L2Solver

class L2Solver(regularizer, lapack_driver: str = 'gesdd')

Solve the Frobenius-norm ordinary least-squares problem with \(L_2\) regularization.

That is, solve

\[\argmin_{\Ohat}\|\D\Ohat\trp - \Z\trp\|_F^2 + \|\lambda\Ohat\trp\|_F^2\]

for some specified \(\lambda \ge 0\).

The exact solution is described by the normal equations:

\[(\D\trp\D + \lambda^2\I)\Ohat\trp = \D\trp\Z\trp,\]

that is,

\[\Ohat = \Z\D(\D\trp\D + \lambda^2\I)^{-\mathsf{T}}.\]

Instead of solving these equations directly, the solution is calculated using the singular value decomposition of the data matrix \(\D\): if \(\D = \bfPhi\bfSigma\bfPsi\trp\), then \(\Ohat\trp = \bfPsi\bfSigma^{*}\bfPhi\trp\Z\trp\) (i.e., \(\Ohat = \Z\bfPhi\bfSigma^{*}\bfPsi\trp\)), where \(\bfSigma^{*}\) is a diagonal matrix with \(i\)-th diagonal entry \(\Sigma_{i,i}^{*} = \Sigma_{i,i}/(\Sigma_{i,i}^{2} + \lambda^2).\)

Parameters:

regularizerfloat

Scalar \(L_2\) regularization constant \(\lambda\).

lapack_driverstr

LAPACK routine for computing the singular value decomposition. See scipy.linalg.svd().

Properties:

d

Number of unknowns in each row of the operator matrix (number of columns of \(\D\) and \(\Ohat\)).

data_matrix

\(k \times d\) data matrix \(\D\).

k

Number of equations in the least-squares problem (number of rows of \(\D\) and number of columns of \(\Z\)).

lhs_matrix

\(r \times k\) left-hand side data \(\Z\).

options

Keyword arguments for scipy.linalg.svd(). These cannot be changed after instantiation.

r

Number of operator matrix rows to learn (number of rows of \(\Z\) and \(\Ohat\))

regularizer

Scalar \(L_2\) regularization constant \(\lambda > 0.\)

Methods:

cond	Compute the \(2\)-norm condition number of the data matrix \(\D\).
copy	Make a copy of the solver.
fit	Verify dimensions and compute the singular value decomposition of the data matrix in preparation to solve the least-squares problem.
load	Load a serialized solver from an HDF5 file, created previously from the save() method.
posterior	Solve the Bayesian operator inference regression, constructing the means and inverse covariances of probability distributions for the rows of an operator matrix posterior.
regcond	Compute the \(2\)-norm condition number of the regularized data matrix \([~\D\trp~~\lambda\I~]\trp.\)
regresidual	Compute the residual of the regularized regression objective for each row of the given operator matrix.
residual	Compute the residual of the \(2\)-norm regression objective for each row of the given operator matrix.
save	Serialize the solver, saving it in HDF5 format.
solve	Solve the Operator Inference regression.
verify	Verify the solver.