Identifiability and Stability of Generative Drifting with Companion-Elliptic Kernel Families

TL;DR

This paper investigates the conditions under which drifting fields in generative models uniquely identify distributions and how stable these identifications are, introducing companion-elliptic kernels to address limitations of standard Gaussian-based methods.

Contribution

It introduces companion-elliptic kernel families, including Gaussian and Matérn kernels, and proves identifiability and stability results for drifting fields within this class.

Findings

Field identifiability holds for arbitrary probability measures.

Mass escaping to infinity can prevent weak convergence without tightness.

A single scalar observable can detect missing mass and ensure weak convergence.

Abstract

This paper studies the identifiability and stability of drifting fields within the framework of Generative Modeling via Drifting. The motivating question is whether a zero-drift equilibrium identifies the target distribution, and whether an approximate zero drift implies weak distributional convergence. Since the original drifting model employs the Laplace kernel by default, we first analyze why standard Gaussian score-based arguments fail to apply. This analysis motivates the introduction of companion-elliptic kernel families, which are characterized by a companion potential satisfying an elliptic closure relation. We show that this class naturally contains the Laplace kernel and consists precisely of Gaussian and Mat\'ern kernels with smoothness parameter $\nu\ge 1/2$. Within this class, we establish field identifiability for arbitrary Borel probability measures on $\mathbb{R}^d$: if the drifting field vanishes identically, then the two measures must coincide. As for stability, we demonstrate that field convergence alone does not guarantee weak convergence, since mass may escape to infinity while remaining invisible to the field. Although tightness of the sequence directly removes this obstruction and restores weak stability, we prove that, even without tightness, every $C_0$-vague cluster point lies exactly on the defect ray $\{cp:0\le c\le1\}$. Consequently, a single scalar $C_0$-observable suffices to detect the missing mass and recover weak convergence.