Decompositions of tensors into factor matrices, which interact through a core tensor, have found numerous applications in signal processing and machine learning. A more general tensor model that represents data as an ordered network of subtensors of order-2 or order-3 has, so far, not been widely considered in these fields, although this so-called tensor network (TN) decomposition has been long studied in quantum physics and scientific computing. In this article, we present novel algorithms and applications of TN decompositions, with a particular focus on the tensor train (TT) decomposition and its variants. The novel algorithms developed for the TT decomposition update, in an alternating way, one or several core tensors at each iteration and exhibit enhanced mathematical tractability and scalability for large-scale data tensors. For rigor, the cases of the given ranks, given approximation error, and the given error bound are all considered. The proposed algorithms provide well-balanced TT-decompositions and are tested in the classic paradigms of blind source separation from a single mixture, denoising, and feature extraction, achieving superior performance over the widely used truncated algorithms for TT decomposition.

中文翻译：

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用于潜在变量分析的Tensor网络：用于Tensor列车逼近的新算法。

张量分解为因子矩阵（通过核心张量进行交互）已经在信号处理和机器学习中得到了广泛的应用。尽管对这种所谓的张量网络（TN）分解进行了长期的研究，但迄今为止，还没有在这些领域中广泛考虑将数据表示为2阶或3阶次张量的有序网络的一般张量模型。在量子物理学和科学计算领域。在本文中，我们介绍了TN分解的新颖算法和应用，特别着重于张量链（TT）分解及其变体。为TT分解开发的新颖算法在每次迭代时以一种交替的方式更新一个或几个核心张量，并且对于大型数据张量表现出增强的数学可处理性和可伸缩性。为了严谨 给定等级，给定近似误差和给定误差界限的情况都被考虑了。所提出的算法提供了均衡的TT分解，并在从单一混合物中分离盲源，去噪和特征提取的经典范例中进行了测试，其性能优于广泛使用的TT分解的截断算法。