Without resorting to FEX/MEX, can the overhead of repeated calls to sparse() be avoided with operations on I, J, V directly?

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I have a time-marching problem bottlenecked by matrix multiplication , where the sparse matrix is the sum of many time-independent sparse matrices with time-dependent scalar coefficients:
I'm aware of FEX solutions that would enable a 3D to be defined once prior to time-marching, though I would prefer to rule out native MATLAB first. My current thinking is to supplement the I, J, V vectors with an index vector K, and do one of the following:
  1. Generate a cell array of sparse matrices along index K, with multiplied with bsxfun, similar to this. (Probably the slowest option)
  2. Sparse matrix construction with B=sparse(I, J, V.*beta(K), N, N)*X
  3. Avoid sparse matrix entirely with B=accumarray(I, V.*beta(K).*X(J), N, N)
I haven't tested option 1, but option 3 with accumarray runs about 50x slower than option 2 (with or without issparse option). This makes sense, as the multiplication with X in accumarray is done before accumulation, hence on a much larger array.
However, option 2 still appears to be wasting performance. For example, sparse(I, J, V, N, N)*X runs about 50x slower than basic A*X. My initial interpretation of this is that much of the time in sparse is spent on unavoidable summation of common terms in I,J,V. I cant profile sparse internally, but tested this by accumulating V prior to sparse(I, J, V, N, N)*X, which comes out to ~7x slower than A*X.
My conclusion is that there is still some significant & (potentially) avoidable overhead from calling sparse many times unrelated to summation of duplicate indices, and the larger piece of summation that is unavoidable due to the summation of time-dependent terms co-located in .
To boil this down, I have two questions I cannot answer:
  1. Considering the pattern of summation of duplicate indices is always the same (static I/J), can the summation overhead in sparse be improved?
  2. What is the source of the remaining overhead in sparse and can it be avoided?
  20 Comments
Bruno Luong
Bruno Luong on 7 Aug 2022
@James Tursa yes you get it right. However
  1. In the range of density/size and N_k (m in your notation) your method is still 50% slower than the accumarray solution we come up with,
  2. The runtime scale with N_k and is not bounded by the assumption of limited "bandwidth" (density of A0) stated by @Joel Lynch. It is probably an important requirement for his application.

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Accepted Answer

Bruno Luong
Bruno Luong on 7 Aug 2022
Edited: Bruno Luong on 7 Aug 2022
EDIT: Here is the final code. Method used in B2 is the fatest;
M = 10000;
N = 10000;
N_k = 100;
A_k = cell(1,N_k);
for k=1:N_k
A_k{k} = sprand(M,N,0.0005);
end
N_k = length(A_k);
[M,N] = size(A_k{1});
beta = rand(N_k,1);
X = rand(N,1);
I = cell(1,N_k);
J = cell(1,N_k);
V = cell(1,N_k);
K = cell(1,N_k);
for k=1:N_k
[I{k},J{k},V{k}] = find(A_k{k});
K{k} = k + zeros(size(I{k}));
end
I = cat(1,I{:});
J = cat(1,J{:});
K = cat(1,K{:});
V = cat(1,V{:});
[IJ,~,ic] = unique([I,J],'rows');
I2 = IJ(:,1);
J2 = IJ(:,2);
Y = sparse(ic, K, V, size(IJ,1), N_k);
tic
B1=sparse(I, J, V.*beta(K), M, N)*X ;
toc
Elapsed time is 0.252217 seconds.
tic
B3 = sparse(I2,J2,Y*beta,M,N)*X;
toc
Elapsed time is 0.195761 seconds.
tic
B2 = accumarray(I2,(Y*beta).*X(J2),[M,1]);
toc
Elapsed time is 0.096573 seconds.
  5 Comments
Joel Lynch
Joel Lynch on 7 Aug 2022
Yeah, at this point I'm just documenting for future viewers. Feel free to post this up top, I cleaned some things up and added comments. Also added integer type casting to reduced memory.
M = 10000; % A_0/A_k rows
N = 10000; % A_0/A_k columns
N_k = 100; % Number of A_k matricies
% Random beta/X
beta = rand(N_k,1);
X = rand(N,1);
% Pick smallest integer type for indices
int_type = 'uint64';
if max(N,M) < intmax("uint16")
int_type = 'uint16';
elseif max(N,M) < intmax("uint32")
int_type = 'uint32';
end
% Generate A_k matricies in single stream
I = []; J=[]; K=[]; V=[];
bandwith = round(M*0.25); % Controls final density of A_0
A_k_ref_density = 1e-4; % density of each A_k
for k=1:N_k
A_k = sprand(M, N, A_k_ref_density);
[Inew, Jnew] = find(A_k);
ind = abs(Inew-Jnew)<bandwith; % Keep i/j's within band
I = [I; cast(Inew(ind), int_type)];
J = [J; cast(Jnew(ind), int_type)];
V = [V; A_k(ind)];
K = [K; repmat(cast(k, int_type), sum(ind), 1)];
% spy(sparse(I,J,K, M,N)) % Spy A_k
end
%spy(sparse(I,J,K, M,N)) % Spy Final A_0 for non-zero Beta's
% Smaller vector of any non-zero i/j's in all A_k's
% numel(ic) = size([I, J], 1)
% ic points to the location in the reduced/unique array of [I, J] pairs
% (i.e. ic holds the row index where each nonzero I/J in A_0 resides)
[IJ, ~, ic] = unique([I, J], 'rows', 'stable');
Iu = IJ(:,1); % unique I in [I, J]
Ju = IJ(:,2); % unique J in [I, J]
Nrow_u = size(IJ, 1);
% Time-Independent Y matrix, such that Y*beta is a column vector of
% non-zero elements in A_0, with I/J defined by Iu, Ju
Y = sparse(ic, double(K), V, Nrow_u, N_k);
% Native approach, construct A_0 within time loop, then A*x
t1 = timeit(@() sparse(I, J, V.*beta(K), M, N)*X)
% Construct sparse, but with the reduced Y
t2 = timeit(@() sparse(Iu, Ju, Y*beta, M, N)*X)
% Optimal Approach, accumarray which avoids sparse()
t3 = timeit(@() accumarray(Iu, (Y*beta).*X(Ju), [M,1]))

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