Robust Automatic Differentiation of Square-Root Kalman Filters via Gramian Differentials

2026-03-13

AI Summary

This work addresses the challenge of gradient computation in square-root Kalman filters when employed within automatic differentiation frameworks, where non-uniqueness in QR decomposition and rank-deficient matrices lead to undefined or divergent gradients, thereby hindering gradient-based parameter learning. To overcome this instability, the authors propose a closed-form chain rule based on the differential of Gramian matrices, which bypasses the problematic QR decomposition and enables exact differentiation of filter outputs. By integrating the differential of Gramian identities with Moore–Penrose pseudoinverses and null-space correction techniques, the method achieves, for the first time, robust automatic differentiation of square-root Kalman filters under rank-deficient and non-unique conditions. The approach yields accurate gradients of both the log marginal likelihood and filtering moments, maintaining numerical stability and correctness.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Square-root Kalman filters propagate state covariances in Cholesky-factor form for numerical stability, and are a natural target for gradient-based parameter learning in state-space models. Their core operation, triangularization of a matrix $M \in \mathbb{R}^{n \times m}$, is computed via a QR decomposition in practice, but naively differentiating through it causes two problems: the semi-orthogonal factor is non-unique when $m > n$, yielding undefined gradients; and the standard Jacobian formula involves inverses, which diverges when $M$ is rank-deficient. Both are resolved by the observation that all filter outputs relevant to learning depend on the input matrix only through the Gramian $MM^\top$, so the composite loss is smooth in $M$ even where the triangularization is not. We derive a closed-form chain-rule directly from the differential of this Gramian identity, prove it exact for the Kalman log-marginal likelihood and filtered moments, and extend it to rank-deficient inputs via a two-component decomposition: a column-space term based on the Moore--Penrose pseudoinverse, and a null-space correction for perturbations outside the column space of $M$.

Problem

• Automatic Differentiation
• Square-Root Kalman Filter
• QR Decomposition
• Rank-Deficiency
• Gradient Ambiguity

Innovation

• Square-root Kalman filter
• Automatic differentiation
• Gramian differential
• Rank-deficient matrix
• Moore–Penrose pseudoinverse