# norm¶

paddle.linalg. norm ( x, p=None, axis=None, keepdim=False, name=None ) [source]

Returns the matrix norm (the Frobenius norm, the nuclear norm and p-norm) or vector norm (the 1-norm, the Euclidean or 2-norm, and in general the p-norm) of a given tensor.

Whether the function calculates the vector norm or the matrix norm is determined as follows:

• If axis is of type int, calculate the vector norm.

• If axis is a two-dimensional array, calculate the matrix norm.

• If axis is None, x is compressed into a one-dimensional vector and the vector norm is calculated.

porder

norm for matrices

norm for vectors

None(default)

frobenius norm

2_norm

fro

frobenius norm

not support

nuc

nuclear norm

not support

inf

max(sum(abs(x), dim=1))

max(abs(x))

-inf

min(sum(abs(x), dim=1))

min(abs(x))

0

not support

sum(x != 0)

1

max(sum(abs(x), dim=0))

as below

-1

min(sum(abs(x), dim=0))

as below

2

The maximum singular value of a matrix consisting of axis.

as below

-2

The minimum singular value of a matrix consisting of axis.

as below

other int

or float

not support

sum(abs(x)^{porder})^ {(1 / porder)}

Parameters
• x (Tensor) – The input tensor could be N-D tensor, and the input data type could be float32 or float64.

• p (int|float|string, optional) – Order of the norm. Supported values are fro, nuc, 0, ±1, ±2, ±inf and any real number yielding the corresponding p-norm. Default value is None.

• axis (int|list|tuple, optional) – The axis on which to apply norm operation. If axis is int or list(int)/tuple(int) with only one element, the vector norm is computed over the axis. If axis < 0, the dimension to norm operation is rank(input) + axis. If axis is a list(int)/tuple(int) with two elements, the matrix norm is computed over the axis. Default value is None.

• keepdim (bool, optional) – Whether to reserve the reduced dimension in the output Tensor. The result tensor will have fewer dimension than the `input` unless `keepdim` is true, default value is False.

• name (str, optional) – The default value is None. Normally there is no need for user to set this property. For more information, please refer to Name.

Returns

results of norm operation on the specified axis of input tensor, it’s data type is the same as input’s Tensor.

Return type

Tensor

Examples

```>>> import paddle
>>> x = paddle.arange(24, dtype="float32").reshape([2, 3, 4]) - 12
>>> print(x)
Tensor(shape=[2, 3, 4], dtype=float32, place=Place(cpu), stop_gradient=True,
[[[-12., -11., -10., -9. ],
[-8. , -7. , -6. , -5. ],
[-4. , -3. , -2. , -1. ]],
[[ 0. ,  1. ,  2. ,  3. ],
[ 4. ,  5. ,  6. ,  7. ],
[ 8. ,  9. ,  10.,  11.]]])

>>> # compute frobenius norm along last two dimensions.
>>> out_fro = paddle.linalg.norm(x, p='fro', axis=[0,1])
>>> print(out_fro)
[17.43559647, 16.91153526, 16.73320007, 16.91153526])

>>> # compute 2-order vector norm along last dimension.
>>> out_pnorm = paddle.linalg.norm(x, p=2, axis=-1)
>>> print(out_pnorm)
[[21.11871147, 13.19090557, 5.47722578 ],
[3.74165750 , 11.22497177, 19.13112640]])

>>> # compute 2-order  norm along [0,1] dimension.
>>> out_pnorm = paddle.linalg.norm(x, p=2, axis=[0,1])
>>> print(out_pnorm)
[15.75857544, 14.97978878, 14.69693947, 14.97978973])

>>> # compute inf-order  norm
>>> print(out_pnorm)
12.)

>>> out_pnorm = paddle.linalg.norm(x, p=float("inf"), axis=0)
>>> print(out_pnorm)
[[12., 11., 10., 9. ],
[8. , 7. , 6. , 7. ],
[8. , 9. , 10., 11.]])

>>> # compute -inf-order  norm