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In this section, we numerically investigate the ability of compressed sensing to recover sparse matrices, before applying the technique to molecular vibrations. The main goal is to elucidate how the extent of undersampling needed for accurate recovery of the entire matrix scales with the sparsity of that matrix. Our approach to sparse matrix recovery is detailed in Fig.~\ref{fig:NumericsScheme}, and each trial works as follows:
\begin{enumerate}
\item We generate an \(N \times N\) matrix \(A\) with \(S\) non-zero entries in random positions. The non-zero entries are drawn uniformly at random on the interval \([-1,1]\).
\item We apply a change-of-basis to \(A\) using the discrete cosine transform matrix \(P\),
\begin{align}
\label{eq:dct}
P_{ij} = \sqrt{\dfrac{2}{N}} \cos \left[ \frac{\pi}{N} \left(i-1\right) \left(j - \frac{1}{2}\right) \right],
\end{align}
(with the first row multiplied by an extra factor of \(1/\sqrt{2}\) to guarantee orthogonality)
in order to generate the matrix \(B = PAP^{\textrm{T}}\).
\item We randomly sample \(M\) entries from \(B\) (\(M < N^2\)), and we use these entries to reconstruct the full sparse matrix \(A\) by solving the compressed sensing basis pursuit problem in eq.~\ref{eq:bpdn}.
\item For each trial, we record the relative 2-norm error
\begin{align}
\dfrac{||A_\textrm{recovered}~-~A_\textrm{original}||_2}{||A_\textrm{original}||_2}
\end{align}
(where the norms denote \emph{vector} norms).
\end{enumerate}
As discussed in the previous section, the discrete cosine transform (DCT) matrix \(P\) ensures that the entries of \(B\) are expressed in an incoherent basis with respect to the entries of \(A\) (roughly speaking, that each entry of \(B\) contains some information about each entry of \(A\)). The coherence associated with the DCT matrix \(P\) can be shown to be \(\mu = \sqrt{2}\), which is near the lower end of the 1 to \(\sqrt{N}\) range, as desired for incoherent sampling.
