mars.tensor.linalg.norm#

mars.tensor.linalg.norm(x, ord=None, axis=None, keepdims=False)[source]#

Matrix or vector norm.

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 (array_like) – Input tensor. 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 mars tensor’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.

• 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 – Norm of the matrix or vector(s).

Return type

float or Tensor

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’

nuclear norm

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

2-norm (largest sing. value)

as below

-2

smallest singular value

as below

other

sum(abs(x)**ord)**(1./ord)

The Frobenius norm is given by 1:

$$||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}$$

The nuclear norm is the sum of the singular values.

References

1

G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15

Examples

>>> from mars.tensor import linalg as LA
>>> import mars.tensor as mt
>>> a = mt.arange(9) - 4
>>> a.execute()
array([-4, -3, -2, -1,  0,  1,  2,  3,  4])
>>> b = a.reshape((3, 3))
>>> b.execute()
array([[-4, -3, -2],
[-1,  0,  1],
[ 2,  3,  4]])

>>> LA.norm(a).execute()
7.745966692414834
>>> LA.norm(b).execute()
7.745966692414834
>>> LA.norm(b, 'fro').execute()
7.745966692414834
>>> LA.norm(a, mt.inf).execute()
4.0
>>> LA.norm(b, mt.inf).execute()
9.0
>>> LA.norm(a, -mt.inf).execute()
0.0
>>> LA.norm(b, -mt.inf).execute()
2.0

>>> LA.norm(a, 1).execute()
20.0
>>> LA.norm(b, 1).execute()
7.0
>>> LA.norm(a, -1).execute()
0.0
>>> LA.norm(b, -1).execute()
6.0
>>> LA.norm(a, 2).execute()
7.745966692414834
>>> LA.norm(b, 2).execute()
7.3484692283495345

>>> LA.norm(a, -2).execute()
0.0
>>> LA.norm(b, -2).execute()
4.351066026358965e-18
>>> LA.norm(a, 3).execute()
5.8480354764257312
>>> LA.norm(a, -3).execute()
0.0


Using the axis argument to compute vector norms:

>>> c = mt.array([[ 1, 2, 3],
...               [-1, 1, 4]])
>>> LA.norm(c, axis=0).execute()
array([ 1.41421356,  2.23606798,  5.        ])
>>> LA.norm(c, axis=1).execute()
array([ 3.74165739,  4.24264069])
>>> LA.norm(c, ord=1, axis=1).execute()
array([ 6.,  6.])


Using the axis argument to compute matrix norms:

>>> m = mt.arange(8).reshape(2,2,2)
>>> LA.norm(m, axis=(1,2)).execute()
array([  3.74165739,  11.22497216])
>>> LA.norm(m[0, :, :]).execute(), LA.norm(m[1, :, :]).execute()
(3.7416573867739413, 11.224972160321824)