Sparse regression-based unmixing has received much attention in recent years; however, its theoretical performance has not been explored in the literature. In this work, we present theoretical guarantees for the performance of a sparse regression based unmixing (in short, sparse unmixing) implemented in the form of a Lasso optimization with non-negativity constraints. We provide a sufficient condition required for the exact recovery of the endmembers and validate it both theoretically and through experiments. In cases in which the condition is not verified, we explore the performance of sparse unmixing in relation to the exact recovery coefficient (ERC).