Log sum of exponentials

Efficient implementation (via Fortran) of the log-sum-exp function.

Usage

logsumexp(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 log-sum-exp (LSE) function calculates $$ \text{LSE}(x_{[i]}) = \log \sum_{j=1}^k \exp(x_j + v_j) = m + \log \sum_{j=1}^k \exp(x_j + v_j - m) $$ where \(m = \max(x_1+v_1, \dots, x_k+v_k)\).

Value

Returns a vector of values of length equal to the number of rows of x.

Author

Luca Scrucca

See also

softmax

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

Examples

x = matrix(rnorm(15), 5, 3)
v = log(c(0.5, 0.3, 0.2))
logsumexp(x, v)
#> [1] -0.6711947 -0.6393459  0.5306414 -0.1252858  0.8242612