L2DecoupledSolver

class L2DecoupledSolver(regularizer, lapack_driver: str = 'gesdd')

Solve \(r\) independent \(2\)-norm ordinary least-squares problems, each with the same data matrix but a different \(L_2\) regularization.

That is, for \(i = 1, \ldots, r\), construct \(\Ohat\) by solving

\[\argmin_{\Ohat}\|\D\ohat_i - \z_i\|_2^2 + \|\lambda_i\Ohat_i\|_2^2\]

where \(\ohat_i\) and \(\z_i\) are the \(i\)-th rows of \(\Ohat\) and \(\Z\), respectively, with corresponding regularization constant \(\lambda_i > 0\).

The exact solution for the \(i\)-th problem is described by the normal equations:

\[(\D\trp\D + \lambda_i^2\I)\ohat_i = \D\trp\z_i.\]

Instead of solving these equations directly, the solution is calculated using the singular value decomposition of the data matrix (see L2Solver).

Parameters:

regularizer(r,) ndarray

Scalar \(L_2\) regularization constants, one for each row of the operator matrix.

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 constants, one for each row of the operator matrix \(\Ohat\).

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.
regcond	Compute the \(2\)-norm condition number of each regularized data matrix, \([~\D\trp~~\lambda_i\I~]\trp\) for \(i = 1, \ldots, r\).
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.