SymmetricMatrixBasedLinearBeliefUpdate

class probnum.linalg.solvers.belief_updates.matrix_based.SymmetricMatrixBasedLinearBeliefUpdate

Bases: LinearSolverBeliefUpdate

Symmetric Gaussian belief update in a matrix-based inference framework assuming linear information.

Updates the belief over the quantities of interest of a linear system \(Ax=b\) given symmetric matrix-variate Gaussian beliefs with symmetric Kronecker covariance structure and linear observations. The belief update computes \(p(M \mid y) = \mathcal{N}(M; M_{i+1}, W_{i+1} \otimes_s W_{i+1})\), [1] [2] such that

\[\begin{split}\begin{align} M_{i+1} &= M_i + (y - M_i s) u^\top + u (y - M_i s)^\top - u s^\top(y - M_i s)u^\top,\\ W_{i+1} &= W_i - W_i s (s^\top W_i s)^\dagger s^\top W_i. \end{align}\end{split}\]

where \(u = W_i s (s^\top W s)^\dagger\).

References

Methods Summary

__call__(solver_state)

Update the belief about the quantities of interest of a linear system.

Methods Documentation

__call__(solver_state)[source]

Update the belief about the quantities of interest of a linear system.

Parameters:

solver_state (LinearSolverState) – Current state of the linear solver.

Return type:

LinearSystemBelief