softmax {mclust}

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)

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(s)

Luca Scrucca

References

Blanchard P., Higham D. J., Higham N. J. (2021). Accurately computing the log-sum-exp and softmax functions. IMA Journal of Numerical Analysis, 41/4:2311–2330. doi:10.1093/imanum/draa038

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)