The so-called block-term decomposition (BTD) tensor model has been recently receiving increasing attention due to its enhanced ability of representing systems and signals that are composed of blocks of rank higher than one, a scenario encountered in numerous and diverse applications. Its uniqueness and approximation have thus been thoroughly studied. Nevertheless, the challenging problem of estimating the BTD model structure, namely the number of block terms and their individual ranks, has only recently started to attract significant attention. In this paper, a novel method of BTD model selection and computation is proposed, based on the idea of imposing column sparsity jointly on the factors and in a hierarchical manner and estimating the ranks as the numbers of factor columns of non-negligible magnitude. Following a block successive upper bound minimization (BSUM) approach for the proposed optimization problem is shown to result in an alternating hierarchical iteratively reweighted least squares (HIRLS) algorithm, which is fast converging and enjoys high computational efficiency, as it relies in its iterations on small-sized sub-problems with closed-form solutions. Simulation results for both synthetic examples and a hyper-spectral image denoising application are reported, which demonstrate the superiority of the proposed scheme over the state-of-the-art in terms of success rate in rank estimation as well as computation time and rate of convergence while attaining a comparable tensor approximation performance.

中文翻译：

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块项张量分解：模型选择和计算

所谓的块项分解（BTD）张量模型由于其增强的表示由以下各项组成的系统和信号的能力而得到了越来越多的关注： 块排名高于1，这是在众多不同应用程序中遇到的情况。因此，已经对其唯一性和近似性进行了深入研究。然而，估计BTD模型结构的挑战性问题，即块项的数量及其各自的等级，直到最近才开始引起人们的广泛关注。本文基于施加列稀疏性的思想，提出了一种新的BTD模型选择与计算方法。共同 在因素和 等级制方式，并以不可忽略的数量级的因子列数来估计等级。遵循针对所提出的优化问题的块连续上界最小化（BSUM）方法，结果显示出交替的分层迭代最小加权平方（HIRLS）算法，该算法收敛速度快且具有较高的计算效率，因为它依赖于迭代封闭式解决方案的小子问题。报告了合成示例和高光谱图像去噪应用的仿真结果，这些结果证明了该方案在秩估计的成功率以及计算时间和速率方面优于最新技术。收敛，同时获得可比的张量逼近性能。