softmax — SciPy v1.15.2 Manual (original) (raw)

scipy.special.

scipy.special.softmax(x, axis=None)[source]#

Compute the softmax function.

The softmax function transforms each element of a collection by computing the exponential of each element divided by the sum of the exponentials of all the elements. That is, if x is a one-dimensional numpy array:

softmax(x) = np.exp(x)/sum(np.exp(x))

Parameters:

xarray_like

Input array.

axisint or tuple of ints, optional

Axis to compute values along. Default is None and softmax will be computed over the entire array x.

Returns:

sndarray

An array the same shape as x. The result will sum to 1 along the specified axis.

Notes

The formula for the softmax function \(\sigma(x)\) for a vector\(x = \{x_0, x_1, ..., x_{n-1}\}\) is

\[\sigma(x)_j = \frac{e^{x_j}}{\sum_k e^{x_k}}\]

The softmax function is the gradient of logsumexp.

The implementation uses shifting to avoid overflow. See [1] for more details.

Added in version 1.2.0.

References

[1]

P. Blanchard, D.J. Higham, N.J. Higham, “Accurately computing the log-sum-exp and softmax functions”, IMA Journal of Numerical Analysis, Vol.41(4), DOI:10.1093/imanum/draa038.

Examples

import numpy as np from scipy.special import softmax np.set_printoptions(precision=5)

x = np.array([[1, 0.5, 0.2, 3], ... [1, -1, 7, 3], ... [2, 12, 13, 3]]) ...

Compute the softmax transformation over the entire array.

m = softmax(x) m array([[ 4.48309e-06, 2.71913e-06, 2.01438e-06, 3.31258e-05], [ 4.48309e-06, 6.06720e-07, 1.80861e-03, 3.31258e-05], [ 1.21863e-05, 2.68421e-01, 7.29644e-01, 3.31258e-05]])

Compute the softmax transformation along the first axis (i.e., the columns).

m = softmax(x, axis=0)

m array([[ 2.11942e-01, 1.01300e-05, 2.75394e-06, 3.33333e-01], [ 2.11942e-01, 2.26030e-06, 2.47262e-03, 3.33333e-01], [ 5.76117e-01, 9.99988e-01, 9.97525e-01, 3.33333e-01]])

m.sum(axis=0) array([ 1., 1., 1., 1.])

Compute the softmax transformation along the second axis (i.e., the rows).

m = softmax(x, axis=1) m array([[ 1.05877e-01, 6.42177e-02, 4.75736e-02, 7.82332e-01], [ 2.42746e-03, 3.28521e-04, 9.79307e-01, 1.79366e-02], [ 1.22094e-05, 2.68929e-01, 7.31025e-01, 3.31885e-05]])

m.sum(axis=1) array([ 1., 1., 1.])