  We propose a novel sparse tensor decomposition method, namely Tensor
Truncated Power (TTP) method, that incorporates variable selection into the
estimation of decomposition components. The sparsity is achieved via an
efficient truncation step embedded in the tensor power iteration. Our method
applies to a broad family of high dimensional latent variable models, including
high dimensional Gaussian mixture and mixtures of sparse regressions. A
thorough theoretical investigation is further conducted. In particular, we show
that the final decomposition estimator is guaranteed to achieve a local
statistical rate, and further strengthen it to the global statistical rate by
introducing a proper initialization procedure. In high dimensional regimes, the
obtained statistical rate significantly improves those shown in the existing
non-sparse decomposition methods. The empirical advantages of TTP are confirmed
in extensive simulated results and two real applications of click-through rate
prediction and high-dimensional gene clustering.
