Generating function handles with large numbers of variable coefficients
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Matthew Reed
el 23 de Mayo de 2019
Comentada: Matthew Reed
el 24 de Mayo de 2019
I've come across a number of types of optimization process which require a parameter search in a high-dimensional space where the number of coefficients to be fit is based on a dataset.
Example: performing PCA on a set of 4k images. The SVD method does not work because the matrix would be ~8 petabytes.
In this context one normalizes each picture, subtracts their average, and renormalizes these difference images. Those images are eigenvectors. One then maximizes the function
sum(a(n)*I(n)).^2
such that
sum(a(n).^2)=1
This requires a function handle of the type
y = @(b,I) b(1)*I(:,:,1)+b(2)*I(:,:,2)+ .. +b(n)*I(:,:,n);
Along with the function to be minimized,
OLS = @(b) (2-sum(sum((y(b)))).^2); % Minimize this to maximize norm(y(b))
Currently i need to write 'y' out explicitly.
Now the question: is there a way to express this in terms of matrix multiplication without explicitly writing it out?
Thank you,
Matt
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Steven Lord
el 23 de Mayo de 2019
I = reshape(1:60, [3 4 5]);
b = [1 2 3 4 5];
R = reshape(b, [1, 1, numel(b)]);
S = sum(I.*R, 3);
The size of I is [3 4 5] and the size of R is [1 1 5] so the product I.*R has size [3 4 5]. Summing that product in the third dimension results in a matrix of size [3 4]. You can compare this with the result of explicitly writing out the multiplication.
S2 = b(1)*I(:,:,1)+b(2)*I(:,:,2)+ b(3)*I(:, :, 3)+b(4)*I(:, :, 4) +b(5)*I(:,:,5);
S-S2
Unfortunately the ability of the sum function to operate on multiple dimensions at a time was introduced in release R2018b, so you can't use that to simplify your OLS function. But if you could upgrade:
OLS = 2-sum(S, 'all').^2
Or combining the expressions together:
OLS = 2-sum(I.*R, 'all').^2
You don't have to define a separate variable with the reshaped b. I did that for clarity of the example.
But going back to your original question, the problem you're trying to solve is taking the pca of a matrix that's larger than can fit in memory, correct? Try storing your data as a tall array and calling pca on that tall array. The documentation for pca in release R2018a says it supports some of the syntaxes for pca on tall arrays.
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Matt J
el 23 de Mayo de 2019
Editada: Matt J
el 23 de Mayo de 2019
I think this is what you want, but am still not totally sure. The way to calculate a sum of scalar/matrix products without explicitly writing out all n terms (if that's the question) would be,
function y=linearOp(b,I)
[p,q,n]=size(I);
y=reshape(I,[],n)*b(:);
y=reshape(y,p,q);
end
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