The nonnegative constraint in NCP will naturally lead to sparse results. For another example, after decomposing EEM tensor, a component in the sample mode denotes the concentrations of a compound in all samples, which is sometimes also sparse. For example, the spectral components from EEG tensor decomposition are usually very sparse, representing the narrow-band frequencies of some brain activities. In many cases, the extracted components by NCP are not only nonnegative but also sparse. Nonnegative CANDECOMP/PARAFAC (NCP), as an important decomposition method, has been widely applied to processing multiway data, such as hyperspectral data, electroencephalograph (EEG) data, fluorescence excitation-emission matrix (EEM) data, neural data, and many other multiway tensor data. Nonnegative tensor decomposition is a powerful tool in signal processing and machine learning. The experimental results demonstrate that our proposed algorithms can efficiently impose sparsity on factor matrices, extract meaningful sparse components, and outperform state-of-the-art methods. We evaluated the proposed sparse NCP methods by experiments on synthetic, real-world, small-scale, and large-scale tensor data. In order to prove the effectiveness and efficiency of the sparse NCP with the proximal algorithm, we employed two optimization algorithms to solve the model, including inexact alternating nonnegative quadratic programming and inexact hierarchical alternating least squares. In addition, we proposed an inexact BCD scheme for sparse NCP, where each subproblem is updated multiple times to speed up the computation. Therefore, the new sparse NCP provides a full column rank condition and guarantees to converge to a stationary point. The subproblems in the new model are strongly convex in the block coordinate descent (BCD) framework. In this paper, we proposed a novel model of sparse NCP with the proximal algorithm. Thus, the sparse NCP is prone to rank deficiency, and the algorithms of sparse NCP may not converge. When high sparsity is imposed, the factor matrices will contain more zero components and will not be of full column rank. In this paper, we investigated the nonnegative CANDECOMP/PARAFAC (NCP) decomposition with the sparse regularization item using \(l_1\)-norm (sparse NCP). Nevertheless, the sparsity is only a side effect and cannot be explicitly controlled without additional regularization. Nonnegative tensor decomposition is a versatile tool for multiway data analysis, by which the extracted components are nonnegative and usually sparse.
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