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

arXiv: https://arxiv.org/abs/2603.13559 · HTML: https://mlsanity.com/paper/2603.13559/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 12 references that are in the map.
- First posted on arXiv 13 Mar 2026.

## Builds on (most cited first, 4 of 4)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [You Only Linearize Once: Tangents Transpose to Gradients](https://mlsanity.com/paper/2204.10923/) (2022)
- [Auto-Differentiating Linear Algebra](https://mlsanity.com/paper/1710.08717/) (2017)
- [Parallel square-root statistical linear regression for inference in nonlinear state space models](https://mlsanity.com/paper/2207.00426/) (2022)

## Abstract (arXiv)

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 ∈ ℝn × 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.