mclust (version 6.1)

softmax: Softmax function

Description

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)

Value

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

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.

Author

Luca Scrucca

Details

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

See Also

logsumexp

Examples

x = matrix(rnorm(15), 5, 3)
v = log(c(0.5, 0.3, 0.2))
(z = softmax(x, v))
rowSums(z)