SymmetricMatrixBasedLinearBeliefUpdate¶
- class probnum.linalg.solvers.belief_updates.matrix_based.SymmetricMatrixBasedLinearBeliefUpdate¶
Bases:
probnum.linalg.solvers.belief_updates.LinearSolverBeliefUpdateSymmetric 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
- 1
Hennig, P., Probabilistic Interpretation of Linear Solvers, SIAM Journal on Optimization, 2015, 25, 234-260
- 2
Wenger, J. and Hennig, P., Probabilistic Linear Solvers for Machine Learning, Advances in Neural Information Processing Systems (NeurIPS), 2020
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