Optimize the weight of vectors given the similarity matrix of mean vectors

Question

I want to solve the optimazation problem to search best weights for groups of vectors. Would you like to give some suggestions about how to solve it by R? Thanks very much.

The problem is as follows.

Given there are N groups, we know their similarity matrix among these N groups. The dimension of S is N*N.

In each group, there are K vectors . There are M elements in each vector which value is 0 or 1. .

we can fit an average vector based on these K vectors. For example, average vector

Based on these avearge vectors in each group, we could calculate the correlation among these avearge vectors.

The object is to minimize the differene between correlation matrix C and known similarity matrix S.

Answer 1

Beacuse you didn't provide any data I will generate random and demonstrate way you can approach your problem.

Similarity matrix:

N <- 6

S <- matrix(runif(N^2, -1, 1), ncol = N, nrow = N)
similarity_matrix <- (S + t(S)) / 2

N is number of groups. Each value of similarity matrix is between -1 and 1 and matrix is symmetric (beacuse you want to compare it to covariance matrix these makes sense).

group vectors:

M <- 10
K <- 8

group_vectors <- replicate(N, replicate(K, sample(c(0, 1), M, TRUE)), FALSE)

M is dimension of vector and K is number of binary vectors in each group.

fitness function

fitness <- function(W, group_vectors, similarity_matrix){
  
  W <- as.data.frame(matrix(W, nrow = K, ncol = N))
  
  SS <- cov(
    mapply(function(x,y)  rowSums(sweep(x, 2, y, "*")), group_vectors, W)
  )
  
  sum(abs(SS - similarity_matrix))
}

fitness for given weights calculates described covariance matrix and its distance from similarity_matrix.

differential evolution approach

res <- DEoptim::DEoptim(
  fn = fitness, 
  lower = rep(-1, K*N), 
  upper = rep(1, K*N),
  group_vectors = group_vectors,
  similarity_matrix = similarity_matrix,
  control = DEoptim::DEoptim.control(VTR = 0, itermax = 1000, trace = 50, NP = 100)
)

W <- matrix(res$optim$bestmem, nrow = K, ncol = N)

genetic algorithm approach

res <- GA::ga(
  type = "real-valued",
  fitness = function(W, ...) -fitness(W, ...),
  lower = rep(-1, K*N), 
  upper = rep(1, K*N),
  group_vectors = group_vectors,
  similarity_matrix = similarity_matrix,
  maxiter = 10000,
  run = 200
)

W <- matrix(res@solution[1,], nrow = K, ncol = N)