Spectral Compression for Cubic Change Detection

Changes in third-order interactions can preserve means and covariances. A change spread across many weak cubic coefficients may nevertheless concentrate in a few singular directions. We study when these directions can be learned and used for inference. For bounded marginal densities, unfolding SVD separates a spectral tail from estimation noise weighted by rank. A geometric inequality converts this error into a retained scalar jump, and a common spectral-energy condition supplies both an initial window and an informative direction. Under geometric absolute regularity, we derive the cubic matrix covariance scale and obtain a uniform concentration bound with an improved leading term. A valid dense dependent model exhibits a strict separation between the resulting sufficient conditions and those from residue-class coupling, including coefficient alternatives. For independent observations, validation and anchor conditions yield a calibrated test and a leading inverse-squared-jump localization bound in the small-signal regime. Related guarantees under dependence follow through analytic scanning and a separate joint-law comparison after thinning. Numerical illustrations examine the spectral construction and selected signal.