mlx.core.linalg.norm

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mlx.core.linalg.norm#

norm(a: array, /, ord: None | int | float | str = None, axis: None | int | list[int] = None, keepdims: bool = False, *, stream: None | Stream | Device = None) → array#

Matrix or vector norm.

This function computes vector or matrix norms depending on the value of the ord and axis parameters.

Parameters:

  • a (array) – Input array. If axis is None, a must be 1-D or 2-D, unless ord is None. If both axis and ord are None, the 2-norm of a.flatten will be returned.

  • ord (int, float or str, optional) – Order of the norm (see table under Notes). If None, the 2-norm (or Frobenius norm for matrices) will be computed along the given axis. Default: None.

  • axis (int or list(int), optional) – If axis is an integer, it specifies the axis of a 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 a is 1-D) or a matrix norm (when a is 2-D) is returned. Default: None.

  • keepdims (bool, optional) – If True, the axes which are normed over are left in the result as dimensions with size one. Default False.

Returns:

The output containing the norm(s).

Return type:

array

Notes

For values of ord < 1, 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.

Both the Frobenius and nuclear norm orders are only defined for matrices and raise a ValueError when a.ndim != 2.

References

Examples

>>> import mlx.core as mx
>>> from mlx.core import linalg as la
>>> a = mx.arange(9) - 4
>>> a
array([-4, -3, -2, ..., 2, 3, 4], dtype=int32)
>>> b = a.reshape((3,3))
>>> b
array([[-4, -3, -2],
       [-1,  0,  1],
       [ 2,  3,  4]], dtype=int32)
>>> la.norm(a)
array(7.74597, dtype=float32)
>>> la.norm(b)
array(7.74597, dtype=float32)
>>> la.norm(b, 'fro')
array(7.74597, dtype=float32)
>>> la.norm(a, float("inf"))
array(4, dtype=float32)
>>> la.norm(b, float("inf"))
array(9, dtype=float32)
>>> la.norm(a, -float("inf"))
array(0, dtype=float32)
>>> la.norm(b, -float("inf"))
array(2, dtype=float32)
>>> la.norm(a, 1)
array(20, dtype=float32)
>>> la.norm(b, 1)
array(7, dtype=float32)
>>> la.norm(a, -1)
array(0, dtype=float32)
>>> la.norm(b, -1)
array(6, dtype=float32)
>>> la.norm(a, 2)
array(7.74597, dtype=float32)
>>> la.norm(a, 3)
array(5.84804, dtype=float32)
>>> la.norm(a, -3)
array(0, dtype=float32)
>>> c = mx.array([[ 1, 2, 3],
...               [-1, 1, 4]])
>>> la.norm(c, axis=0)
array([1.41421, 2.23607, 5], dtype=float32)
>>> la.norm(c, axis=1)
array([3.74166, 4.24264], dtype=float32)
>>> la.norm(c, ord=1, axis=1)
array([6, 6], dtype=float32)
>>> m = mx.arange(8).reshape(2,2,2)
>>> la.norm(m, axis=(1,2))
array([3.74166, 11.225], dtype=float32)
>>> la.norm(m[0, :, :]), LA.norm(m[1, :, :])
(array(3.74166, dtype=float32), array(11.225, dtype=float32))
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