TikhonovSolver

class TikhonovSolver(regularizer, method: str = 'lstsq', cond: float = None, lapack_driver: str = None)

Solve the Frobenius-norm ordinary least-squares problem with Tikhonov regularization.

That is, solve

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

for a specified symmetric-positive-definite regularization matrix \(\bfGamma \in \RR^{d \times d}\). This is equivalent to solving the following stacked least-squares problem:

\[\begin{split}\argmin_{\Ohat} \left\| \left[\begin{array}{c}\D \\ \bfGamma\end{array}\right]\Ohat\trp - \left[\begin{array}{c}\Z\trp \\ \0\end{array}\right] \right\|_F^2.\end{split}\]

The exact solution is described by the normal equations:

\[(\D\trp\D + \bfGamma\trp\bfGamma)\Ohat\trp = \D\trp\Z\trp,\]

that is,

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

Parameters:

regularizer(d, d) or (d,) ndarray

Symmetric semi-positive-definite regularization matrix \(\bfGamma\) or, if regularizer is a one-dimensional array, the diagonal entries of \(\bfGamma\). Here, d is the number of columns in the data matrix.

methodstr

Strategy for solving the regularized least-squares problem. Options:

• "lstsq": solve the stacked least-squares problem via scipy.linalg.lstsq(); by default, this computes and uses the singular value decomposition of the stacked data matrix \([~D\trp~~\bfGamma\trp~]\trp\).

• "normal": directly solve the normal equations \((\D\trp\D + \bfGamma\trp\bfGamma) \Ohat\trp = \D\trp\Z\trp\) via scipy.linalg.solve().

condfloat or None

Cutoff for ‘small’ singular values of the data matrix, see scipy.linalg.lstsq(). Ignored if method = "normal".

lapack_driverstr or None

Which LAPACK driver is used to solve the least-squares problem, see scipy.linalg.lstsq(). Ignored if method = "normal".

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 \(\Z\).

method

Strategy for solving the regularized least-squares problem, either "lstsq" (default) or "normal".

options

Keyword arguments for scipy.linalg.lstsq().

r

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

regularizer

Symmetric semi-positive-definite \(d \times d\) regularization matrix \(\bfGamma\).

Methods:

cond	Compute the \(2\)-norm condition number of the data matrix \(\D\).
copy	Make a copy of the solver.
fit	Verify dimensions and precompute quantities in preparation to solve the least-squares problem.
get_operator_regularizer	Construct a regularizer so that each operator is regularized separately.
load	Load a serialized solver from an HDF5 file, created previously from the save() method.
regcond	Compute the \(2\)-norm condition number of the regularized data matrix \([~\D\trp~~\bfGamma\trp~]\trp.\)
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.