pyproximal.QuadraticEnvelopeRankL2

class pyproximal.QuadraticEnvelopeRankL2(dim: Tuple[int, ...], r0: int, M: ndarray[tuple[Any, ...], dtype[_ScalarT]])

Quadratic envelope of the rank function with an L2 misfit term.

The penalty \(p\) is given by

\[p(X) = \mathcal{R}_{r_0}(X) + \frac{1}{2}\|X - M\|_F^2\]

where \(\mathcal{R}_{r_0}\) is the quadratic envelope of the hard-rank function.

Parameters:

dimtuple

Size of input matrix \(X\).

r0int

Threshold parameter, encouraging matrices with rank lower than or equal to r0.

Mnumpy.ndarray

L2 misfit term (must be the same size as the input matrix).

See also

SingularValuePenalty

Proximal operator of a penalty acting on the singular values

QuadraticEnvelopeCardIndicator

Quadratic envelope of the indicator function of \(\ell_0\)-penalty

Notes

The proximal operator solves the minimization problem

\[\argmin_Z \mathcal{R}_{r_0}(Z) + \frac{1}{2}\|Z - M\|_F^2 + \frac{1}{2\tau}\| Z - X \|_F^2\]

which is a convex-concave min-max problem, see [1] for details.

References

[1]

Larsson, V. and Olsson, C. “Convex Low Rank Approximation”, In International Journal of Computer Vision (IJCV), 120:194–214, 2016.

Methods

__init__(dim, r0, M)
affine_addition(v)	Affine addition
chain(g)	Chain
grad(x)	Gradient of the Moreau envelope of the function.
postcomposition(sigma)	Postcomposition
precomposition(a, b)	Precomposition
prox(**kwargs)
proxdual(**kwargs)

Examples using pyproximal.QuadraticEnvelopeRankL2

Hankel matrix estimation

Hankel matrix estimation