Coseparable nonnegative tensor factorization with T-CUR decomposition

报告摘要：Nonnegative Matrix Factorization (NMF) is an important unsupervised learning method to extract meaningful features from data. To address the NMF problem within a polynomial time framework, researchers have introduced a separability assumption, which has recently evolved into the concept of coseparability. This advancement offers a more ecient core representation for the original data. However, in the real world, data is more naturally represented as a multi-dimensional array, such as images or videos. The NMF's application to high-dimensional data involves vectorization, which risks losing essential multi-dimensional correlations. To retain these inherent correlations, we turn to tensors (multi-dimensional arrays) and leverage the tensor t-product. This approach extends the coseparable NMF to the tensor setting, creating what we term coseparable Nonnegative Tensor Factorization (NTF). In this work, we provide an alternating index selection method to select the coseparable core. Furthermore, we validate the t-CUR sampling theory and integrate it with the tensor Discrete Empirical Interpolation Method (t-DEIM) to introduce an alternative, randomized index selection process. These methods have been tested on both synthetic and facial analysis datasets. The results demonstrate the eciency of coseparable NTF when compared to coseparable NMF.

报告人简介：魏益民，复旦大学教授、博士生导师。曾获得上海市高校优秀青年教师和上海市“曙光”学者的称号。长期从事矩阵计算的理论和应用研究工作，迄今已在《SIAM J. Matrix Anal. Appl.》《SIAM J. Numer. Anal.》《SIAM J. Sci. Comput.》《J. Sci. Comput.》等国际重要学术期刊发表论文一百余篇，并在科学出版社出版中英文专著《Generalized Inverses: Theory and Calculations》(2004)、《Effective Condition Number for Numerical Partial Differential Equations》(2013)、《广义逆的符号模式》(2014)三部和英文教材《Numerical Linear Algebra and Its Applications》(2004)一部。曾多次主持国家自然科学基金面上项目、教育部博士点基金项目和973子课题等项目。目前正主持国家自然科学基金项目，担任国际学术期刊《Comput. Appl. Math.》和《J. Appl. Math. Comput.》和《Communications in Mathematical Research》、《高校计算数学学报》的编委。