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Jacob Sanders deleted file sectionCompressibili.tex
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\section{Compressibility and finding a description of the problem}
This ability of compressed sensing to recover a sparse matrix with a
number of measurements that scales just linearly with the non-zero elements of the matrix opens new possibilities for the calculations of matrices, even if a basis where the basis is sparse is not known.
The idea of finding a basis where the matrix is approximately diagonal is the basis of perturbation theory, for example. However in general this methods involve approximations with an error that depends on how good the eigenvalues are approximated.
By using compressed sensing, however, the approximated basis does not influence the error in the result. Only the computational cost.
begs the obvious question: for a given scientific application, how can
we find a basis in which the matrix we wish to calculate is sparse?
Of course, every diagonalizable matrix has a sparse basis, namely the
basis of its eigenvectors. The catch, of course, is that finding the
eigenvectors of a matrix is at least as hard as calculating the whole
matrix itself. This is where the compressed sensing approach we
presented in the last section can help.
In many scientific applications, notably computational chemistry, it
turns out to be possible to cheaply \emph{approximate} the
eigenvectors of a matrix, and thus obtain a basis in which the matrix
is sparse (and nearly diagonal). This is the approach we take to
estimating the vibrational modes and frequencies of molecules, where
it enables a reduction of about 75\% of the expensive quantum
mechanical computations.
