Era of Big Data Processing: A New Approach via Tensor Networks and Tensor Decompositions
Era of Big Data Processing: A New Approach via Tensor Networks and Tensor Decompositions
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发表时间：
2014-03
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通讯作者：
A. Cichocki
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作者：
A. Cichocki
计算神经科学、神经信息学、模式/图像识别、信号处理和机器学习中的许多问题产生了大量多方面、高维的多维数据。张量（即，多路数组）通常通过适当的低秩近似为这种大量多维数据提供自然和紧凑的表示。大数据分析需要新的技术来在可容忍的时间内有效地处理庞大的数据集。这种用于多维大数据的新兴技术是通过张量网络（TN）和张量分解（TD）的多路分析，张量网络和张量分解通过因子（分量）矩阵和低阶（核心）张量的集合来表示张量。动态张量分析允许我们发现复杂数据的有意义的隐藏结构，并通过捕获多线性和多方面的关系来执行泛化。我们将讨论一些基本的TN模型，它们的数学和图形描述以及用于大规模TD和TN的相关学习算法，具有许多潜在的应用，包括：异常检测，特征提取，分类，聚类分析，数据融合和集成，模式识别，预测建模，回归，时间序列分析和多路分量分析。保留字：大规模HOSVD，张量分解，CPD，Tucker模型，分层Tucker（HT）分解，低秩张量近似（LRA），张量化/量化，张量训练（TT/QTT）-矩阵乘积状态（MPS），矩阵乘积算子（MPO），DMRG，强克罗内克乘积（SKP）。
Many problems in computational neuroscience, neuroinformatics, pattern/image recognition, signal processing and machine learning generate massive amounts of multidimensional data with multiple aspects and high dimensionality. Tensors (i.e., multi-way arrays) provide often a natural and compact representation for such massive multidimensional data via suitable low-rank approximations. Big data analytics require novel technologies to efficiently process huge datasets within tolerable elapsed times. Such a new emerging technology for multidimensional big data is a multiway analysis via tensor networks (TNs) and tensor decompositions (TDs) which represent tensors by sets of factor (component) matrices and lower-order (core) tensors. Dynamic tensor analysis allows us to discover meaningful hidden structures of complex data and to perform generalizations by capturing multi-linear and multi-aspect relationships. We will discuss some fundamental TN models, their mathematical and graphical descriptions and associated learning algorithms for large-scale TDs and TNs, with many potential applications including: Anomaly detection, feature extraction, classification, cluster analysis, data fusion and integration, pattern recognition, predictive modeling, regression, time series analysis and multiway component analysis. Keywords: Large-scale HOSVD, Tensor decompositions, CPD, Tucker models, Hierarchical Tucker (HT) decomposition, low-rank tensor approximations (LRA), Tensorization/Quantization, tensor train (TT/QTT) - Matrix Product States (MPS), Matrix Product Operator (MPO), DMRG, Strong Kronecker Product (SKP).