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

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.