torch.linalg.norm — PyTorch 2.7 documentation (original) (raw)
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
dimis an int, the vector norm will be computed. - If
dimis a 2-tuple, the matrix norm will be computed. - If
dim= None andord= None,Awill be flattened to 1D and the 2-norm of the resulting vector will be computed. - If
dim= None andord!= None,Amust 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
dtypebefore 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))