Author : Rungang Han
Publisher :
ISBN 13 :
Total Pages : 322 pages
Book Rating : 4.:/5 (128 download)
Book Synopsis Statistical Learning for High-order Tensors by : Rungang Han
Download or read book Statistical Learning for High-order Tensors written by Rungang Han and published by . This book was released on 2021 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: Statistical learning for high-dimensional high-order tensor data has attracted increasing interests in recent years. The challenges arise as many classic statistical methods suffer from either statistical sub-optimality or computational limitation due to the fundamental complicated tensor algebra. In this thesis, we introduce three statistical tensor inference frameworks under different low-dimensional structures driven by real data applications: low-rankness, clustering, and smoothness. In Chapter 2, we describe a flexible framework for generalized low-rank tensor estimation problems that includes many important instances arising from applications in computational imaging, genomics, and network analysis. We propose an estimator which consists of finding a low-rank tensor fit to the data under generalized parametric models. To overcome the difficulty of non-convexity in these problems, we introduce a unified approach of projected gradient descent that adapts to the underlying low-rank structure. Under mild conditions on the loss function, we establish both an upper bound on statistical error and the linear rate of computational convergence through a general deterministic analysis. These algorithmic and theoretical results are then applied to a suite of generalized tensor estimation problems, including sub-Gaussian tensor PCA, tensor regression, and Poisson and binomial tensor PCA. We prove that the proposed algorithm achieves the minimax optimal rate of convergence in estimation error. Finally, we demonstrate the superiority of the proposed framework via extensive experiments on both simulated and real data. The main target of Chapter 3 is to propose a statistical tensor model for high-order clustering, which aims to identify heterogeneous substructure in multiway dataset that arises commonly in multilayer network studies. In addition to the non-convexity that widely appears in statistical tensor problem, this model is more complicated because of its discontinuous nature. In that chapter, we propose a tensor block model and the computationally efficient methods, high-order Lloyd algorithm (HLloyd) and high-order spectral clustering (HSC), for the high-order clustering task. We similarly establish the convergence of the proposed procedure, and we show that our method achieves exact clustering under some reasonable assumptions. We also give the complete characterization for the statistical-computational trade-off in high-order clustering based on three different signal-to-noise ratio regimes. Finally, we show the merits of the proposed procedures via extensive experiments on both synthetic and real datasets. Finally, Chapter 4 introduces the functional tensor singular value decomposition model (FTSVD), a novel dimension reduction framework for tensors with one functional mode and several tabular modes. This problem is motivated by high-order longitudinal data analysis. Our model assumes the observed data to be a random realization of an approximate CP low-rank functional tensor measured on a discrete time grid. Incorporating tensor algebra and the theory of Reproducing Kernel Hilbert Space (RKHS), we propose a novel RKHS-based constrained power iteration with spectral initialization. Our method can successfully estimate both singular vectors and functions of the low-rank structure in the observed data. With mild assumptions, we establish the non-asymptotic contractive error bounds for the proposed algorithm. We also perform extensive experiments on both simulated and real data to illustrate the superiority of the proposed framework.
