def norm_nuclear(X):
r"""Compute the nuclear norm
.. math::
\| X \|_* = \sum_i \sigma_i
where :math:`\sigma_i` are the singular values of matrix :math:`X`.
Parameters
----------
X : array_like
Input array :math:`X`
Returns
-------
nncl : float
Nuclear norm of `X`
"""
return np.sum(np.linalg.svd(sl.promote16(X), compute_uv=False))
def norm_nuclear(X):
r"""Compute the nuclear norm
.. math::
\| X \|_* = \sum_i \sigma_i
where :math:`\sigma_i` are the singular values of matrix :math:`X`.
Parameters
----------
X : array_like
Input array :math:`X`
Returns
-------
nncl : float
Nuclear norm of `X`
"""
return np.sum(np.linalg.svd(sl.promote16(X), compute_uv=False))
def prox_nuclear(v, alpha):
r"""Proximal operator of the nuclear norm :cite:`cai-2010-singular`.
Parameters
----------
v : array_like
Input array :math:`V`
alpha : float
Parameter :math:`\alpha`
Returns
-------
x : ndarray
Output array
s : ndarray
Singular values of `x`
"""
U, s, V = sl.promote16(v, fn=np.linalg.svd, full_matrices=False)
ss = np.maximum(0, s - alpha)
return np.dot(U, np.dot(np.diag(ss), V)), ss
def prox_nuclear(V, alpha):
r"""Proximal operator of the nuclear norm :cite:`cai-2010-singular`
with parameter :math:`\alpha`.
Parameters
----------
v : array_like
Input array :math:`V`
alpha : float
Parameter :math:`\alpha`
Returns
-------
X : ndarray
Output array
s : ndarray
Singular values of `X`
"""
Usvd, s, Vsvd = sl.promote16(V, fn=np.linalg.svd, full_matrices=False)
ss = np.maximum(0, s - alpha)
return np.dot(Usvd, np.dot(np.diag(ss), Vsvd)), ss
def nucnorm(x):
"""Nuclear norm"""
return np.sum(np.linalg.svd(sl.promote16(x), compute_uv=False))
def shrinksv(v, alpha):
"""Shrinkage of singular values"""
U, s, V = sl.promote16(v, fn=np.linalg.svd, full_matrices=False)
ss = np.maximum(0, s - alpha)
return np.dot(U, np.dot(np.diag(ss), V)), ss