TotalLeastSquaresSolver

class TotalLeastSquaresSolver(lapack_driver: str = 'gesdd')

Solve the total least-squares problem without any regularization, i.e.,

\[\argmin_{\Ohat} \left\|[~\Delta_\D~~\Delta_{\Z\trp}~]\right\|_{F}^{2} \quad\text{such that}\quad (\D + \Delta_\D)\Ohat\trp = \Z\trp + \Delta_{\Z\trp}.\]

The solution is computed via the singular value decomposition of the augmented \(k \times (d + r)\) matrix \([~\D~~\Z\trp~]\): if \(\bfPhi\bfSigma\bfPsi\trp = [~\D~~\Z\trp~]\), write the \((d + r) \times (d + r)\) right singular vector matrix \(\bfPsi\) as

\[\begin{split}\bfPsi = \left[\begin{array}{cc} \bfPsi_{\D\D} & \bfPsi_{\D\Z} \\ \bfPsi_{\Z\D} & \bfPsi_{\Z\Z} \end{array}\right],\end{split}\]

where \(\bfPsi_{\D\D}\) is \(d \times d\), \(\bfPsi_{\D\Z}\) is \(d \times r\), \(\bfPsi_{\Z\D}\) is \(r \times d\), and \(\bfPsi_{\Z\Z}\) is \(r \times r\). Then the optimal solution is given by \(\Ohat = -\bfPsi_{\Z\Z}^{-\mathsf{T}}\bfPsi_{\D\Z}\trp\).

Parameters:

lapack_driverstr

LAPACK routine for computing the SVD. See scipy.linalg.svd().

Properties:

augcond

\(2\)-norm condition number of the augmented data matrix \([~\D~~\Z\trp~]\).

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 \(\D\).

error

Frobenius norm of the error matrices, \(\|[~\Delta_\D~~\Delta_{\Z\trp}~\|_F\).

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().

r

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

Methods:

cond	Compute the \(2\)-norm condition number of the data matrix \(\D\).
copy	Make a copy of the solver.
fit	Verify dimensions, compute the singular value decomposition of the data matrix, and solve the problem.
load	Load a serialized solver from an HDF5 file, created previously from the save() method.
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	Return the total least-squares solution to the Operator Inference regression.
verify	Verify the solver.