1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
|
"""Sparse matrix norms.
"""
from __future__ import division, print_function, absolute_import
import numpy as np
from scipy.sparse import issparse
from numpy.core import Inf, sqrt, abs
__all__ = ['norm']
def _sparse_frobenius_norm(x):
if np.issubdtype(x.dtype, np.complexfloating):
sqnorm = abs(x).power(2).sum()
else:
sqnorm = x.power(2).sum()
return sqrt(sqnorm)
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/master/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 Not implemented
-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
"""
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:
raise TypeError(msg)
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):
raise ValueError('Invalid axis %r for an array with shape %r' %
(axis, x.shape))
if row_axis % nd == col_axis % nd:
raise ValueError('Duplicate axes given.')
if ord == 2:
raise NotImplementedError
#return _multi_svd_norm(x, row_axis, col_axis, amax)
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 == 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 == -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):
raise ValueError('Invalid axis %r for an array with shape %r' %
(axis, x.shape))
if ord == Inf:
M = abs(x).max(axis=a)
elif ord == -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:
raise ValueError('Invalid norm order for vectors.')
M = np.power(abs(x).power(ord).sum(axis=a), 1 / ord)
return M.A.ravel()
else:
raise ValueError("Improper number of dimensions to norm.")
|
