grf_gp.inference module

grf_gp.inference.pathwise_conditioning(x_train, x_test, phi, y_train, noise_std, batch_size, device)

Perform pathwise conditioning to sample from the Gaussian Process posterior.

This function implements the Matheron’s rule (or pathwise update) to produce exact posterior samples by perturbing prior samples with a correction term derived from the residuals of the observations.

The update follows the identity:

\[\mathbf{g}_{|\mathbf{y}}(\cdot) = \mathbf{g}(\cdot) + \hat{\mathbf{K}}_{(\cdot)\mathbf{x}} \left( \hat{\mathbf{K}}_{\mathbf{x}\mathbf{x}} + \sigma_n^2 \mathbf{I} \right)^{-1} \left( \mathbf{y} - (\mathbf{g}(\mathbf{x}) + \boldsymbol{\varepsilon}) \right)\]

Parameters:

• x_train – Training node indices.

• x_test – Test node indices.

• phi – Feature matrix whose rows correspond to graph nodes.

• y_train – Training targets.

• noise_std – Observation noise standard deviation.

• batch_size – Number of posterior samples to draw.

• device – Device on which the computation is performed.

Returns:

Posterior function samples at x_test with shape (batch_size, len(x_test)).

grf_gp.inference.woodbury_pathwise_conditioning(x_train, x_test, phi, y_train, noise_std, batch_size, device)

Perform pathwise conditioning using a Woodbury solve in feature space.

This is the low-rank analogue of pathwise_conditioning for kernels of the form \(\hat{\mathbf{K}} = \mathbf{\Phi}\mathbf{\Phi}^\top\), where \(\mathbf{\Phi}\) has a small feature dimension. It applies the Woodbury identity to solve an equivalent system in small feature space.

The update follows the identity:

\[\left( \mathbf{\Phi}_{\mathbf{x}}\mathbf{\Phi}_{\mathbf{x}}^\top + \sigma_n^2 \mathbf{I} \right)^{-1} = \sigma_n^{-2}\mathbf{I} - \sigma_n^{-4}\mathbf{\Phi}_{\mathbf{x}} \left( \mathbf{I} + \sigma_n^{-2} \mathbf{\Phi}_{\mathbf{x}}^\top \mathbf{\Phi}_{\mathbf{x}} \right)^{-1} \mathbf{\Phi}_{\mathbf{x}}^\top\]

Parameters:

• x_train – Training node indices.

• x_test – Test node indices.

• phi – Low-rank feature matrix whose rows correspond to graph nodes.

• y_train – Training targets.

• noise_std – Observation noise standard deviation.

• batch_size – Number of posterior samples to draw.

• device – Device on which the computation is performed.

Returns:

Posterior function samples at x_test with shape (batch_size, len(x_test)).