TikhonovSolver

class TikhonovSolver(regularizer=None, 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.
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 \([~\D\trp~~\bfGamma\trp~]\trp.\)
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.