symmetry S S Article Incremental Nonnegative Tucker Decomposition with Block-Coordinate Descent and Recursive Approaches Rafal Zdunek * and Krzysztof Fonal   Citation: Zdunek, R.; Fonal, K. Incremental Nonnegative Tucker Decomposition with Block- Coordinate Descent and Recursive Approaches. Symmetry 2022, 14, 113. https://doi.org/10.3390/ sym14010113 Academic Editor: Dumitru Baleanu Received: 4 December 2021 Accepted: 5 January 2022 Published: 9 January 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Faculty of Electronics,Photonics, and Microsystems, Wroclaw University of Science and Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland; krzysiekfonal@gmail.com * Correspondence: rafal.zdunek@pwr.edu.pl; Tel.: +48-71-320-3215 Abstract: Nonnegative Tucker decomposition (NTD) is a robust method used for nonnegative multilinear feature extraction from nonnegative multi-way arrays. The standard version of NTD assumes that all of the observed data are accessible for batch processing. However, the data in many real-world applications are not static or are represented by a large number of multi-way samples that cannot be processing in one batch. To tackle this problem, a dynamic approach to NTD can be explored. In this study, we extend the standard model of NTD to an incremental or online version, assuming volatility of observed multi-way data along one mode. We propose two computational approaches for updating the factors in the incremental model: one is based on the recursive update model, and the other uses the concept of the block Kaczmarz method that belongs to coordinate descent methods. The experimental results performed on various datasets and streaming data demonstrate high efficiently of both algorithmic approaches, with respect to the baseline NTD methods. Keywords: nonnegative tucker decomposition; incremental algorithm; recursive update; block Kaczmarz method; signal processing 1. Introduction Tensor decompositions are robust tools for multi-linear feature extraction and multi- modal dimensionality reduction [1,2]. There are many models of tensor decompositions. Examples include CANDECOMP/PARAFAC (CP) [3,4], nonnegative CP [5–7], Tucker decomposition [8,9], sparse Tucker decomposition [10,11], higher-order singular value decomposition (HOSVD) [12], higher-order orthogonal iterations (HOOI) [13], hierarchical Tucker (HT) decomposition [14], tensor train (TT) [15], smooth tensor tree [16,17], tensor ring [18], compact tensor ring [19], etc. The Tucker decomposition model assumes that an input multi-way array, which will be referred to as a tensor, is decomposed into a core tensor and a set of lower-rank matrices, capturing the multilinear features associated with all the modes of the input tensor. The baseline model was proposed by L. R. Tucker [9] in 1966 as a multi-linear extension to principal component analysis (PCA). Currently, many versions of this model are available across multiple applications in various areas of science and technology, among others, facial image representation [20–24], hand-written digit recognition [25], data clustering and segmentation [26–29], communication [28,30], hyperspectral image compression [31], muscle activity analysis [32]. A survey of its applications can be found in [10,33,34]. Nonnegative Tucker decomposition (NTD) is a particular case of the Tucker decomposition in which the nonnegativity constraints are imposed onto the core tensor and all the factor matrices. Due to such constraints, the multi-way features are parts-based representations, easier for interpretation and may have a physical meaning if an input multi-way array contains only nonnegative entries. NTD has been used in multiple applications, including image classification [35–38], clustering [39], hyperspectral image denoising and compression [40, 41], audio pattern extraction [42], image fusion [43], EEG signal analysis [44,45], etc. Symmetry 2022, 14, 113. https://doi.org/10.3390/sym14010113 https://www.mdpi.com/journal/symmetry