Non-negative matrix factorization (NMF) has been used in a variety of real-world applications such as blind signal processing, spectral recovery, pattern recognition, text mining and bioinformatics. NMF is to recover hidden structures or bases from the non-negative input data and compute the linear projection of the data onto the bases keeping with non-negativity property. By adding orthogonal constraints into the factorization, orthogonal non-negative matrix factorization (ONMF) is developed, and ONMF is more challenging and useful because of its uniqueness and rigorous pattern interpretation. However, it is challenging to design efficient solution methods for solving the solution of ONMF especially for large-scale data matrices. Existing techniques include multiplicative update algorithms, gradient calculation in Stiefel manifold, and augmented Lagrangian approach by enforcing orthogonality constraints. The main aim of this project to propose a new formulation of ONMF and design new efficient algorithms for ONMF. Our idea is to analyze the structure of orthogonal constraints, we study similarity matrices in the new formulation and design efficient and structured optimization algorithms to compute ONMF. We will analyze solutions of the new formulation and their error bounds by using matrix theory and probability in perturbation analysis of eigenvectors. Multi-dimensional data is becoming prevalent in many data science applications. The proposed approach and eigenvector analysis will be further investigated into orthogonal non-negative tensor factorization (ONTF).
In the project, we plan to apply the proposed ONMF and ONTF models to (i) supervised learning, transfer learning and hyperspectral unmixing frameworks. In supervised learning, prior information can be incorporated in the proposed models and they can used for classification purpose. In transfer learning, knowledge from multiple sources can be transferred to a target domain via the latent features learnt by ONTF, and this new framework can enhance learning capability. (iii) Based on our ONTF model, we can classify pixels in hyperspectral images according to factorized latent features. We will test the numerical performance of the proposed models and algorithms for these applications. For these specific applications, learning error analysis can be studied and derived.
This project is supported by the Research Grants Council (RGC), Hong Kong SAR, China (Project 12300218).
For further information on this research topic, please contact Prof. Michael NG.