Softmax function

Efficient implementation (via Fortran) of the softmax (aka multinomial logistic) function converting a set of numerical values to probabilities summing to 1.

Usage

softmax(x, v = NULL)

Arguments

x

a matrix of dimension \(n \times k\) of numerical values. If a vector is provided, it is converted to a single-row matrix.

v

an optional vector of length \(k\) of numerical values to be added to each row of x matrix. If not provided, a vector of zeros is used.

Details

Given the matrix x, for each row \(x_{[i]} = [x_1, \dots, x_k]\) (with \(i=1,\dots,n\)), the softmax function calculates $$ \text{softmax}(x_{[i]})_j = \dfrac{\exp{x_j + v_j}}{\sum_{l=1}^k \exp(x_l + v_l)} \qquad \text{for } j = 1,\dots,k $$

Value

Returns a matrix of the same dimension as x with values in the range \((0,1)\) that sum to 1 along the rows.

Author

Luca Scrucca

See also

logsumexp

Examples

x = matrix(rnorm(15), 5, 3)
v = log(c(0.5, 0.3, 0.2))
(z = softmax(x, v))
#>           [,1]       [,2]       [,3]
#> [1,] 0.4039612 0.04453269 0.55150615
#> [2,] 0.5363225 0.34888841 0.11478912
#> [3,] 0.7237697 0.18876193 0.08746842
#> [4,] 0.4567840 0.52021530 0.02300074
#> [5,] 0.3457171 0.24608575 0.40819713
rowSums(z)
#> [1] 1 1 1 1 1