Provable Sparse Tensor Decomposition

We propose a novel sparse tensor decomposition algorithm that incorporates sparsity into the estimation of decomposition components. Our method encourages the sparsity structure via a new truncation procedure. A thorough theoretical analysis of the proposed method is provided. In particular, we show that the final decomposition estimator is guaranteed to achieve a local convergence rate. We further strengthen this result to the global convergence by showing a newly introduced initialization procedure is appropriate. In high dimensional regimes, the obtained rate of convergence significantly improves those shown in existing non-sparse decomposition algorithms. Moreover, our method is general to solve a broad family of high dimensional latent variable models, including high dimensional gaussian mixture model and mixtures of sparse regressions. The theoretical superiority of our procedure is backed up by extensive numerical results.