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Topics in learning sparse and low-rank models of non-negative data

Advances in information and measurement technology have led to a surge in prevalence of high-dimensional data. Sparse and low-rank modeling can both be seen as techniques of dimensionality reduction, which is essential for obtaining compact and interpretable representations of such data. In this thesis, we investigate aspects of sparse and low-rank modeling in conjunction with non-negative data or non-negativity constraints. The first part is devoted to the problem of learning sparse non-negative representations, with a focus on how non-negativity can be taken advantage of. We work out a detailed analysis of non-negative least squares regression, showing that under certain conditions sparsity-promoting regularization, the approach advocated paradigmatically over the past years, is not required. Our results have implications for problems in signal processing such as compressed sensing and spike train deconvolution. In the second part, we consider the problem of factorizing a given matrix into two factors of low rank, out of which one is binary. We devise a provably correct algorithm computing such factorization whose running time is exponential only in the rank of the factorization, but linear in the dimensions of the input matrix. Our approach is extended to noisy settings and applied to an unmixing problem in DNA methylation array analysis. On the theoretical side, we relate the uniqueness of the factorization to Littlewood-Offord theory in combinatorics.