NuclearNorm¶

class odl.solvers.functional.default_functionals.NuclearNorm(*args, **kwargs)[source]¶

Bases: Functional

Nuclear norm for matrix valued functions.

Notes

For a matrix-valued function $f : \Omega \rightarrow \mathbb{R}^{n \times m}$ , the nuclear norm with parameters $p$ and $q$ is defined by

$\left( \int_\Omega \|\sigma(f(x))\|_p^q d x \right)^{1/q},$

where $\sigma(f(x))$ is the vector of singular values of the matrix $f(x)$ and $\| \cdot \|_p$ is the usual $p$ -norm on $\mathbb{R}^{\min(n, m)}$ .

For a detailed description of its properties, e.g, its proximal, convex conjugate and more, see [Du+2016].

References

[Du+2016] J. Duran, M. Moeller, C. Sbert, and D. Cremers. Collaborative Total Variation: A General Framework for Vectorial TV Models SIAM Journal of Imaging Sciences 9(1): 116--151, 2016.

Attributes:

adjoint: Adjoint of this operator (abstract).
convex_conj: Convex conjugate of the nuclear norm.
domain: Set of objects on which this operator can be evaluated.
grad_lipschitz: Lipschitz constant for the gradient of the functional.
gradient: Gradient operator of the functional.
inverse: Return the operator inverse.
is_functional: True if this operator's range is a Field.
is_linear: True if this operator is linear.
proximal: Return the proximal operator.
range: Set in which the result of an evaluation of this operator lies.

Methods

`__call__`(x[, out])	Return `self(x[, out, **kwargs])`.
`bregman`(point, subgrad)	Return the Bregman distance functional.
`derivative`(point)	Return the derivative operator in the given point.
`norm`([estimate])	Return the operator norm of this operator.
`translated`(shift)	Return a translation of the functional.

__init__(space, outer_exp=1, singular_vector_exp=2)[source]¶

Initialize a new instance.

Parameters:

spaceProductSpace of ProductSpace of TensorSpace: Domain of the functional.
outer_exp{1, 2, inf}, optional: Exponent for the outer norm.
singular_vector_exp{1, 2, inf}, optional: Exponent for the norm for the singular vectors.

Examples

Simple example, nuclear norm of matrix valued function with all ones in 3 points. The singular values are [2, 0], which has squared 2-norm 2. Since there are 3 points, the expected total value is 6.

>>> r3 = odl.rn(3)
>>> space = odl.ProductSpace(odl.ProductSpace(r3, 2), 2)
>>> norm = NuclearNorm(space)
>>> norm(space.one())
6.0