Finite-Probe Total-Variation Certificates for Finite-Basis Drifting Models

2026-08-02

AI Summary

This work addresses the challenge of reliably quantifying the discrepancy between a target and a model distribution when vector field information is accessible only through noisy observations at limited locations. The paper introduces the first confidence upper bound framework based on total variation (TV), integrating probe-dependent operators, an L¹ residual radius, and observability boundaries, while supporting a reject option mechanism. Leveraging the linear structure of unnormalized densities, Gaussian/RBF and Laplacian kernels, empirical Bernstein inequalities, and Gram matrix analysis, the method reduces to mean matching in the high-bandwidth regime. Experiments demonstrate that the proposed TV certificate is robust to preset or adaptive radii, non-zero residuals, rounding bounds, and active rejection on synthetic data, and remains stable under joint growth of dimensionality and basis size.

Abstract

Drifting objectives compare a target and model distribution through a vector field observed noisily at finitely many locations. We ask what distributional conclusion such a frozen measurement system warrants. For integrable antisymmetric interactions and absolutely continuous laws in a declared finite density basis, the unnormalized sampled numerator satisfies $\operatorname{vec}(V_X)=Mc$, where $c$ is an antisymmetric mismatch and $M$ is probe-dependent. This identity yields an a posteriori total-variation (TV) upper confidence bound accounting for held-out field noise, estimated-operator error, and externally validated $L^1$ residual radii around normalized density approximants in the span; a nonpositive observability margin returns the trivial TV bound and abstains. The audit recomputes this numerator from held-out samples; a normalized drift statistic requires a separate joint numerator--denominator analysis. For Gaussian-RBF interactions, a global envelope supports distribution-free and empirical-Bernstein radii without truncation, with companion bounds for the Laplace similarity in the original drifting objective. We characterize random-probe observability by a population Gram matrix, identify rank and symmetry degeneracies, and prove large-bandwidth collapse toward mean matching. Synthetic studies exercise Gaussian and Laplace numerators, separately prespecified bounded-vector and variance-adaptive radii, Monte Carlo-calibrated operators, nonzero residual radii around normalized finite-basis approximants, outward-rounded observability bounds, and designed abstention. A joint basis-size/dimension stress path extends evaluation through $m=8$. The result is a conditional diagnostic for a finite density class, or for normalized finite-basis density approximants with external residual radii, not a universal guarantee from small training drift.

Problem

• drifting objectives
• finite-basis models
• total-variation certificates
• distributional inference
• observability

Innovation

• total-variation certificate
• finite-basis drifting model
• observability margin
• distribution-free radius
• antisymmetric interaction