scipy.special.softmax

scipy.special.softmax(x, axis=None)

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.

New 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

>>> 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.])