LinearBasis

class LinearBasis(entries, weights=None, check_orthogonality: bool = True, name: str = None)

Linear low-dimensional state approximation.

This class approximates high-dimensional states \(\q\in\RR^n\) as a linear combination of \(r\) basis vectors \(\v_1,\ldots,\v_r\in\RR^n\). The basis matrix \(\Vr = [~\v_1~~\cdots~~\v_r~]\in \RR^{n \times r}\) and an (optional) weighting matrix \(\W\in\RR^{n \times n}\) define the approximation.

The encoding from the high-dimensional space \(\RR^n\) to the low-dimensional space \(\RR^r\) is given by

\[\q \mapsto \qhat = \Vr\trp\W\q,\]

while the decoding from low-dimensional space \(\RR^r\) to the high-dimensional space \(\RR^n\) is defined as

\[\qhat \mapsto \breve{\q} = \Vr\qhat = \sum_{i=1}^r \hat{q}_i \v_i,\]

where \(\qhat = [\hat{q}_1,\ldots,\hat{q}_r]\trp\in\RR^r\).

Basis entries \(\Vr\) and the weights \(\W\) are specified explicitly in the constructor, not learned from state data.

Parameters:

entries(n, r) ndarray

Basis entries \(\Vr\in\RR^{n\times r}\).

weights(n, n) ndarray or (n,) ndarray None

Weight matrix \(\W\in\RR^n\) or its diagonal entries. If None (default), set \(\W\) to the identity.

check_orthogonalitybool

If True, raise a warning if the basis is not orthogonal, i.e., if \(\Vr\trp\W\Vr\) is not the identity.

namestr or None

Label for the state variable that this basis approximates.

Notes

Pair with a opinf.pre.ShiftScaleTransformer to do centered approximations of the form \(\q \approx\Vr\qhat + \bar{\q}\).

Properties:

entries

Entries of the basis matrix \(\Vr\in\RR^{n \times r}\). Also accessible via indexing (basis[:]).

full_state_dimension

Dimension \(n\) of the full state.

name

Label for the state variable that this basis approximates.

reduced_state_dimension

Dimension \(r\) of the reduced (compressed) state.

shape

Dimensions \((n, r)\) of the basis.

weights

Weight matrix \(\W \in \RR^{n \times n}\).

Methods:

compress	Map high-dimensional states to low-dimensional latent coordinates.
decompress	Map low-dimensional latent coordinates to high-dimensional states.
fit	Do nothing, the basis entries are set in the constructor.
load	Load a basis from an HDF5 file.
plot1D	Plot the basis vectors over a one-dimensional domain.
project	Project a high-dimensional state vector to the subset of the high-dimensional space that can be represented by the basis.
projection_error	Compute the error of the basis representation of a state or states.
save	Save the basis to an HDF5 file.
verify	Verify that compress() and decompress() are consistent in the sense that the range of decompress() is in the domain of compress() and that project() defines a projection operator, i.e., project(project(Q)) = project(Q).