Learning Scalar Quadratic Contrasts for High-Dimensional Change Point Localization

A high-dimensional distribution can change through many weak interactions while every coordinate mean and variance remains fixed. We study how to aggregate this signal into one quadratic contrast for change point localization. A normalized matrix representation identifies the Frobenius jump with the $L^2$ change in the degree-two density projection. For bounded densities with low-rank jumps in their degree-two coefficient matrices, a sufficiently strong pilot signal allows the matrix scan to learn a quadratic contrast retaining at least half of the Frobenius jump. With enough data to estimate its two means, independent evaluation observations then yield a small-jump localization error proportional to $kapp{a}^{-2}$, up to confidence factors, for a fixed density bound. The pilot pays for learning the aggregate contrast, while its retained signal governs scalar precision. Explicit matrix bounds extend detection and isolation to three temporal dependence models. Under geometric absolute regularity, deterministic spacing also yields a complete refinement guarantee in original time units. Experiments connect retained signal to localization accuracy, explain the effect of working rank, and quantify the sampling and resolution costs of spacing.