Low rank tensor representation (LRTR) methods are very useful for hyperspectral anomaly detection (HAD). To overcome the limitations that they often overlook spectral anomaly and rely on large-scale matrix singular value decomposition, we first apply non-negative matrix factorization (NMF) to alleviate spectral dimensionality redundancy and extract spectral anomaly and then employ LRTR to extract spatial anomaly while mitigating spatial redundancy, yielding a highly efffcient layered tensor decomposition (LTD) framework for HAD. An iterative algorithm based on proximal alternating minimization is developed to solve the proposed LTD model, with convergence guarantees provided. Moreover, we introduce a rank reduction strategy with validation mechanism that adaptively reduces data size while preventing excessive reduction. Theoretically, we rigorously establish the equivalence between the tensor tubal rank and tensor group sparsity regularization (TGSR) and, under mild conditions, demonstrate that the relaxed formulation of TGSR shares the same global minimizers and optimal values as its original counterpart. Experimental results on the Airport-Beach-Urban and MVTec datasets demonstrate that our approach outperforms state-of-the-art methods in the HAD task.

翻译：低秩张量表示（LRTR）方法在高光谱异常检测（HAD）中具有重要应用价值。为克服现有方法常忽略光谱异常且依赖大规模矩阵奇异值分解的局限，本研究首先应用非负矩阵分解（NMF）以缓解光谱维度冗余并提取光谱异常，继而采用LRTR提取空间异常同时抑制空间冗余，从而构建了一个高效的分层张量分解（LTD）框架用于HAD。我们开发了基于邻近交替最小化的迭代算法来求解所提出的LTD模型，并提供了收敛性保证。此外，我们引入了一种带有验证机制的秩约简策略，能够自适应降低数据规模并防止过度约简。在理论层面，我们严格证明了张量管秩与张量群稀疏正则化（TGSR）的等价性，并在温和条件下论证了TGSR松弛形式与其原始形式具有相同的全局最小点和最优值。在Airport-Beach-Urban和MVTec数据集上的实验结果表明，本方法在HAD任务中优于当前最先进的技术。