低秩张量网络用于降维和大规模优化问题：视角与挑战 第1部分
Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges PART 1
AI总结：
本文探讨低秩张量网络在处理大规模数据和优化问题中的应用，重点介绍张量分解方法及其扩展，分析其在降维和数据挖掘中的核心贡献与挑战。
摘要：
机器学习和数据挖掘算法在分析大规模、多关系和多模态数据集时日益重要，这些数据集通常可方便地表示为多维数组或张量。因此，多学科研究社区及时回顾张量分解和张量网络作为大规模数据分析和数据挖掘新兴工具具有重要价值。本文提供张量网络的数学和图形表示及其解释，主要聚焦于Tucker和张量列车（TT）分解及其扩展或泛化。关键词：张量网络，与函数相关的张量，CP分解，Tucker模型，张量列车（TT）分解，矩阵乘积态（MPS），矩阵乘积算子（MPO），基本张量运算，多维成分分析，多线性盲源分离，张量补全，线性/多线性降维，大规模优化问题，对称特征值分解（EVD），PCA/SVD，大规模线性方程组，大矩阵的伪逆，Lasso和典型相关分析（CCA）（第一部分）
英文摘要：
Machine learning and data mining algorithms are becoming increasingly important in analyzing large volume, multi-relational and multi--modal datasets, which are often conveniently represented as multiway arrays or tensors. It is therefore timely and valuable for the multidisciplinary research community to review tensor decompositions and tensor networks as emerging tools for large-scale data analysis and data mining. We provide the mathematical and graphical representations and interpretation of tensor networks, with the main focus on the Tucker and Tensor Train (TT) decompositions and their extensions or generalizations. Keywords: Tensor networks, Function-related tensors, CP decomposition, Tucker models, tensor train (TT) decompositions, matrix product states (MPS), matrix product operators (MPO), basic tensor operations, multiway component analysis, multilinear blind source separation, tensor completion, linear/multilinear dimensionality reduction, large-scale optimization problems, symmetric eigenvalue decomposition (EVD), PCA/SVD, huge systems of linear equations, pseudo-inverse of very large matrices, Lasso and Canonical Correlation Analysis (CCA) (This is Part 1)
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
