Hybrid Kronecker Product Decomposition and Approximation
Hybrid Kronecker Product Decomposition and Approximation
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DOI：
10.1080/10618600.2022.2134873
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发表时间：
2019-12
影响因子：
2.4
通讯作者：
Chencheng Cai;Rong Chen;Han Xiao
中科院分区：
文献类型：
--
作者：
Chencheng Cai;Rong Chen;Han Xiao
摘要发现高维矩阵的潜在低维结构传统上是通过以一阶矩阵和的形式进行的低阶矩阵近似来完成的。在本文中，我们提出了一种新的方法。我们假设高维矩阵可以由具有潜在不同构型的矩阵的少量Kronecker积之和来近似，称为混合Kronecker外积近似(HKoPA)。与低阶矩阵近似相比，它提供了一种非常灵活的降维方法。当组件Kronecker产品的配置不同或未知时，在估计hKoPA时会出现挑战。在给定构型集的情况下，我们提出了一种估计过程，当构型未知时，我们提出了一种联合构型确定和分量估计过程。具体地说，当给定配置时，使用最小二乘回代算法。在结构未知的情况下，提出了一种迭代贪婪算法。仿真和真实图像实例都表明，所提出的算法具有良好的性能。还给出了一些可辨识性条件。混合Kronecker乘积近似在高维数据的低维表示中具有潜在的更广泛的应用。这篇文章的补充材料可以在网上找到。
Abstract Discovering underlying low dimensional structure of a high-dimensional matrix is traditionally done through low rank matrix approximations in the form of a sum of rank-one matrices. In this article, we propose a new approach. We assume a high-dimensional matrix can be approximated by a sum of a small number of Kronecker products of matrices with potentially different configurations, named as a hybrid Kronecker outer Product Approximation (hKoPA). It provides an extremely flexible way of dimension reduction compared to the low-rank matrix approximation. Challenges arise in estimating a hKoPA when the configurations of component Kronecker products are different or unknown. We propose an estimation procedure when the set of configurations are given, and a joint configuration determination and component estimation procedure when the configurations are unknown. Specifically, a least squares backfitting algorithm is used when the configurations are given. When the configurations are unknown, an iterative greedy algorithm is developed. Both simulation and real image examples show that the proposed algorithms have promising performances. Some identifiability conditions are also provided. The hybrid Kronecker product approximation may have potentially wider applications in low dimensional representation of high-dimensional data. Supplementary materials for this article are available online.