The elementwise Huber function, \(Huber(x, M) = \)

  • \(2M|x|-M^2\)for \(|x| \geq |M|\)

  • \(|x|^2\)for \(|x| \leq |M|.\)

huber(x, M = 1)

Arguments

x

An Expression, vector, or matrix.

M

(Optional) A positive scalar value representing the threshold. Defaults to 1.

Value

An Expression representing the Huber function evaluated at the input.

Examples

set.seed(11)
n <- 10
m <- 450
p <- 0.1    # Fraction of responses with sign flipped

# Generate problem data
beta_true <- 5*matrix(stats::rnorm(n), nrow = n)
X <- matrix(stats::rnorm(m*n), nrow = m, ncol = n)
y_true <- X %*% beta_true
eps <- matrix(stats::rnorm(m), nrow = m)

# Randomly flip sign of some responses
factor <- 2*rbinom(m, size = 1, prob = 1-p) - 1
y <- factor * y_true + eps

# Huber regression
beta <- Variable(n)
obj <- sum(huber(y - X %*% beta, 1))
prob <- Problem(Minimize(obj))
result <- solve(prob)
result$getValue(beta)
#>             [,1]
#>  [1,] -2.8809598
#>  [2,]  0.1535914
#>  [3,] -7.4980147
#>  [4,] -6.8611676
#>  [5,]  5.8162832
#>  [6,] -4.6080355
#>  [7,]  6.5681235
#>  [8,]  3.1077124
#>  [9,] -0.2235099
#> [10,] -4.8797994