分段代数曲线的连续动态时间规整（CDTW）距离近似
Approximating CDTW Distance of Piecewise Algebraic Curves
浏览论文内容
中文总结 AI 辅助
针对现有CDTW算法的局限，本文提出一种适用于欧几里得范数下分段高阶代数曲线的CDTW距离FPTAS，给出了其复杂度，填补了相关算法的空白。
中文摘要 AI 辅助
曲线作为输入数据自然出现在金融、地震学、医学、时空数据挖掘、恶意活动检测等多个领域。分析这些数据集的常用方法是相似度匹配或聚类，用于衡量曲线相似度的最常见指标是动态时间规整（DTW）和弗雷歇（Fréchet）距离，这两个指标分别对采样率和异常值敏感，无法产生鲁棒的结果。连续动态时间规整（CDTW）是一种更鲁棒的距离指标，对DTW和Fréchet距离进行了改进。现有的CDTW算法要么是针对非欧几里得范数和分段线性曲线的精确算法，要么是仅适用于分段线性曲线的近似算法。本文提出了一种在欧几里得范数下计算分段（高阶）代数曲线CDTW距离的近似算法，即提出了一个乘法误差为ε的全多项式时间近似方案（FPTAS），其复杂度为O((m+n)^(19/6)·(1/ε)^(10/3)·log((m+n)/ε²))，其中m和n分别是两条输入曲线的分段数量。
英文摘要
Curves as input data naturally arise in a variety of fields including finance, seismology, medicine, spatio-temporal data mining, malicious activity detection, and more. A common way to analyze these data sets is to do similarity matching or clustering. The most common metrics used for measuring similarity of curves are Dynamic Time Warping (DTW) and Fréchet distance. These metrics are sensitive to sampling rate and outliers respectively, and do not yield robust outcomes. Continuous Dynamic Time Warping (CDTW) is a more robust distance metric that improves upon DTW and Fréchet distances. Existing algorithms for CDTW are either exact algorithms that focus on non-Euclidean norms and piecewise linear curves, or approximation algorithms limited to piecewise linear curves. We present an approximation algorithm for computing the CDTW distance under Euclidean norm between piecewise (higher degree) algebraic curves. That is, we present a fully polynomial-time approximation scheme (FPTAS) of multiplicative error $\varepsilon$, with $O \left( (m+n)^{\frac{19}{6}} (\frac{1}{\varepsilon})^{\frac{10}{3}} \log \left( \frac{ (m+n) }{\varepsilon^2} \right) \right)$ complexity, where $m$ and $n$ are the number of pieces of the two input curves.
发表机构
- UT San Antonio（圣安东尼奥大学）
- The Harker School（哈克学校）
机构由 AI 辅助整理，请以论文原文为准。
