Robust Automatic Differentiation of Square-Root Kalman Filters via Gramian Differentials
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...
| Authors | Adrien Corenflos |
|---|---|
| Primary category | stat.ML |
| Published | 13 March 2026 |
| Last updated at source | 13 March 2026 |
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