Iterating in a matrix avoiding loop in MATLAB - matlab

Question

I am posing an interesting and useful question that needs to be carried out in MATLAB. It is about efficiency of programming by avoiding using Loops"
Assume a matrix URm whose columns are products and rows are people. The matrix entries are rating of people to these products, and this matrix is sparse as each person normally rates only few products.
URm [n_u, n_i]
Another matrix of interest is F, which contains attribute for each of the products and the attribute is of fixed length:
F [n_f,n_i]
We divide the URm into two sub-matrices randomly: URmTrain and URmTest where the former is used for training the system and the latter for test. These two matrices have similar rows (users) but they could have different number of columns (products).
We can find the similarity between items very fast using pdist() or Matrix transpose:
S = F * F' ;
For each row (user) in URmTest:
URmTestp = zeros(size(URmTest));
u = 1 ; %% Example user 1
for i = 1 : size(URmTest,2)
indTrain = find(URmTrain(u,:)) ; % For each user, search for items in URmTrain that have been rated by the the user (i.e. the have a rating greater than zero)
for j = 1 : length(indTrain)
URmTestp(u,i) = URmTestp(u,i) + S(i,indTrain(j))*URmTrain(u,indTrain(j))
end
end
where URmp is the predicted version of URm and we can compute an error on how good our prediction has been.
Example
Lets's make a simple example. Let's assume the items user 1 has rated items 3 , 5 and 17:
indTrain = [3 5 17]
For each item j in URmTest, I want to predict the rating using the following formula:
URmTestp(u,j) = S(j,3)*URmTrain(u,3) + S(j,5)*URmTrain(u,5) + S(j,17)*URmTrain(u,17)
Once completed this process needs to be repeated for all users.
As URm is typically very big, I prefer options which use least amount of 'loops'. We may be able to take advantage of bsxfun but I am not sure if we can.
Please suggest me ides that can help on accelerating this process as rapid as possible. Thank you

Answer 1

I'm still not sure I completely understand your problem. But it seems to me that if you pre-compute s_ij as
s_ij = F.' * F %'// [ni x ni] matrix
then what you're after is simply
URmTestp(u,indTest) = URmTrain(u,indTrain) * s_ij(indTrain,indTest);
% or
%URmTestp(u,:) = URmTrain(u,indTrain) * s_ij(indTrain,:);
or if you only compute a smaller s_ij block only for the necessary arrays,
s_ij = F(:,indTrain).' * F(:,indTest);
then
URmTestp(u,indTest) = URmTrain(u,indTrain) * s_ij;
Alternatively, you can always compute the necessary subblock of s_ij on the fly:
URmTestp(u,indTest) = URmTrainp(u,indTrain) * F(:,indTrain).'*F(:,indTest);
If I understand correctly that indTest and indTrain are functions of u, such as
URmTestp = zeros(n_u,n_i); %// pre-allocate here!
for u=1:n_u
indTest = testCell{u};
indTrain = trainCell{u};
URmTestp(u,indTest) = URmTrainp(u,indTrain) * F(:,indTrain).'*F(:,indTest); %'
...
end
then probably not much can be vectorized on this loop, unless there's a very tricky indexing scheme that allows you to use linear indices. I'd stick with this setup.