We introduce a novel algorithm to compute nonnegative sparse principal components of positive semidefinite (PSD) matrices. Our algorithm comes with approximation guarantees contingent on the spectral profile of the input matrix A: the sharper the eigenvalue decay, the better the quality of the approximation. If the eigenvalues decay like any asymptotically vanishing function, we can approximate nonnegative sparse PCA within any accuracy in time polynomial in the matrix dimension n and desired

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... arsity k, but not in 1 / . Further, we obtain a data dependent bound that is computed by executing an algorithm on a given data set. This bound is significantly tighter than a-priori bounds and can be used to show that for all tested datasets our algorithm is provably within 40% − 90% from the unknown optimum. Our algorithm is combinatorial and explores a subspace defined by the leading eigenvectors of A. We test our scheme on several data sets, showing that it matches or outperforms the previous state of the art.