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Using additional information is an essential concept in compressed sensing and it is intimately related to the concepts of sparsity and compressibility, but it is also a concept that is routinely used in numerical simulations. For example, in quantum chemistry it is customary to represent the orbitals of a molecule in a basis formed by the orbitals of the atoms in the molecule~\cite{Szabo1996}, which allows for an efficient and compact representation and a controlled discretization error. This choice comes from the notion that the electronic structure of the molecule is roughly described by ``patching together'' the electronic structure of the constituent atoms.  An ideal basis in which to reconstruct a matrix is the basis of its eigenvectors, or \textit{eigenbasis}, as this basis  only requires the evaluation of the diagonal elements to obtain the whole matrix. Of course, finding the eigenbasis requires knowing the matrix in the first place, so reconstructing a matrix in its eigenbasis is not useful for practical purposes. However, in many cases it is possible to obtain a set of reasonable approximations to the eigenvectors (an idea which also forms the basis of perturbation theory in quantum mechanics). The approximate eigenbasis probably constitutes a good sparsifying basis for many problems, as we expect the matrix to be diagonally dominant, with a large fraction of the off-diagonal elements equal  tobe  zero or at least quite small. Since the determination of an approximate eigenbasis depends on the specific problem, it is difficult to give a general prescription for how to do it. There are some general ideas that could work in some cases, for example in iterative or propagative simulations, results from previous iterations or steps could be used as a guess for the next one. Alternatively low-accuracy low-cost methods can be used to guess an approximate eigenbasis. In this case, our the  procedure we propose  provides a framework for bootstraping the results of a low-cost calculation in order to reduce the required number of costly high-accuracy calculations. This last strategy is the one we apply to the case of molecular vibrations. What makes looking for sparsifying basis attractive, even at some computational cost and code-complexity overhead, are the properties of the recovery method (detailed in the next section). method.  First, the cost of recovering the matrix is roughly proportional to its sparsity. Second, the reconstruction of the matrix is always exact up to a desired precision; even if the sparsifying basis is not a good one, we eventually obtain converge to  the correct result. The penalty for a bad sparsifying basis is additional computations, computation,  which in the worst case make makes  the calculation as costly as if compressed sensing were not used at all. This feature implies that the method will almost certainly offer some performance gain. There is an important consideration to make. For some matrices there is a preferred basis where the matrix is cheaper to compute, to compute it in a different basis might offset the reduction in cost offered by compressed sensing.