Collaborative Filtering

This is sample R code for collaborative filtering.

First, we define the function matrix_factorization which is used to decompose a matrix R (m*n) into the product of two matrices P (m*K) and Q (K*n). alpha and beta are coefficients for the gradient descent algorithm, which is used to find the optimal minimum/maximum. Steps denotes the number of interations.

matrix_factorization <- function(R, P, Q, K, steps=5000, alpha=0.0002, beta=0.02) {
  Q <- t(Q)
  for (step in 1:steps) {
    for (i in 1:nrow(R)) {
      for (j in 1:ncol(R)) {
        if (R[i, j] > 0) {
          # calculate error between the true rating matrix and the approximated rating matrix
          eij <- R[i, j] - sum(P[i,] * Q[,j])
          for (k in 1:K) {
            # calculate gradient with alpha and beta parameter
            P[i, k] <- P[i, k] + alpha * (2 * eij * Q[k, j] - beta * P[i, k])
            Q[k, j] <- Q[k, j] + alpha * (2 * eij * P[i, k] - beta * Q[k, j])
          }}}}
    
    eR <- P %*% Q
    e <- 0
    
    for (i in 1:nrow(R)) {
      for (j in 1:ncol(R)) {
        if (R[i, j] > 0) {
          e <- e + (R[i, j] - sum(P[i,] * Q[,j]))^2
            for (k in 1:K) {
            e <- e + (beta/2) * (P[i, k]^2 + Q[k, j]^2)
          }}}}
    
    if (e < 0.001){break}}
  
  return(list(P = P, Q = t(Q)))
}

This is used to generate random numbers.

set.seed(123) 

Matrix R is the rating matrix to be decomposed; 0 means no rating.

R <- matrix(c(5, 3, 0, 1,
              4, 0, 0, 1,
              1, 1, 0, 5,
              1, 0, 0, 4,
              0, 1, 5, 4,
              2, 1, 3, 0), nrow = 6, ncol = 4, byrow = TRUE)

# N: num of User
N <- nrow(R)
# M: num of Movie
M <- ncol(R)
# Num of Features
K <- 3

Starting with two random matrices P and Q:

P <- matrix(runif(N * K), nrow = N, ncol = K)
Q <- matrix(runif(M * K), nrow = M, ncol = K)

Running the algorithm to decompose R into the product of P and Q:

result <- matrix_factorization(R, P, Q, K)
nP <- result$P
nQ <- result$Q

nR <- nP %*% t(nQ)

Printing P, Q, and the predicted rating matrix (nR):

print(nP)

##           [,1]      [,2]         [,3]
## [1,] 0.2463714 1.2785584  1.884240882
## [2,] 0.2058249 1.1525078  1.385007010
## [3,] 1.8185968 1.1550282 -0.255033789
## [4,] 1.7291866 0.5349949  0.223180524
## [5,] 1.3502813 1.2750629 -0.002350487
## [6,] 0.5163619 1.0115940  0.310546768

print(nQ)

##             [,1]      [,2]       [,3]
## [1,] -0.09132514 1.4309696  1.6931080
## [2,]  0.27092979 0.5848768  1.1271429
## [3,]  1.93218953 1.8271619  0.5202290
## [4,]  2.04541001 0.9883406 -0.4065921

print(nR)

##           [,1]      [,2]     [,3]      [,4]
## [1,] 4.9973016 2.9383573 3.792406 1.0014643
## [2,] 3.9753730 2.2909401 3.224032 0.9969337
## [3,] 1.0549269 0.8808018 5.491621 4.9650321
## [4,] 0.9855119 1.0329506 4.434744 3.9749094
## [5,] 1.6972819 1.1089368 4.937523 4.0230309
## [6,] 1.9261927 1.0815863 3.007611 1.9297053