gpflow.models.cglb

Classes

gpflow.models.cglb.NystromPreconditioner

class gpflow.models.cglb.NystromPreconditioner(A, LB, sigma_sq)

Bases: object

Preconditioner of the form $Q=(Q_{f}f+\sigma ²I)⁻¹$, where L is lower triangular with :math: LLᵀ = Kᵤᵤ $A=\sigma ⁻²L⁻¹Kᵤₓ$ and $B=AAᵀ+I=LᵦLᵦᵀ$

Parameters:

• A (Tensor) –

• LB (Tensor) –

• sigma_sq (float) –

Functions

gpflow.models.cglb.cglb_conjugate_gradient

gpflow.models.cglb.cglb_conjugate_gradient(K, b, initial, preconditioner, cg_tolerance, max_steps, restart_cg_step)

Conjugate gradient algorithm used in CGLB model. The method of conjugate gradient (Hestenes and Stiefel, 1952) produces a sequence of vectors $v_0,v_1,v_2,...,v_N$ such that $v_0$ = initial, and (in exact arithmetic) $K{v}_n=b$. In practice, the v_i often converge quickly to approximate $K^{-1}b$, and the algorithm can be stopped without running N iterations.

We assume the preconditioner, $Q$, satisfies $Q\prec K$, and stop the algorithm when $r_i=b-K{v}_i$ satisfies $||rᵢᵀ|{|}_{Q⁻¹r}^2=rᵢᵀQ⁻¹rᵢ<=ϵ$.

Parameters:

• K (Union[ndarray[Any, Any], Tensor, Variable, Parameter]) –

  • K has shape [N, N].

  Matrix we want to backsolve from. Must be PSD.

• b (Union[ndarray[Any, Any], Tensor, Variable, Parameter]) –

  • b has shape [B, N].

  Vector we want to backsolve.

• initial (Union[ndarray[Any, Any], Tensor, Variable, Parameter]) –

  • initial has shape [P, N].

  Initial vector solution.

• preconditioner (NystromPreconditioner) – Preconditioner function.

• cg_tolerance (float) – Expected maximum error. This value is used as a decision boundary against stopping criteria.

• max_steps (int) – Maximum number of CG iterations.

• restart_cg_step (int) – Restart step at which the CG resets the internal state to the initial position using the currect solution vector $v$. Can help avoid build up of numerical errors.

Return type:

Tensor

Returns:

• return has shape [P, N].

v where v approximately satisfies $Kv=b$.