# On the Sequentially Semiseparable Structure of SDE-Induced Kernels and Efficient Algorithms for Kernel-Based Estimation Problems

arXiv: https://arxiv.org/abs/2610.05856 · HTML: https://mlsanity.com/paper/2610.05856/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 41 references that are in the map.
- First posted on arXiv 5 Oct 2026.

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

- [Pathwise Conditioning of Gaussian Processes](https://mlsanity.com/paper/2011.04026/) (2020)
- [Parallelizable sparse inverse formulation Gaussian processes (SpInGP)](https://mlsanity.com/paper/1610.08035/) (2016)
- [On kernel design for regularized LTI system identification](https://mlsanity.com/paper/1612.03542/) (2016)
- [Numerically Efficient and Stable Algorithms for Kernel-Based Regularized System Identification Using Givens-Vector Representation](https://mlsanity.com/paper/2511.01534/) (2025)

## Abstract (arXiv)

We study kernel functions induced by a linear time-invariant (LTI) stochastic differential equation (SDE), an approach that embeds prior knowledge about the function to be estimated directly into the state-space description of the SDE. This kernel-design perspective is relevant across the many settings where kernels play a central role, including kernel methods in statistics, Gaussian process (GP) regression in machine learning, and kernel-based regularization methods in system identification. The resulting class of SDE-induced kernels includes several kernels of practical interest, such as Mat'ern kernels with half-integer smoothness, spline kernels, and related kernels used in system identification. In contrast to existing linear-time GP regression methods for these kernels, which are typically based on Kalman filtering and Rauch--Tung--Striebel smoothing on the latent state, we work directly with the SSS representation of the observed covariance matrix. This yields linear-complexity recursive algorithms for the matrix operations required in GP regression, including factorization, linear solves, log determinants, posterior covariance, log-marginal-likelihood gradients, and posterior sampling; the same generator-level recursions extend, with only minor modifications, to the analogous computations in kernel-based regularization methods for system identification. Compared with previous uses of SSS representations in GP regression, we derive efficient recursive algorithms for analytic marginal-likelihood gradients in three application-relevant cases, and we develop new recursions for posterior computations and Matheron-type sampling. Numerical experiments verify the accuracy of the proposed recursions and illustrate posterior inference and sampling for representative SDE-induced kernels.