haojin2 commented on a change in pull request #17014: [NumPy] Add NumPy support
for norm
URL: https://github.com/apache/incubator-mxnet/pull/17014#discussion_r364561278
##########
File path: python/mxnet/ndarray/numpy/linalg.py
##########
@@ -24,49 +24,168 @@
__all__ = ['norm', 'svd', 'cholesky', 'inv', 'det', 'slogdet', 'solve',
'tensorinv']
+# pylint: disable=too-many-return-statements
def norm(x, ord=None, axis=None, keepdims=False):
r"""Matrix or vector norm.
-
- This function can only support Frobenius norm for now.
- The Frobenius norm is given by [1]_:
-
- :math:`||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
-
+ This function is able to return one of eight different matrix norms,
+ or one of an infinite number of vector norms (described below), depending
+ on the value of the ``ord`` parameter.
Parameters
----------
x : ndarray
- Input array.
- ord : {'fro'}, optional
- Order of the norm.
+ Input array. If `axis` is None, `x` must be 1-D or 2-D.
+ ord : {non-zero int, inf, -inf, 'fro', 'nuc'}, 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, the norm of the whole ndarray is
- returned.
-
+ 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.
keepdims : bool, optional
If this is set to True, the axes which are normed over are left in the
result as dimensions with size one. With this option the result will
broadcast correctly against the original `x`.
-
Returns
-------
- n : float or ndarray
+ n : ndarray
Norm of the matrix or vector(s).
-
+ Notes
+ -----
+ For values of ``ord <= 0``, the result is, strictly speaking, not a
+ mathematical 'norm', but it may still be useful for various numerical
+ purposes.
+ The following norms can be calculated:
+ ===== ============================ ==========================
+ ord norm for matrices norm for vectors
+ ===== ============================ ==========================
+ None Frobenius norm 2-norm
+ 'fro' Frobenius norm --
+ 'nuc' -- --
+ inf max(sum(abs(x), axis=1)) max(abs(x))
+ -inf min(sum(abs(x), axis=1)) min(abs(x))
+ 0 -- sum(x != 0)
+ 1 max(sum(abs(x), axis=0)) as below
+ -1 min(sum(abs(x), axis=0)) as below
+ 2 -- as below
+ -2 -- as below
+ other -- sum(abs(x)**ord)**(1./ord)
+ ===== ============================ ==========================
+ The Frobenius norm is given by [1]_:
+ :math:`||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
+ The nuclear norm is the sum of the singular values.
+ When you want to operate norm for matrices,if you ord is (-1, 1, inf,
-inf),
+ you must give you axis, it is not support default axis.
References
----------
.. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
+ Examples
+ --------
+ >>> from mxnet import np
+ >>> 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.]])
+ >>> np.linalg.norm(a)
+ array(7.745967)
+ >>> np.linalg.norm(b)
+ array(7.745967)
+ >>> np.linalg.norm(b, 'fro')
+ array(7.745967)
+ >>> np.linalg.norm(a, 'inf')
+ array(4.)
+ >>> np.linalg.norm(b, 'inf', axis=(0, 1))
+ array(9.)
+ >>> np.linalg.norm(a, '-inf')
+ array(0.)
+ >>> np.linalg.norm(b, '-inf', axis=(0, 1))
+ array(2.)
+ >>> np.linalg.norm(a, 1)
+ array(20.)
+ >>> np.linalg.norm(b, 1, axis=(0, 1))
+ array(7.)
+ >>> np.linalg.norm(a, -1)
+ array(0.)
+ >>> np.linalg.norm(b, -1, axis=(0, 1))
+ array(6.)
+ >>> np.linalg.norm(a, 2)
+ array(7.745967)
+ >>> np.linalg.norm(a, -2)
+ array(0.)
+ >>> np.linalg.norm(a, 3)
+ array(5.8480353)
+ >>> np.linalg.norm(a, -3)
+ array(0.)
+ Using the `axis` argument to compute vector norms:
+ >>> c = np.array([[ 1, 2, 3],
+ ... [-1, 1, 4]])
+ >>> np.linalg.norm(c, axis=0)
+ array([1.4142135, 2.236068 , 5. ])
+ >>> np.linalg.norm(c, axis=1)
+ array([3.7416573, 4.2426405])
+ >>> np.linalg.norm(c, ord=1, axis=1)
+ array([6., 6.])
+ Using the `axis` argument to compute matrix norms:
+ >>> m = np.arange(8).reshape(2,2,2)
+ >>> np.linalg.norm(m, axis=(1,2))
+ array([ 3.7416573, 11.224973 ])
+ >>> np.linalg.norm(m[0, :, :]), np.linalg.norm(m[1, :, :])
+ (array(3.7416573), array(11.224973))
"""
- if ord is not None and ord != 'fro':
- raise ValueError('only support Frobenius norm for now, received
ord={}'.format(str(ord)))
- if isinstance(axis, tuple) and len(axis) > 2:
- raise ValueError('Improper number of dimensions to norm')
- if ord == 'fro' and x.ndim > 2 and axis is None:
- raise ValueError('Improper number of dimensions to norm')
- return _mx_nd_np.sqrt(_mx_nd_np.sum(x * x, axis=axis, keepdims=keepdims))
+ if axis is None and ord is None:
+ return _npi.norm(x, ord=2, axis=None, keepdims=keepdims, flag=-2)
+ if axis is None or isinstance(axis, (int, tuple)): # pylint:
disable=too-many-nested-blocks
+ if axis is not None:
+ if isinstance(axis, int):
+ axis = (axis, )
+ if len(axis) == 2:
+ if ord in ['inf', '-inf']:
+ row_axis, col_axis = axis
+ if not keepdims:
+ if row_axis > col_axis:
+ row_axis -= 1
+ if ord == 'inf':
+ return _mx_nd_np.sum(_mx_nd_np.abs(x), axis=col_axis,
keepdims=keepdims).max(axis=row_axis, keepdims=keepdims) # pylint:
disable=line-too-long
+ else:
+ return _mx_nd_np.sum(_mx_nd_np.abs(x), axis=col_axis,
keepdims=keepdims).min(axis=row_axis, keepdims=keepdims) # pylint:
disable=line-too-long
+ if ord in [1, -1]:
+ row_axis, col_axis = axis
+ if not keepdims:
+ if row_axis < col_axis:
+ col_axis -= 1
+ if ord == 1:
+ return _mx_nd_np.sum(_mx_nd_np.abs(x), axis=row_axis,
keepdims=keepdims).max(axis=col_axis, keepdims=keepdims) # pylint:
disable=line-too-long
+ elif ord == -1:
+ return _mx_nd_np.sum(_mx_nd_np.abs(x), axis=row_axis,
keepdims=keepdims).min(axis=col_axis, keepdims=keepdims) # pylint:
disable=line-too-long
+ if ord in [2, -2]:
+ return _npi.norm(x, ord=ord, axis=axis, keepdims=keepdims,
flag=0)
+ if ord is None:
+ return _npi.norm(x, ord=2, axis=axis, keepdims=keepdims,
flag=1)
+ if ord == 'inf':
+ return _mx_nd_np.max(_mx_nd_np.abs(x), axis=axis,
keepdims=keepdims)
+ #return _npi.norm(x, ord=float('inf'), axis=axis,
keepdims=keepdims, flag=3)
Review comment:
remove un-used code.
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