Project onto a simplex where observations in the unit simplex x

Minimizes: \(1/2 ||w-v||^2_2 \quad\) s.t. \(sum_i w_i = 1, w_i \geq 0\)

project_onto_simplex(y)

Arguments

y	n dimensional vector to be projected onto the simplex

Value

proj_y projection of y onto the unit simplex of dimension n

Details

This code replicates a solution referenced on stackexchange, which linked to following Matlab code.

Examples

library(ggplot2)
x1 <- runif(2, -5, 5)
x2 <- c(.1, 1.6)
x3 <- c(.1, 1.1)
x4 <- c(.1,.1)

x_vals <- list(x1, x2, x3, x4)

proj_xs <- lapply(x_vals, project_onto_simplex)

vis_list <- list()
for (idx in 1:4){
  x <- x_vals[[idx]]
  proj_x <- proj_xs[[idx]]
  data1 <- data.frame(X = x[1], Y = x[2],
                      X_proj = proj_x[1],
                      Y_proj = proj_x[2])

  data_simplex <- data.frame(X_low = 0,
                             Y_low = 1,
                             X_high = 1,
                             Y_high = 0)

  vis_list[[idx]] <- ggplot() + geom_segment(data = data1,
                                        aes(x = X, y = Y,
                                            xend = X_proj,
                                            yend = Y_proj)) +
    geom_point(data = data1, aes(x = X, y = Y)) +
    geom_point(data = data1, aes(x = X_proj, y = Y_proj), color = "blue") +
    geom_segment(data = data_simplex, aes(x = X_low, y = Y_low,
                                          xend = X_high,
                                          yend = Y_high), color = "blue") +
    coord_fixed()
}
#> Coordinate system already present. Adding new coordinate system, which will replace the existing one.
#> Coordinate system already present. Adding new coordinate system, which will replace the existing one.
#> Coordinate system already present. Adding new coordinate system, which will replace the existing one.
#> Coordinate system already present. Adding new coordinate system, which will replace the existing one.
gridExtra::grid.arrange(grobs = vis_list, nrow = 2)