# High-Dimensional Nonparametric Change-Point Detection via Low-Rank Degree-Three Density Projection

arXiv: https://arxiv.org/abs/2608.15466 · HTML: https://mlsanity.com/paper/2608.15466/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 20 references that are in the map.
- First posted on arXiv 16 Aug 2026.

## Builds on (most cited first, 9 of 9)

- [Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression](https://mlsanity.com/paper/1804.02605/) (2018)
- [High-dimensional changepoint estimation via sparse projection](https://mlsanity.com/paper/1606.06246/) (2016)
- [Narrowest-Over-Threshold Detection of Multiple Change-points and Change-point-like Features](https://mlsanity.com/paper/1609.00293/) (2016)
- [Univariate Mean Change Point Detection: Penalization, CUSUM and Optimality](https://mlsanity.com/paper/1810.09498/) (2018)
- [Optimal nonparametric multivariate change point detection and localization](https://mlsanity.com/paper/1910.13289/) (2019)
- [Seeded Binary Segmentation: A general methodology for fast and optimal change point detection](https://mlsanity.com/paper/2002.06633/) (2020)
- [Localizing Changes in High-Dimensional Regression Models](https://mlsanity.com/paper/2010.10410/) (2020)
- [Change-point Detection for Sparse and Dense Functional Data in General Dimensions](https://mlsanity.com/paper/2205.09252/) (2022)
- [Estimation and Inference for Change Points in Functional Regression Time Series](https://mlsanity.com/paper/2405.05459/) (2024)

## Abstract (arXiv)

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 H3(X)∈Sym3(Rd+1) such that A(f)=EfH3(X) is an exact isometric encoding of the degree-three density projection: |A(f)-A(g)|F=|P3(f-g)|L2. Instead, fixed tensor contractions are degree-three polynomial chaoses with psi2/3 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 √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 OPp(kappa-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∈20,50,100,200 and a three-change d=100 sequence demonstrate the intended high-dimensional regime without materializing a (d+1)3 tensor.