ott.geometry.regularizers.Quadratic

class ott.geometry.regularizers.Quadratic(A=None, b=None, *, is_complement=False, is_orthogonal=False, is_factor=False, solver=None)

Quadratic operator \(\frac{1}{2} \left<x, Q x\right> + b\).

The matrix \(Q\) is defined as:

• \(Q := A\) if not factored and not an orthogonal complement.

• \(Q := A^{\perp}\) if not factored and a complement.

• \(Q := A^TA\) if factored and not a complement.

• \(Q := \left(A^{\perp}\right)^TA^{\perp}\) if factored and a complement.

Parameters:

• A (Union[Array, AbstractLinearOperator, None]) – Linear operator \(A\). If None, use identity.

• b (Optional[Array]) – Offset \(b\). If None, use array of 0s.

• is_complement (bool) – Whether to regularize in the orthogonal complement of \(A\), defined as \(A^{\perp} := I - A^T (AA^T)^{-1} A\).

• is_orthogonal (bool) – Whether \(AA^T = I\).

• is_factor (bool) – Whether to factor the matrix \(Q\) as mentioned above.

• solver (Optional[Callable[[AbstractLinearOperator, Array], Array]]) – Linear solver. If None, use lineax.linear_solve().

Methods

moreau_envelope(x[, tau])	Moreau Envelope.
prox(v[, tau])	Proximal operator.
prox_dual(v[, tau])	Proximal operator of the convex conjugate.

Attributes

A_comp	Orthogonal complement \(A^{\perp}\) of \(A\).
Q	Linear operator \(Q\).
is_complement	Whether Q is defined using \(A_{\perp}\) or \(A\).
is_factor	Whether Q is factored.
is_orthogonal	Whether AA^T = I.