C++ eigenvalue/vector decomposition, only need first n vectors fast

Question

I have a ~3000x3000 covariance-alike matrix on which I compute the eigenvalue-eigenvector decomposition (it\'s a OpenCV matrix, and I use cv::eigen() to get the job

Answer 1

In this article, Simon Funk shows a simple, effective way to estimate a singular value decomposition (SVD) of a very large matrix. In his case, the matrix is sparse, with dimensions: 17,000 x 500,000.

Now, looking here, describes how eigenvalue decomposition closely related to SVD. Thus, you might benefit from considering a modified version of Simon Funk's approach, especially if your matrix is sparse. Furthermore, your matrix is not only square but also symmetric (if that is what you mean by covariance-like), which likely leads to additional simplification.

... Just an idea :)

Answer 2

It seems that Spectra will do the job with good performances.

Here is an example from their documentation to compute the 3 first eigen values of a dense symmetric matrix M (likewise your covariance matrix):

#include <Eigen/Core>
#include <Spectra/SymEigsSolver.h>
// <Spectra/MatOp/DenseSymMatProd.h> is implicitly included
#include <iostream>
using namespace Spectra;
int main()
{
    // We are going to calculate the eigenvalues of M
    Eigen::MatrixXd A = Eigen::MatrixXd::Random(10, 10);
    Eigen::MatrixXd M = A + A.transpose();
    // Construct matrix operation object using the wrapper class DenseSymMatProd
    DenseSymMatProd<double> op(M);
    // Construct eigen solver object, requesting the largest three eigenvalues
    SymEigsSolver< double, LARGEST_ALGE, DenseSymMatProd<double> > eigs(&op, 3, 6);
    // Initialize and compute
    eigs.init();
    int nconv = eigs.compute();
    // Retrieve results
    Eigen::VectorXd evalues;
    if(eigs.info() == SUCCESSFUL)
        evalues = eigs.eigenvalues();
    std::cout << "Eigenvalues found:\n" << evalues << std::endl;
    return 0;
}