TikhonovDecoupledSolver

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

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

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

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

where \(\ohat_i\) and \(\z_i\) are the \(i\)-th rows of \(\Ohat\) and \(\Z\), respectively, with corresponding symmetric-positive-definite regularization matrix \(\bfGamma_i\).

This is equivalent to solving the following stacked least-squares problems:

\[\begin{split}\argmin{\Ohat} \left\| \left[\begin{array}{c}\D \\ \bfGamma_i\end{array}\right]\Ohat\trp - \left[\begin{array}{c}\Z\trp \\ \0\end{array}\right] \right\|_F^2, \quad i = 1, \ldots, r.\end{split}\]

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

\[(\D\trp\D + \bfGamma_i\trp\bfGamma_i)\ohat_i = \D\trp\z_i.\]

Parameters:

regularizerlist of r (d, d) or (d,) ndarrays

Symmetric semi-positive-definite regularization matrices \(\bfGamma_1,\ldots,\bfGamma_r\). If the i``th entry of ``regularizer is a one-dimensional array, it is intepreted as the diagonal entries of \(\bfGamma_i\). 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 regularization matrices \(\bfGamma_1,\ldots,\bfGamma_r\), one for each row of the operator matrix.

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 each regularized data matrix \([~\D\trp~~\bfGamma_i\trp~]\trp,~~i = 1, \ldots, r\).
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.