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 andord
= None,A
will be flattened to 1D and the 2-norm of the resulting vector will be computed.If
dim
= None andord
!= 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: Nonekeepdim (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 todtype
before performing the operation, and the returned tensor’s type will bedtype
. 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))