We present an algorithm for the unsupervised learning of latent variable models based on the method of moments. We give efficient estimates of the moments for two models that are well known, e.g., in text mining, the single-topic model and latent Dirichlet allocation, and we provide a tensor decomposition algorithm for the moments that proves to be robust both in theory and in practice. Experiments on synthetic data show that the proposed estimators outperform the existing ones in terms of reconstruction accuracy, and that the proposed tensor decomposition technique achieves the learning accuracy of the state-of-the-art method with significantly smaller running times. We also provide examples of applications to real-world text corpora for both single-topic model and LDA, obtaining meaningful results