How to efficiently retrieve top K-similar vectors by cosine similarity using R?

Question

I'm working on a high-dimensional problem (~4k terms) and would like to retrieve top k-similar (by cosine similarity) and can't afford to do a pair-wise calculation.

My training set is 6million x 4k matrix and I would like to make predictions for 600k x 4k matrix.

What is the most efficient way to retrieve the k-similar items for each item in my 600k x 4k matrix?

Ideally, I would like to get a matrix which is 600k x 10 (i.e., top 10-similar items for each of the 600k items).

ps: I've researched the SO website and found almost all "cosine similarity in R" questions refer to cosine_sim(vector1, vector2). But this question refers to cosine_sim(matrix1, matrix2).

Update The following code uses a naive method to find the cosine similarity between each row in the testset and every row in the training set.

set.seed(123)
train<-matrix(round(runif(30),0),nrow=6,ncol=5)
set.seed(987)
test<-matrix(round(runif(20),0),nrow=4,ncol=5)
train

[1,]    0    1    1    0    1    
[2,]    1    1    1    1    1    
[3,]    0    1    0    1    1    
[4,]    1    0    1    1    1    
[5,]    1    1    0    1    0    
[6,]    0    0    0    1    0

test

[1,]    0    1    1    0    0
[2,]    1    0    1    0    1
[3,]    1    0    0    0    0
[4,]    1    0    0    1    1

coSim<-function(mat1, mat2, topK){
require(plyr)
#mat2: is the testset
#mat1: is the training set. We will find cosine similarity between each row in testset and every row in trainingset.
#topK: user-input. for each row in testset we will return 'topk' similar rows(index) from the testset

#set up an empty result matrix. nrow(result) will be the same as the cartesian product between mat1 & mat2.
result<-matrix(rep(NA, nrow(mat1)*nrow(mat2)), nrow=nrow(mat1)*nrow(mat2), ncol=3)
k=1
for(i in 1:nrow(mat2)){
    for(j in 1:nrow(mat1)){
    result[k,1]<-i
    result[k,2]<-j
    result[k,3]<-crossprod(mat1[j,], mat2[i,])/sqrt(crossprod(mat1[j,]) * crossprod(mat2[i,]))
    k<-k+1
        }
    }
#sort the result matrix by cosine similarity found for each row in testset. not sure how to keep topK from each group so convert to df
result<-as.data.frame(result)
colnames(result)<-c("testRowId", "trainRowId","CosineSimilarity")
result<-ddply(result, "testRowId", function(x) head(x[order(x$CosineSimilarity, decreasing = TRUE) , ], topK))
resultMat<-matrix(result$trainRowId, nrow=nrow(mat2), ncol=topK,byrow=T)
finalResult<-list(similarity=result, index=resultMat)
}

system.time(cosineSim<-coSim(train, test, topK=2)) #0.12 secs
cosineSim
$similarity
  testRowId trainRowId CosineSimilarity
1         1          1        0.8164966
2         1          2        0.6324555
3         2          4        0.8660254
4         2          2        0.7745967
5         3          5        0.5773503
6         3          4        0.5000000
7         4          4        0.8660254
8         4          2        0.7745967

$index
     [,1] [,2]
[1,]    1    2
[2,]    4    2
[3,]    5    4
[4,]    4    2


set.seed(123)
train<-matrix(round(runif(1000000),0),nrow=5000,ncol=200)
set.seed(987)
test<-matrix(round(runif(400000),0),nrow=2000,ncol=200)
system.time(cosineSim<-coSim(train, test, topK=50)) #380secs

When I run the same function with 5000x200 matrix for training and 2000x200 matrix for testing, it took over 380secs.

Ideally, I would like to see some ideas where I do not have to calculate the similarity between each and every row. If that is not possible, some pointers on how to vectorise the above code will be helpful.

Answer 1

No need to compute the similarity for every row. You can use this instead:

coSim2<-function(mat1, mat2, topK){
    #similarity computation:

    xy <- tcrossprod(mat1, mat2)
    xx <- rowSums(mat1^2)
    yy <- rowSums(mat2^2)
    result <- xy/sqrt(outer(xx,yy))

    #top similar rows from train (per row in test):

    top <- apply(result, 2, order, decreasing=TRUE)[1:topK,]
    result_df <- data.frame(testRowId=c(col(top)), trainRowId=c(top))
    result_df$CosineSimilarity <- result[as.matrix(result_df[,2:1])]
    list(similarity=result_df, index=t(top))
}

Test data (I've reduced your train matrix)

set.seed(123)
train<-matrix(round(runif(100000),0),nrow=500,ncol=200)
set.seed(987)
test<-matrix(round(runif(400000),0),nrow=2000,ncol=200)

Result:

> system.time(cosineSim<-coSim(train, test, topK=50)) #380secs
   user  system elapsed 
  41.71    1.59   43.72 

> system.time(cosineSim2<-coSim2(train, test, topK=50)) #380secs
   user  system elapsed 
   0.46    0.02    0.49 

Using your full 5000 x 200 train matrix, coSim2 runs in 7.8 sec.

Also note:

> any(cosineSim$similarity != cosineSim2$similarity)
[1] FALSE
> any(cosineSim$index != cosineSim2$index)
[1] FALSE

You can't use identical because my function returns integers instead of doubles for row IDs.