Alternatives to For/Do Loop to optimize indexing Matrix

Question

I am trying to optimize my code that searches for elements in the rows of matrix that are greater than 0. Then use that position to grab the element from other matrix. I use the do loop to iterate through each row. Then have another function that creates a new vector to replace that row of the vector. I have read these two questions on stackexchange (Question 1,Question 2) that say to use Map, Scan or NestList instead of for loops, but I am having difficulty applying it to my program. Any suggestions on how to speed this up would be greatly appreciated.

These are my function to search the matrix

positionAB = Function[{dataM, AB, row}, AB[[Flatten[Position[
            dataM[[All, row]], x : _ /; x > 0]]]]];


Rij = Function[{dataM, row}, dataM[[Flatten[Position[dataM[[All, row]], 
      x : _ /; x > 0]], row]]];


randomnFunction=Function[{x,y},(x.y)^2]

My loop is

indexM[data0_, initA0_, initB0_] := Block[

      {initA = initA0,
       initB = initB0,
       data = data0,
       iA0, iB0, iAB, id, ir},
       iA0 = initA;
       iB0 = initB;
       id = data;

         Do[

           iAB = positionAB[id, iA0, m];
           ir = Rij[id, m];
           ii=randomFunction[ir,iAB];
           iB0[[m]]=ii;,

               {m, 0, Length[iB0]}

           ];
 ]

Everyone says not to use for loops, which is what I originally had so switch to a Do but it has the exact same time to evaluate. Not sure how to use the suggestion in (Question 1,Question 2) for my iteration. Any advice would be greatly appreciated. I have many other numerical iterations using Euler's Algorithm that I have the same issue with.

It pretty large matrices. I have

dataM = RandomInteger[{0, 10}, {2000, 3000}];
iA = RandomReal[{-1, 1}, {2000, 10}];
iB = RandomReal[{-1, 1}, {3000, 10}];

So i have

 indexM[dataM,iA,iB]//AbsoluteTiming

Which for each iteration of m creates a 1x10 array. So after the loop the row of iB0 i.e. iB0[[m]] will be update with the new vector.

Update: I have tried to do the same thing as this numerical method on stack exchange but doesn't help with my row replacement. I have tried MarcoB's suggestion but has gotten very convoluted to to the dot product and of the ir and iAB which is a giant array of arrays of different sizes. Ill update my code i have tried with that soon.

I also tried to use ParallelDo but doesn't update my iB0 unless I use SetSharedVariable which makes the computation time longer than my original. Any suggestions to get ParallelDo to work if i can't use Functional methods?

Update: MarcoB method

dataMatrix = RandomInteger[{0, 2}, {3, 4}];
iA = RandomReal[{-1, 1}, {3, 2}];
iB = RandomReal[{-1, 1}, {4, 2}];

positions = Position[dataMatrix, _?(# > 0 &)]
c = SortBy[GatherBy[positions, Last], Last@*Last][[;; , ;; , 1]]
rij = Select[DeleteCases[0] /@ Transpose[dataMatrix], UnsameQ[#, {}] &]
f = Function[{n, m}, n.m];

AB = Map[(iB[[#, All]]) &, c]
MapThread[f, {rij, AB}]

So I have gotten this but still have trouble dealing with a column that has all zeros.

Update: In some good news if I can over come the zero elements it takes the time down from 69.1293 to 36.64. Not great but something!!! Open for any suggestions!!!

Answer 1

You may be going to too much trouble here. Consider that Position works on a matrix just as well as a vector.

You did not include values for your parameters, so I can't use your code, but take a look at the following example. I have a matrix selector that contains $0$ or $1$ entries, and I want to grab the elements of another matrix target that correspond to $1$ elements in the first matrix.

SeedRandom[2016]
selector = RandomChoice[{0, 1}, {10, 10}];
target = RandomInteger[{100, 200}, {10, 10}];

positions = Position[selector, _?Positive]
Extract[target, positions]

This is quite similar to the functionality of Pick, which you may also be interested in.