scipy.special.

# softmax#

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.

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]])

>>> m.sum()
1.0


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.])