ai-content-maker/.venv/Lib/site-packages/scipy/sparse/linalg/_norm.py

194 lines
5.9 KiB
Python

"""Sparse matrix norms.
"""
import numpy as np
from scipy.sparse import issparse
from scipy.sparse.linalg import svds
import scipy.sparse as sp
from numpy import sqrt, abs
__all__ = ['norm']
def _sparse_frobenius_norm(x):
data = sp._sputils._todata(x)
return np.linalg.norm(data)
def norm(x, ord=None, axis=None):
"""
Norm of a sparse matrix
This function is able to return one of seven different matrix norms,
depending on the value of the ``ord`` parameter.
Parameters
----------
x : a sparse matrix
Input sparse matrix.
ord : {non-zero int, inf, -inf, 'fro'}, optional
Order of the norm (see table under ``Notes``). inf means numpy's
`inf` object.
axis : {int, 2-tuple of ints, None}, optional
If `axis` is an integer, it specifies the axis of `x` along which to
compute the vector norms. If `axis` is a 2-tuple, it specifies the
axes that hold 2-D matrices, and the matrix norms of these matrices
are computed. If `axis` is None then either a vector norm (when `x`
is 1-D) or a matrix norm (when `x` is 2-D) is returned.
Returns
-------
n : float or ndarray
Notes
-----
Some of the ord are not implemented because some associated functions like,
_multi_svd_norm, are not yet available for sparse matrix.
This docstring is modified based on numpy.linalg.norm.
https://github.com/numpy/numpy/blob/main/numpy/linalg/linalg.py
The following norms can be calculated:
===== ============================
ord norm for sparse matrices
===== ============================
None Frobenius norm
'fro' Frobenius norm
inf max(sum(abs(x), axis=1))
-inf min(sum(abs(x), axis=1))
0 abs(x).sum(axis=axis)
1 max(sum(abs(x), axis=0))
-1 min(sum(abs(x), axis=0))
2 Spectral norm (the largest singular value)
-2 Not implemented
other Not implemented
===== ============================
The Frobenius norm is given by [1]_:
:math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
References
----------
.. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
Examples
--------
>>> from scipy.sparse import *
>>> import numpy as np
>>> from scipy.sparse.linalg import norm
>>> a = np.arange(9) - 4
>>> a
array([-4, -3, -2, -1, 0, 1, 2, 3, 4])
>>> b = a.reshape((3, 3))
>>> b
array([[-4, -3, -2],
[-1, 0, 1],
[ 2, 3, 4]])
>>> b = csr_matrix(b)
>>> norm(b)
7.745966692414834
>>> norm(b, 'fro')
7.745966692414834
>>> norm(b, np.inf)
9
>>> norm(b, -np.inf)
2
>>> norm(b, 1)
7
>>> norm(b, -1)
6
The matrix 2-norm or the spectral norm is the largest singular
value, computed approximately and with limitations.
>>> b = diags([-1, 1], [0, 1], shape=(9, 10))
>>> norm(b, 2)
1.9753...
"""
if not issparse(x):
raise TypeError("input is not sparse. use numpy.linalg.norm")
# Check the default case first and handle it immediately.
if axis is None and ord in (None, 'fro', 'f'):
return _sparse_frobenius_norm(x)
# Some norms require functions that are not implemented for all types.
x = x.tocsr()
if axis is None:
axis = (0, 1)
elif not isinstance(axis, tuple):
msg = "'axis' must be None, an integer or a tuple of integers"
try:
int_axis = int(axis)
except TypeError as e:
raise TypeError(msg) from e
if axis != int_axis:
raise TypeError(msg)
axis = (int_axis,)
nd = 2
if len(axis) == 2:
row_axis, col_axis = axis
if not (-nd <= row_axis < nd and -nd <= col_axis < nd):
message = f'Invalid axis {axis!r} for an array with shape {x.shape!r}'
raise ValueError(message)
if row_axis % nd == col_axis % nd:
raise ValueError('Duplicate axes given.')
if ord == 2:
# Only solver="lobpcg" supports all numpy dtypes
_, s, _ = svds(x, k=1, solver="lobpcg")
return s[0]
elif ord == -2:
raise NotImplementedError
#return _multi_svd_norm(x, row_axis, col_axis, amin)
elif ord == 1:
return abs(x).sum(axis=row_axis).max(axis=col_axis)[0,0]
elif ord == np.inf:
return abs(x).sum(axis=col_axis).max(axis=row_axis)[0,0]
elif ord == -1:
return abs(x).sum(axis=row_axis).min(axis=col_axis)[0,0]
elif ord == -np.inf:
return abs(x).sum(axis=col_axis).min(axis=row_axis)[0,0]
elif ord in (None, 'f', 'fro'):
# The axis order does not matter for this norm.
return _sparse_frobenius_norm(x)
else:
raise ValueError("Invalid norm order for matrices.")
elif len(axis) == 1:
a, = axis
if not (-nd <= a < nd):
message = f'Invalid axis {axis!r} for an array with shape {x.shape!r}'
raise ValueError(message)
if ord == np.inf:
M = abs(x).max(axis=a)
elif ord == -np.inf:
M = abs(x).min(axis=a)
elif ord == 0:
# Zero norm
M = (x != 0).sum(axis=a)
elif ord == 1:
# special case for speedup
M = abs(x).sum(axis=a)
elif ord in (2, None):
M = sqrt(abs(x).power(2).sum(axis=a))
else:
try:
ord + 1
except TypeError as e:
raise ValueError('Invalid norm order for vectors.') from e
M = np.power(abs(x).power(ord).sum(axis=a), 1 / ord)
if hasattr(M, 'toarray'):
return M.toarray().ravel()
elif hasattr(M, 'A'):
return M.A.ravel()
else:
return M.ravel()
else:
raise ValueError("Improper number of dimensions to norm.")