We investigate a generalized framework to estimate a latent low-rank plus sparse tensor, where the low-rank tensor often captures the multi-way principal components and the sparse tensor accounts for potential model mis-specifications or heterogeneous signals that are unexplainable by the low-rank part. The framework is flexible covering both linear and non-linear models, and can easily handle continuous or categorical variables. We propose a fast algorithm by integrating the Riemannian gradient descent and a novel gradient pruning procedure. Under suitable conditions, the algorithm converges linearly and can simultaneously estimate both the low-rank and sparse tensors. The statistical error bounds of final estimates are established in terms of the gradient of loss function. The error bounds are generally sharp under specific statistical models, e.g., the robust tensor PCA and the community detection in hypergraph networks with outlier vertices. Moreover, our method achieves non-trivial error bounds for heavy-tailed tensor PCA whenever the noise has a finite $2+\varepsilon$ moment. We apply our method to analyze the international trade flow dataset and the statistician hypergraph co-authorship network, both yielding new and interesting findings.
Kinetic Energy Plus Penalty Functions for Sparse Estimation
本文提出并研究了一类稀疏诱导罚函数。由于罚函数与狭义相对论中的动能有关，我们称之为\emph{kinetic energy plus}（KEP）函数。我们利用$χ^2$ -距离函数的凹共轭构造了KEP函数，并提出了利用$q=1$构造KEP函数的一些新的见解.特别地，我们推导了一个基于KEP函数的阈值算子，并证明了它的数学性质和在稀疏建模中的渐近性质。此外，我们表明，坐标下降算法是特别适合的KEP功能。此外，我们还讨论了KEP与罚函数$\ell_{1/2}$和MCP之间的关系。理论和实证分析验证了KEP函数在高维数据建模中的有效性和高效性。
Transfer Denoising of Low-Rank Plus Sparse Matrices under Ambient and Rank Growth
Department of Operations Research & Financial Engineering, Princeton University（普林斯顿大学运筹学与金融工程系）；Department of Technology, Operations, & Statistics, New York University（纽约大学技术、运营与统计系）；Department of Computer Science, Dartmouth College（达特茅斯学院计算机科学系）
Extrinsic Methods for Coding and Dictionary Learning on Grassmann Manifolds
College of Engineering and Computer Science, Australian National University（澳大利亚国立大学工程与计算机学院）；NICTA（澳大利亚国家信息与通信技术研究院）；School of Computer Science, The University of Adelaide（阿德莱德大学计算机科学学院）；The University of Queensland（昆士兰大学）；University of Queensland（昆士兰大学）
