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 $H_3(X)\in {\text{Sym}}^3({\mathbb{R}}^{d+1})$ such that $A(f)={\text{E}}_f{H}_3(X)$ is an exact isometric encoding of the degree-three density projection: $\mathrm{\parallel }A(f)-A(g){\mathrm{\parallel }}_{\text{F}}=\mathrm{\parallel }{P}_3(f-g){\mathrm{\parallel }}_{L^2}$. Instead, fixed tensor contractions are degree-three polynomial chaoses with ${\psi }_{2\mathrm{/}3}$ 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 $O_{\text{Pp}}({\kappa }^{-2})$ 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 $d\in \{20,50,100,200\}$ and a three-change $d=100$ sequence demonstrate the intended high-dimensional regime without materializing a $(d+1{)}^3$ tensor.