Combinatorial QPs via a low-dimensional subspace sampling

Several large scale data processing problems can be formulated as quadratic programs subject to combinatorial constraints. We present a novel low-rank approximation framework for problems such as sparse PCA, nonnegative PCA, or finding the k-densest submatrix. Our framework comes with provable approximation guarantees, that dependent on the spectrum of the data-set matrix. Our algorithm operates by solving a number of QP instances, which are randomly sampled from a low-dimensional subspace of the input matrix.