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

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))
x : array_like

Input array.

axis : int or tuple of ints, optional

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

s : ndarray

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


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.

New in version 1.2.0.


>>> 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()

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.])

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