torch.linalg.norm

torch.linalg.norm(A, ord=None, dim=None, keepdim=False, *, out=None, dtype=None) → Tensor

Computes a vector or matrix norm.

Supports input of float, double, cfloat and cdouble dtypes.

Whether this function computes a vector or matrix norm is determined as follows:

• If dim is an int, the vector norm will be computed.
• If dim is a 2-tuple, the matrix norm will be computed.
• If dim= None and ord= None, A will be flattened to 1D and the 2-norm of the resulting vector will be computed.
• If dim= None and ord != None, A must be 1D or 2D.

ord defines the norm that is computed. The following norms are supported:

ord	norm for matrices	norm for vectors
None (default)	Frobenius norm	2-norm (see below)
‘fro’	Frobenius norm	– not supported –
‘nuc’	nuclear norm	– not supported –
inf	max(sum(abs(x), dim=1))	max(abs(x))
-inf	min(sum(abs(x), dim=1))	min(abs(x))
0	– not supported –	sum(x != 0)
1	max(sum(abs(x), dim=0))	as below
-1	min(sum(abs(x), dim=0))	as below
2	largest singular value	as below
-2	smallest singular value	as below
other int or float	– not supported –	sum(abs(x)^{ord})^{(1 / ord)}

where inf refers to float(‘inf’), NumPy’s inf object, or any equivalent object.

See also

torch.linalg.vector_norm() computes a vector norm.

torch.linalg.matrix_norm() computes a matrix norm.

The above functions are often clearer and more flexible than using torch.linalg.norm(). For example, torch.linalg.norm(A, ord=1, dim=(0, 1)) always computes a matrix norm, but with torch.linalg.vector_norm(A, ord=1, dim=(0, 1)) it is possible to compute a vector norm over the two dimensions.

Parameters

• A (Tensor) – tensor of shape (*, n) or (*, m, n) where * is zero or more batch dimensions
• ord (int, float, inf, -inf, 'fro', 'nuc', optional) – order of norm. Default: None
• dim (int, Tuple[int], optional) – dimensions over which to compute the vector or matrix norm. See above for the behavior when dim= None. Default: None
• keepdim (bool, optional) – If set to True, the reduced dimensions are retained in the result as dimensions with size one. Default: False

Keyword Arguments

• out (Tensor, optional) – output tensor. Ignored if None. Default: None.
• dtype (torch.dtype, optional) – If specified, the input tensor is cast to dtype before performing the operation, and the returned tensor’s type will be dtype. Default: None

Returns

A real-valued tensor, even when A is complex.

Examples:

>>> from torch import linalg as LA
>>> a = torch.arange(9, dtype=torch.float) - 4
>>> a
tensor([-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.])
>>> B = a.reshape((3, 3))
>>> B
tensor([[-4., -3., -2.],
        [-1.,  0.,  1.],
        [ 2.,  3.,  4.]])

>>> LA.norm(a)
tensor(7.7460)
>>> LA.norm(B)
tensor(7.7460)
>>> LA.norm(B, 'fro')
tensor(7.7460)
>>> LA.norm(a, float('inf'))
tensor(4.)
>>> LA.norm(B, float('inf'))
tensor(9.)
>>> LA.norm(a, -float('inf'))
tensor(0.)
>>> LA.norm(B, -float('inf'))
tensor(2.)

>>> LA.norm(a, 1)
tensor(20.)
>>> LA.norm(B, 1)
tensor(7.)
>>> LA.norm(a, -1)
tensor(0.)
>>> LA.norm(B, -1)
tensor(6.)
>>> LA.norm(a, 2)
tensor(7.7460)
>>> LA.norm(B, 2)
tensor(7.3485)

>>> LA.norm(a, -2)
tensor(0.)
>>> LA.norm(B.double(), -2)
tensor(1.8570e-16, dtype=torch.float64)
>>> LA.norm(a, 3)
tensor(5.8480)
>>> LA.norm(a, -3)
tensor(0.)

Using the dim argument to compute vector norms:

>>> c = torch.tensor([[1., 2., 3.],
...                   [-1, 1, 4]])
>>> LA.norm(c, dim=0)
tensor([1.4142, 2.2361, 5.0000])
>>> LA.norm(c, dim=1)
tensor([3.7417, 4.2426])
>>> LA.norm(c, ord=1, dim=1)
tensor([6., 6.])

Using the dim argument to compute matrix norms:

>>> A = torch.arange(8, dtype=torch.float).reshape(2, 2, 2)
>>> LA.norm(A, dim=(1,2))
tensor([ 3.7417, 11.2250])
>>> LA.norm(A[0, :, :]), LA.norm(A[1, :, :])
(tensor(3.7417), tensor(11.2250))