Quasiseparable kernels produce incorrect covariances for non-reversible state-space models

[burke86]

I am using tinygp to implement quasiseparable Gaussian-process models for AGN reverberation mapping. These models can have finite-dimensional state-space representations while being non-reversible.

tinygp’s current quasiseparable construction implicitly relies on

$$F P_\infty = P_\infty F^{\mathsf T},$$

where $F$ is the physical forward state transition and $P_\infty$ is the stationary covariance. This identity holds for reversible stationary processes, including tinygp’s existing quasiseparable kernels, but not for general causal state-space models.

Minimal example

A minimal two-state non-reversible kernel and regression test are included in PR #276, in tests/kernels/test_quasisep_nonreversible.py.

The test constructs the covariance in three ways:

• directly from the state-space transition;
• using QuasisepSolver;
• using KalmanSolver.

On the current main branch, the quasiseparable covariance has incorrectly oriented cross-covariances and disagrees with the direct and Kalman formulations.

The discrepancy arises because tinygp’s transition_matrix(t1, t2) returns the adjoint of the physical forward transition. If

$$x(t_2) = F_{21}x(t_1),$$

then transition_matrix(t1, t2) returns

$$A_{21} = F_{21}^{\mathsf T}.$$

The current QSM construction effectively uses

$$P_\infty F^{\mathsf T},$$

while forward state propagation requires

$$F P_\infty.$$

These expressions are equal for reversible processes but differ for the test kernel.

Expected behavior

For (t_i > t_j), the quasiseparable representation should produce

$$K_{ij} = h_i F_i F_{i-1}\cdots F_{j+1} P_\infty h_j^{\mathsf T}.$$

The quasiseparable covariance and likelihood should agree with the direct state-space and Kalman formulations for both reversible and non-reversible kernels.

Proposed fix

The quasiseparable generators should use the physical forward-transition orientation. The corresponding changes need to cover:

• symmetric QSM construction;
• scalar kernel evaluation;
• generalized QSM construction used for cross-coordinate matrix multiplication and prediction;
• documentation of the transition_matrix convention.

I implemented the proposed change and regression tests in PR #276. Existing reversible kernels retain their current covariance and likelihood behavior.