The aim of this paper is to present a mathematical framework for tensor PCA.
The proposed approach is able to overcome the limitations of previous methods
that extract a low dimensional subspace by iteratively solving an optimization
problem. The core of the proposed approach is the derivation of a basis in
tensor space from a real self-adjoint tensor operator, thus reducing the
problem of deriving a basis to an eigenvalue problem. Three different cases
have been studied to derive: i) a basis from a self-adjoint tensor operator;
ii) a rank-1 basis; iii) a basis in a subspace. In particular, the equivalence
between eigenvalue equation for a real self-adjoint tensor operator and
standard matrix eigenvalue equation has been proven. For all the three cases
considered, a subspace approach has been adopted to derive a tensor PCA.
Experiments on image datasets validate the proposed mathematical framework.

针对张量 PCA 问题，本文提出了一种数学框架，通过从实自伴张量算子中导出张量空间的基础来减少问题的导出；研究了三种不同的情况来导出：从自伴张量算子中的基础，一个秩 - 1 的基础，以及一个子空间中的基础，并采用子空间方法导出张量 PCA。实验验证了该数学框架的有效性。

张量空间中的基张量主成分分析

Tensor PCA from basis in tensor space

This paper describes 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. The
proposed estimator 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. Then we further consider 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.

该论文提出了一种灵活的通用低秩张量估计问题的框架，包括计算成像、基因组学和网络分析等应用中的许多重要实例，并通过投影梯度下降的统一方法来克服这些问题的非凸性，以适应底层低秩结构，并证明该算法在估计误差的收敛速度上达到极小值最优率。

广义张量估计的最优统计和计算框架

An Optimal Statistical and Computational Framework for Generalized  Tensor Estimation

These notes survey and explore an emerging method, which we call the
low-degree method, for predicting and understanding
statistical-versus-computational tradeoffs in high-dimensional inference
problems. In short, the method posits that a certain quantity -- the second
moment of the low-degree likelihood ratio -- gives insight into how much
computational time is required to solve a given hypothesis testing problem,
which can in turn be used to predict the computational hardness of a variety of
statistical inference tasks. While this method originated in the study of the
sum-of-squares (SoS) hierarchy of convex programs, we present a self-contained
introduction that does not require knowledge of SoS. In addition to showing how
to carry out predictions using the method, we include a discussion
investigating both rigorous and conjectural consequences of these predictions.
These notes include some new results, simplified proofs, and refined
conjectures. For instance, we point out a formal connection between spectral
methods and the low-degree likelihood ratio, and we give a sharp low-degree
lower bound against subexponential-time algorithms for tensor PCA.

本文介绍了一种新的方法，称为低次方法，用于预测和理解高维推断问题中的统计与计算权衡，并给出了一些新的结果、简化的证明和改进的猜想。该方法假定一个特定数量 —— 低次似然比的第二时刻 —— 可以揭示解决给定假设检验问题所需的计算时间的信息，进而可用于预测各种统计推断任务的计算复杂度。