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\section{Numerical foundation: compressed sensing for matrices}  In this section we develop a method to calculate the coefficients of sparse matrices without knowing \textit{a priori} where those elements are  in the matrix. This method results from the application of the  compressed sensing approach to matrices. Related work has been  presented in the field of compressive principal component  pursuit~\cite{Baraniuk2011,Candes2011,Zhou2010,Wright2013}, which  focuses on reconstructing matrices that are the sum of a low-rank  component and a sparse component. Our work instead outlines a general  procedure for reconstructing any sparse matrix by measuring it in a  different basis.  Suppose we wish to recover an \(N \times N\) matrix \(A\) known to be sparse in a particular orthonormal basis $\{\psi_i\}$(for simplicity we restrict ourselves to square matrices an orthonormal bases). Without no prior knowledge of where the \(S\) non-zero  entries of \(A\) are located, to recover the matrix we need to calculate all the elements of of the matrix.  In a second basis \(\{\phi_i\}\), the matrix \(A\) has a second representation \(B\) given by   \begin{equation}  \label{eq:cob_matrix}  B = PAP^{T}\ ,  \end{equation}  where \(P\) is the orthogonal matrix that transforms any vector from  the basis $\{\psi_i\}$ to $\{\phi_i\}$. Note that in general, \(B\) is not a sparse matrix.  If we regard \(A\) and \(B\) as \(N^2\)-element \emph{vectors}  formed by stacking the columns of the matrices, we can note that the change-of-basis  transformation between \(A\) and \(B\) given by eq.~(\ref{eq:cob_matrix}) is \emph{linear}. This fact enables us to use of the machinery of compressed sensing for the problem of determining \(A\).  Given the matrix \(P\) or equivalently the basis \(\{\phi_i\}\), compressed sensing allows to reconstruct the full matrix \(A\) sampling \texit{some} of the entries of \(B\). The reconstruction is done by solving the basis pursuit problem (BP),  \begin{equation}  \label{eq:bpdn}  \min_{A} ||A||_1 \quad \textrm{subject to} \quad (PAP^T)_{ij} = B_{ij}\quad \forall  \ i,j \in W \ ,  \end{equation}  where the 1-norm is considered as a \emph{vector} norm  (\(||A||_1 = \sum_{i,j} \left|A_{ij}\right|\)), and \(W\) is a set of randomly chosen matrix entries.   The size of \(W\), a number that we call \(M\), is the number of matrix elements of \(B\) that we need to sample and determines the quality of the reconstruction. From compressed sensing theory we can find a lower bound for \(M\), as a function of \(P\), \(S\) and \(N\).  One important requirement for compressed sensing is that the basis  $\{\psi_i\}$ for \(A\) and the basis $\{\phi_i\}$ for \(B\) should be  %  \begin{equation}  \label{eq:coherence}  \mu = \sqrt{N} \max_{i,j} \langle \psi_i | \phi_j \rangle  \end{equation}  %  should be as \emph{small} as possible (in general \(\mu\) ranges from  1 to \(\sqrt{N}\)). Intuitively speaking, this incoherence condition  simply means that the change-of-basis matrix \(P\) should thoroughly  scramble the entries of \(A\) to generate \(B\).   It can be proven~\cite{Candes2006,Donoho2006,Candes2008} that the  number of entries of \(B\) which must be measured in order to fully  recover \(A\) by solving the BP problem in eq.~\ref{eq:bpdn} scales  as  \begin{align}  \label{eq:csscaling}  M^* \propto \mu^2 S \log N^2\ .  \end{align}  This scaling equation encapsulates the important aspects of compressed  sensing applied to sparse matrices: if a proper basis is chosen, the  number of entries which must be measured scales linearly with the  the matrix.