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.

本研究提出了一种新的稀疏张量分解方法 - Tensor Truncated Power（TTP）方法，它将变量选择纳入分解组件估计中，通过张量幂迭代中嵌入的有效截断步骤来实现稀疏性。我们证明该方法可以应用于高维潜变量模型，包括高维高斯混合和稀疏回归混合。经过深入的理论研究，我们发现最终的分解估计器有助于实现局部统计速率，并通过引入适当的初始化过程进一步加强全局统计速率。在高维度情况下，所获得的统计速率显着优于现有的非稀疏分解方法。TTP 的实证优势在广泛的仿真结果和 Click-through 率预测和高维基因聚类的两个真实应用中得到了证实。

可证明的稀疏张量分解

Provable Sparse Tensor Decomposition

We provide a general theory of the expectation-maximization (EM) algorithm
for inferring high dimensional latent variable models. In particular, we make
two contributions: (i) For parameter estimation, we propose a novel high
dimensional EM algorithm which naturally incorporates sparsity structure into
parameter estimation. With an appropriate initialization, this algorithm
converges at a geometric rate and attains an estimator with the (near-)optimal
statistical rate of convergence. (ii) Based on the obtained estimator, we
propose new inferential procedures for testing hypotheses and constructing
confidence intervals for low dimensional components of high dimensional
parameters. For a broad family of statistical models, our framework establishes
the first computationally feasible approach for optimal estimation and
asymptotic inference in high dimensions. Our theory is supported by thorough
numerical results.

本文提供了一个关于期望最大化 (EM) 算法应用于高维潜变量模型推断的一般理论。作者提出了一个新的高维 EM 算法，自然地融入了稀疏结构到参数估计中。基于估计值，作者提出了新的推论程序来测试假设并构建置信区间。这个算法针对广泛的统计模型，提供了高维最先进的估计和渐近推断的第一个可计算的方法。