We introduce a simple and computationally efficient method for fitting the admixture model of genetic population structure, called ALStructure. The strategy of ALStructure is to first estimate the low-dimensional linear subspace of the population admixture components and then search for a model within this subspace that is consistent with the admixture model's natural probabilistic constraints. Central to this strategy is the observation that all models belonging to this constrained space of solutions are risk-minimizing and have equal likelihood, rendering any additional optimization unnecessary. The low-dimensional linear subspace is estimated through a recently introduced principal components analysis method that is appropriate for genotype data, thereby providing a solution that has both principal components and probabilistic admixture interpretations. Our approach differs fundamentally from other existing methods for estimating admixture, which aim to fit the admixture model directly by searching for parameters that maximize the likelihood function or the posterior probability. We observe that ALStructure typically outperforms existing methods both in accuracy and computational speed under a wide array of simulated and real human genotype datasets. Throughout this work we emphasize that the admixture model is a special case of a much broader class of models for which algorithms similar to ALStructure may be successfully employed.