linalg.norm in PyTorch

#pytorch #matrixnorm #vectornorm #regularization

*Memos:

My post explains linalg.vector_norm().
My post explains linalg.matrix_norm().

linalg.norm() can get the 0D or more D tensor of the zero or more elements computed with norm from the 0D or more D tensor of zero or more elements as shown below:

*Memos:

linalg.norm() can be used with torch but not with a tensor. *A tensor can use torch.norm which is deprecated but not linalg.norm().
The 1st argument with torch is input(Required-Type:tensor of float or complex): *Memos:
- It must be the 0D or more D tensor of zero or more elements.
- A complex tensor returns a float tensor even if dtype=torch.complex64 is set to linalg.norm() which is a bug.
The 2nd argument with torch is ord(Optional-Default:None-Type:int, float or str). *It sets a norm.
The 3rd argument with torch is dim(Optional-Default:None-Type:int or tuple or list of int). *The length of tuple or list of int must be 1 or 2.
The 4th argument with torch is keepdim(Optional-Default:False-Type:bool). *My post explains keepdim argument.
There is out argument with torch(Optional-Default:None-Type:tensor): *Memos:
- out= must be used.
- My post explains out argument.
There is dtype argument with torch(Optional-Default:None-Type:dtype): *Memos:
- If it's None, it's inferred from input.
- dtype= must be used.
- My post explains dtype argument.
If dim is int or the length 1 of tuple or list of int, a vector norm is computed.
If dim is the length 2 of tuple or list of int, a matrix norm is computed.
If ord and dim are None, the input tensor is flattened to a 1D tensor and the L2 norm of the resulting vector will be computed.
If ord is not None and dim is None, input must be 1D or 2D tensor.
ord supports the following norms: *Memos:
- inf can be torch.inf, float('inf'), etc:
- For vector, there are also L3 norm(ord=3), L4 norm(ord=4) etc.

`ord`	Norm for vector	Norm for matrix
`None`(Default)	L2 norm	Frobenius norm
`'fro'`	Not supported	Frobenius norm
`'nuc'`	Not supported	Nuclear norm
`inf`	`max(abs(x))`	`max(sum(abs(x), dim=1))`
`-inf`	`min(abs(x))`	`min(sum(abs(x), dim=1))`
`0`	`sum(x != 0)`(L0 norm)	Not supported
`1`	`sum(abs(x)^{ord})^{(1/ord)}`(L1 norm)	`max(sum(abs(x), dim=0))`
`-1`	Same as above	`min(sum(abs(x), dim=0))`
`2`	Same as above(L2 norm)	The largest singular value of SVD(Singular Value Decomposition)
`-2`	Same as above	The smallest singular value of SVD
Other `int` or `float`	Same as above	Not supported

import torch
from torch import linalg

my_tensor = torch.tensor([-3., -2., -1., 0., 1., 2., 3., 4.])

linalg.norm(input=my_tensor) # L2 norm
# tensor(6.6332)

linalg.norm(input=my_tensor, ord=0) # L0 norm
# tensor(7.)

linalg.norm(input=my_tensor, ord=1) # L1 norm
# tensor(16.)

linalg.norm(input=my_tensor, ord=-1)
# tensor(0.)

linalg.norm(input=my_tensor, ord=2) # L2 norm
# tensor(6.6332)

linalg.norm(input=my_tensor, ord=-2)
# tensor(0.)

linalg.norm(input=my_tensor, ord=torch.inf)
# tensor(4.)

linalg.norm(input=my_tensor, ord=-torch.inf)
# tensor(0.)

my_tensor = torch.tensor([[-3., -2., -1., 0.],
                          [1., 2., 3., 4.]])
linalg.norm(input=my_tensor) # L2 norm
# tensor(6.6332)

linalg.norm(input=my_tensor, ord=1)
# tensor(4.)

linalg.norm(input=my_tensor, ord=-1)
# tensor(4.)

linalg.norm(input=my_tensor, ord=2)
# tensor(5.8997)

linalg.norm(input=my_tensor, ord=-2)
# tensor(3.0321)

linalg.norm(input=my_tensor, ord=torch.inf)
# tensor(10.)

linalg.norm(input=my_tensor, ord=-torch.inf)
# tensor(6.)

linalg.norm(input=my_tensor, ord='fro') # Frobenius norm
# tensor(6.6332)

linalg.norm(input=my_tensor, ord='nuc') # Nuclear norm
# tensor(8.9318)

my_tensor = torch.tensor([[[-3., -2.], [-1., 0.]],
                          [[1., 2.], [3., 4.]]])
linalg.norm(input=my_tensor) # L2 norm
# tensor(6.6332)

linalg.norm(input=my_tensor, ord=0, dim=2) # L0 norm
linalg.norm(input=my_tensor, ord=0, dim=-1) # L0 norm
# tensor([[2., 1.],
#         [2., 2.]])

linalg.norm(input=my_tensor, ord=1, dim=2) # L1 norm
linalg.norm(input=my_tensor, ord=1, dim=-1) # L1 norm
# tensor([[5., 1.],
#         [3., 7.]])

linalg.norm(input=my_tensor, ord=-1, dim=2)
linalg.norm(input=my_tensor, ord=-1, dim=-1)
# tensor([[1.2000, 0.0000],
#         [0.6667, 1.7143]])

linalg.norm(input=my_tensor, ord=2, dim=2) # L2 norm
linalg.norm(input=my_tensor, ord=2, dim=-1) # L2 norm
# tensor([[3.6056, 1.0000],
#         [2.2361, 5.0000]])

linalg.norm(input=my_tensor, ord=-2, dim=2)
linalg.norm(input=my_tensor, ord=-2, dim=-1)
# tensor([[1.6641, 0.0000],
#         [0.8944, 2.4000]])

linalg.norm(input=my_tensor, ord=torch.inf, dim=2)
linalg.norm(input=my_tensor, ord=torch.inf, dim=-1)
# tensor([[3., 1.],
#         [2., 4.]])

linalg.norm(input=my_tensor, ord=-torch.inf, dim=2)
linalg.norm(input=my_tensor, ord=-torch.inf, dim=-1)
# tensor([[2., 0.],
#         [1., 3.]])

# Frobenius norm
linalg.norm(input=my_tensor, ord='fro', dim=(1, 2))
# tensor([3.7417, 5.4772])

# Nuclear norm
linalg.norm(input=my_tensor, ord='nuc', dim=(1, 2))
# tensor([4.2426, 5.8310])

my_tensor = torch.tensor([[[-3.+0.j, -2.+0.j], [-1.+0.j, 0.+0.j]],
                          [[1.+0.j, 2.+0.j], [3.+0.j, 4.+0.j]]])
linalg.norm(input=my_tensor, dtype=torch.complex64) # L2 norm
# tensor(6.6332)

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linalg.norm in PyTorch

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