基于低秩三阶密度投影的高维非参数变点检测
High-Dimensional Nonparametric Change-Point Detection via Low-Rank Degree-Three Density Projection
- North Carolina State University（北卡罗来纳州立大学）
- Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对高维数据中均值、协方差无法检测到的分布变化，提出基于低秩三阶密度投影的非参数变点检测方法，通过种子最短区间算法实现精确变点定位，在多维度实验中验证了方法有效性。
AI中文摘要：
分布变化可能在均值和协方差层面不可见，却会出现在偏度、非对称交互作用或其他三阶结构中。我们开发了一种非参数变点检测方法，该方法保留密度的所有不超过三阶的系数，同时避免直接进行密度估计。对于取值在 $[-1,1]^d$ 中的观测值，我们构造了对称三阶勒让德特征张量 $H_3(X)\in\Sym^3(\R^{d+1})$，使得 $A(f)=\E_fH_3(X)$ 是三阶密度投影的精确等距编码：$\\|A(f)-A(g)\\|_{\F}=\\|P_3(f-g)\\|_{L^2}$。相反，固定张量收缩是具有 $\psi_{2/3}$ 尾的三阶多项式混沌。这两个项具有特征性的三阶张量缩放，并与简单随机张量的尖锐集中结果中的幂次相匹配。对于坐标正交的特殊情况，该界改进为 $\sqrt{\log d}$，并支持在数百维空间中实现前缀和。我们推导了精确的总体帐篷形状和定位裕度，引入了带填充局部重中心化步骤的种子最短区间算法，并通过归纳法证明了精确恢复：空递归段保持不活跃，每个未检测到的变点都保留一个平衡隔离区间，且最短活跃种子在重中心化前恰好包含一个变点。双向交叉拟合标量细化在小跳变区域达到 $O_{\Pp}(\kappa^{-2})$ 的定位性能，与纯三次族的 Le Cam 下界匹配，该族的二阶投影跳变恰好为零。在 $d\in\{20,50,100,200\}$ 上的可复现实验以及 $d=100$ 的三变点序列实验，证明了无需实现 $(d+1)^3$ 张量即可达到预期的高维区域。
英文摘要：
Distributional changes can be invisible to means and covariances yet appear in skewness, asymmetric interactions, or other third-order structure. We develop a nonparametric change-point method that retains every degree-at-most-three coefficient of a density while avoiding direct density estimation. For observations in $[-1,1]^d$, we construct a symmetric order-three Legendre feature tensor $H_3(X)\in\Sym^3(\R^{d+1})$ such that $A(f)=\E_fH_3(X)$ is an exact isometric encoding of the degree-three density projection: $\|A(f)-A(g)\|_{\F}=\|P_3(f-g)\|_{L^2}$. Instead, fixed tensor contractions are degree-three polynomial chaoses with $ψ_{2/3}$ tails. The two terms have the characteristic order-three tensor scaling and match the powers in sharp concentration results for simple random tensors. For a coordinate-orthogonal specialization, the bound improves to $\sqrt{\log d}$ and enables a prefix-sum implementation in hundreds of dimensions. We derive the exact population tent shape and localization margin, introduce a seeded shortest-interval algorithm with a padded local recentering step, and prove exact recovery by induction: null recursive segments remain inactive, every undetected change retains a balanced isolating interval, and the shortest active seed contains exactly one change before recentering. A two-way cross-fitted scalar refinement attains $O_{\Pp}(κ^{-2})$ localization in the small-jump regime, matching a Le Cam lower bound on a pure cubic family whose degree-two projection jump is exactly zero. Reproducible experiments at $d\in\{20,50,100,200\}$ and a three-change $d=100$ sequence demonstrate the intended high-dimensional regime without materializing a $(d+1)^3$ tensor.
