Identifiability and Stability of Generative Drifting in the Companion-Elliptic Kernel Family

Abstract

A drifting model is a one-step generator trained by moving each sample along a field of kernel-weighted attraction toward data samples and repulsion between model samples; training halts once this field vanishes. The soundness of this scheme rests on two questions: whether a zero-field equilibrium guarantees agreement with the data distribution, and how the error is controlled when the field is small. We answer both questions. We introduce the companion-elliptic kernel class, which contains the Laplace kernel, and prove that a vanishing field identifies arbitrary Borel probability measures within this class and its blockwise product extension. The scalar companion-elliptic kernels are exactly the Gaussian and Mat\'ern families, and deconvolution, the recovery of a measure from its kernel-smoothed density, is possible precisely when the zero set of the kernel's Fourier transform has empty interior. At approximate equilibria, a mass-escape counterexample shows that field information on a bounded region cannot control the global error. We therefore prove an explicit stability inequality bounding the error by the field residual on the observation region plus the convolution mass outside it. In this inequality the Gaussian kernel requires only field values but amplifies high-frequency errors exponentially, whereas the Mat\'ern kernel requires one additional order of derivative information in exchange for polynomial amplification. Since this gap widens exponentially with frequency, the balance favors the Mat\'ern family and hence the Laplace kernel.