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The authors present a new matrix compression algorithm to improve the efficiency of the fast multipole method (FMM). The method is based on the application of the singular value decomposition (SVD) to the plane wave FMM aggregation matrices. These matrices are low-ranked, which is exploited to provide alternative sets of orthonormal singular basis functions, obtained as linear combinations of the original basis. By choosing only the most relevant singular functions, a much more compact representation is obtained to accurately handle the interactions between the FMM groups. The new formulation provides a reduction close to one order of magnitude both in computational cost and memory requirements, with a moderate impact on the accuracy of the solution.