Support non-reversible quasiseparable kernels

[burke86]

Summary

This PR adds support for non-reversible state-space models in tinygp’s quasiseparable kernel machinery.

The previous QSM generator construction works for tinygp’s existing reversible kernels, but it relies implicitly on a reversibility identity that does not hold for general causal systems. In a non-reversible model, the ordering of the stationary covariance and transition matrices matters. Using the reversible ordering can produce incorrectly oriented cross-covariances and disagreement between the quasiseparable, Kalman, and direct dense formulations.

Motivation

I encountered this while implementing a causal filtered-response kernel in which a latent stochastic process drives a downstream response through a one-way linear filter. This structure appears in many settings beyond that particular application, including delayed-response systems, causal multivariate Gaussian processes. These models are generally non-reversible because information propagates from the driver into the response without an equivalent reverse coupling.

Previous behavior

tinygp defines transition_matrix(t1, t2) using an adjoint convention. If $F_{21}$ is the physical forward column-state transition,

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

then transition_matrix(t1, t2) returns

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

The previous QSM construction used

$$p_i=h_iP_\infty A_i,\qquad q_i=h_i,\qquad a_i=A_i.$$

For $t_i>t_j$, this represents the lower-triangular covariance as

$$K_{ij}^{\mathrm{old}} = h_iP_\infty A_iA_{i-1}\cdots A_{j+1} h_j^{\mathsf{T}}.$$

Because $A_k=F_k^{\mathsf{T}}$, this is equivalent to

$$K_{ij}^{\mathrm{old}} = h_iP_\infty \left(F_{j+1}\cdots F_i\right)^{\mathsf{T}} h_j^{\mathsf{T}}.$$

The covariance obtained by propagating the column state forward in time is instead

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

The essential difference is therefore

$$\boxed{ P_\infty F^{\mathsf{T}} \quad\text{versus}\quad F P_\infty }$$

For a reversible stationary process, these expressions are equal:

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

This is why the previous implementation works for tinygp’s existing reversible kernels and why the issue was not exposed by the existing test suite.

For a non-reversible process, such as a one-way driver-to-response system,

$$F P_\infty\ne P_\infty F^{\mathsf{T}}.$$

The previous QSM construction can therefore produce incorrectly oriented cross-covariances. This is especially visible when different observations load different components of a multivariate latent state.

New behavior

This PR stores the physical forward transition in the QSM generators:

$$p_i=h_iA_i^{\mathsf{T}}=h_iF_i,$$ $$a_i=A_i^{\mathsf{T}}=F_i,$$ $$q_i=h_iP_\infty.$$

The resulting lower-triangular covariance element is

$$\begin{aligned} K_{ij} &= p_i a_{i-1}\cdots a_{j+1}q_j^{\mathsf{T}}\\\ &= h_iF_iF_{i-1}\cdots F_{j+1} P_\infty h_j^{\mathsf{T}}, \end{aligned}$$

which is the standard causal state-space covariance.

For reversible kernels, the old and new constructions represent the same covariance. The change therefore preserves existing behavior while extending the QSM machinery to the more general non-reversible case.

Changes

This PR:

• constructs the symmetric QSM generators using the physical forward-transition orientation;
• updates scalar kernel evaluation to use the same causal covariance ordering;
• updates generalized QSM construction used for cross-coordinate matrix multiplication and prediction;
• documents that transition_matrix returns the adjoint of the physical forward column-state transition;
• adds a minimal two-state, non-reversible driver-to-response kernel as a regression test.

The test kernel has one-way coupling of the form

$$\text{driver}\longrightarrow\text{response},$$

so its stationary state satisfies

$$F P_\infty\ne P_\infty F^{\mathsf{T}}.$$

It therefore exposes transition-orientation errors that reversible test kernels cannot detect.

Testing

The new tests compare:

• an explicitly constructed dense state-space covariance;
• the covariance produced by QuasisepSolver;
• likelihoods from QuasisepSolver;
• likelihoods from KalmanSolver;
• cross-coordinate quasiseparable matrix multiplication;
• conditioning at the training coordinates;
• conditioning at new coordinates.

The existing quasiseparable and Kalman tests continue to pass.

Validation on the clean branch:

• 133 tests passed;
• 1 optional test was skipped;
• all pre-commit hooks passed.

User impact

This enables custom quasiseparable kernels for causal and coupled state-space systems where one latent component drives another without reciprocal coupling.

Existing reversible kernels retain their current covariance and likelihood behavior, while non-reversible kernels now agree across the direct dense, Kalman, and quasiseparable implementations.