Goggin's corrected Kalman Filter: Guarantees and Filtering Regimes

2025-02-19

AI Summary

This paper addresses the convergence rate of nonlinear filters under non-Gaussian noise, focusing on the Goggin filter in the pre-limit regime. We propose a correction scheme that cascades a score-function transformation with standard Kalman filtering, yielding— for the first time under non-Gaussian, non-asymptotic conditions—an explicit finite-sample convergence rate that depends quantitatively on the observation noise level. Theoretical contributions include: (1) the first rigorous quantification of the Goggin filter’s convergence rate in the pre-limit setting; (2) a precise characterization of “degenerate” versus “balanced” filtering regimes in linear models, demonstrating its strict superiority over trivial estimators in the balanced regime; and (3) derivation of an information-theoretic, noise-adaptive filtering error bound via the posterior Cramér–Rao lower bound and the Fisher information central limit theorem.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

In this paper we revisit a non-linear filter for {em non-Gaussian} noises that was introduced in [1]. Goggin proved that transforming the observations by the score function and then applying the Kalman Filter (KF) to the transformed observations results in an asymptotically optimal filter. In the current paper, we study the convergence rate of Goggin's filter in a pre-limit setting that allows us to study a range of signal-to-noise regimes which includes, as a special case, Goggin's setting. Our guarantees are explicit in the level of observation noise, and unlike most other works in filtering, we do not assume Gaussianity of the noises. Our proofs build on combining simple tools from two separate literature streams. One is a general posterior Cram'er-Rao lower bound for filtering. The other is convergence-rate bounds in the Fisher information central limit theorem. Along the way, we also study filtering regimes for linear state-space models, characterizing clearly degenerate regimes -- where trivial filters are nearly optimal -- and a {em balanced} regime, which is where Goggin's filter has the most value. footnote{This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.

Problem

• non-Gaussian noise filtering
• convergence rate analysis
• signal-to-noise regimes exploration

Innovation

• Non-Gaussian noise filtering
• Score function transformation
• Convergence rate analysis