pyproximal.utils.bilinear.BilinearOperator

class pyproximal.utils.bilinear.BilinearOperator

Common interface for bilinear operator of a function.

Bilinear operator template class. A user must subclass it and implement the following methods:

• gradx: a method evaluating the gradient over \(\mathbf{x}\): \(\nabla_x H\)

• grady: a method evaluating the gradient over \(\mathbf{y}\): \(\nabla_y H\)

• grad: a method returning the stacked gradient vector over \(\mathbf{x},\mathbf{y}\): \([\nabla_x H\), [nabla_y H]`

• lx: Lipschitz constant of \(\nabla_x H\)

• ly: Lipschitz constant of \(\nabla_y H\)

Two additional methods (updatex and updatey) are provided to update the \(\mathbf{x}\) and \(\mathbf{y}\) internal variables. It is user responsability to choose when to invoke such method (i.e., when to update the internal variables).

Notes

A bilinear operator is defined as a differentiable nonlinear function \(H(x,y)\) that is linear in each of its components indipendently, i.e, \(\mathbf{H_x}(y)\mathbf{x}\) and \(\mathbf{H_y}(y)\mathbf{x}\).

Methods

__init__()
grad(y)
gradx(x)
grady(y)
lx(x)
ly(y)
updatex(x)	Update x variable (to be used to update the internal variable x)
updatexy(xy)
updatey(y)	Update y variable (to be used to update the internal variable y)

Examples using pyproximal.utils.bilinear.BilinearOperator

Low-Rank completion via Matrix factorization

Low-Rank completion via Matrix factorization