张量网络在大数据分析和大规模优化问题中的应用
Tensor Networks for Big Data Analytics and Large-Scale Optimization Problems
AI总结：
本文探讨了大规模张量网络的基本模型和算法，特别是通过新颖的数学和图形表示方法实现张量列车（TT）分解。重点介绍了张量化（从低阶数据生成高阶张量）和通过量化张量网络实现的数据超压缩，展示了张量网络在解决大数据优化问题中的应用。
摘要：
本文回顾了大规模张量网络的基本和新兴模型及相关算法，特别是通过新颖的数学和图形表示方法实现的张量列车（TT）分解。讨论了张量化（即从低阶原始数据生成非常高的阶张量）的概念以及通过量化张量网络（QTT）实现的数据超压缩。张量化和量化的目的在于通过低秩张量近似实现“超”压缩，从而获得有意义且紧凑的结构数据表示。本文的主要目的是展示如何通过张量化和使用相对较小的矩阵和张量进行所有操作，并通过迭代优化和近似张量收缩来解决广泛的大数据优化问题（这些问题远非传统数值方法能够处理）。关键词：张量网络，张量列车（TT）分解，矩阵积态（MPS），矩阵积算子（MPO），基本张量运算，张量化，分布式表示数据优化问题，用于非常大规模问题：广义特征值分解（GEVD），PCA/SVD，典型相关分析（CCA）
英文摘要：
In this paper we review basic and emerging models and associated algorithms for large-scale tensor networks, especially Tensor Train (TT) decompositions using novel mathematical and graphical representations. We discus the concept of tensorization (i.e., creating very high-order tensors from lower-order original data) and super compression of data achieved via quantized tensor train (QTT) networks. The purpose of a tensorization and quantization is to achieve, via low-rank tensor approximations "super" compression, and meaningful, compact representation of structured data. The main objective of this paper is to show how tensor networks can be used to solve a wide class of big data optimization problems (that are far from tractable by classical numerical methods) by applying tensorization and performing all operations using relatively small size matrices and tensors and applying iteratively optimized and approximative tensor contractions. Keywords: Tensor networks, tensor train (TT) decompositions, matrix product states (MPS), matrix product operators (MPO), basic tensor operations, tensorization, distributed representation od data optimization problems for very large-scale problems: generalized eigenvalue decomposition (GEVD), PCA/SVD, canonical correlation analysis (CCA).
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。
