NuclearNorm

class odl.functionals.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.

__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