pyproximal.QuadraticEnvelopeRankL2#
- class pyproximal.QuadraticEnvelopeRankL2(dim, r0, M)[source]#
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
- dim
tuple Size of input matrix \(X\).
- r0
int Threshold parameter, encouraging matrices with rank lower than or equal to r0.
- M
numpy.ndarray L2 misfit term (must be the same size as the input matrix).
- dim
See also
SingularValuePenaltyProximal operator of a penalty acting on the singular values
QuadraticEnvelopeCardIndicatorQuadratic 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)Compute gradient
postcomposition(sigma)Postcomposition
precomposition(a, b)Precomposition
prox(**kwargs)proxdual(**kwargs)
