarhus University（图宾根大学; 苏黎世联邦理工学院; 国家科学研究中心及巴黎cité大学，IRIF; 奥胡斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究针对时间序列聚类等复杂结构降维问题，将终端嵌入推广到仿射线段，利用保线终端嵌入结合约翰逊 - 林登施特劳斯嵌入获得无维核心集，实验表明其在时间序列降维上表现良好且有独特优势。
AI 中文摘要
终端嵌入已成为一种强大的降维工具。给定一组点\(P\subset \mathbb{R}^d\)，终端嵌入是一个映射\(f:\mathbb{R}^d\rightarrow \mathbb{R}^t\)，在该映射下，任意两点\(p\in P\)和\(q\in \mathbb{R}^d\)的成对距离在小失真范围内得以保留。终端嵌入在构建\(k\)均值和\(k\)中位数核心集方面成果丰硕。但这些技术尚未扩展到更复杂的结构，如在测量之间的普通直线插值下对时间序列数据进行聚类。主要问题是终端嵌入不能是线性的，不适合保留线性结构。在这项工作中，我们将终端嵌入推广到仿射线段以克服此问题。我们使用保线终端嵌入获得了在弗雷歇距离下聚类时间序列的首个无维核心集，展示了其适用性。基础降维使用约翰逊 - 林登施特劳斯（JL）嵌入，实验表明终端嵌入在合成和真实世界时间序列上与JL表现相似且优于PCA，同时只有终端嵌入将成对距离保留扩展到整个环境空间。
英文摘要
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.
CommentsICML 2026
