Abstract
Nonnegative matrix factorization (NMF) is a widely used method for extracting interpretable latent features from high-dimensional data in signal processing. Despite its advantages, NMF is inherently NP-hard and ill-posed, and its solutions are often sensitive to the choice of rank. As a result, conventional NMF algorithms frequently converge to suboptimal solutions and produce unstable components that are either mixed or fragmented, thereby reducing interpretability. This thesis develops a unified framework, termed NMFMerge, to improve the robustness, adaptability, and scalability of NMF by deliberately expanding the rank space, followed by an analytically solvable pairwise component merge strategy. We theoretically demonstrate the possibility of escaping poor local optima through this rank expansion and recombination process. Experimental results indicate that this strategy consistently improves solution stability across different initializations and guides non-ideal factorizations toward better local optima. In addition, it enhances solution consistency, reduces the occurrence of ``plateau phenomena'' near saddle points, and maintains computational efficiency comparable to standard NMF methods while remaining compatible with a wide range of existing algorithms. In addition, the thesis further develops GSVD-NMF, an adaptive rank-expansion technique that enable NMF to be performed in an interactive manner. By exploiting discrepancies between under-complete NMF solutions and the SVD of the data matrix, missing components can be efficiently initialized and refined without restarting the factorization from scratch, leading to improved optimization outcomes and substantial computational savings. To address scalability and spatial structure in large imaging datasets, we further propose a localized NMF framework, termed TiledNMF, based on a novel sparse representation of the data. Combined with efficient component expansion and pairwise recombination strategies, this approach enables global factorization while isolating independent features, resulting in more localized and unimodal components. Experiments on LC–MS and large-scale calcium imaging datasets demonstrate improved reconstruction fidelity, reduced component overlap, and more effective separation of meaningful signals with fewer spurious components. In conclusion, these results establish adaptive rank expansion, component merge, and localized sparse representations as a practical and coherent framework for enhancing the performance, interpretability, and scalability of NMF, bridging theoretical analysis with large-scale practical applications.
Committee Chair
Timothy Holy
Committee Members
Dennis Barbour; Geoffrey Goodhill; Joseph Culver; Joseph O’Sullivan
Degree
Doctor of Philosophy (PhD)
Author's Department
Interdisciplinary Programs
Document Type
Dissertation
Date of Award
2-24-2026
Language
English (en)
Recommended Citation
Guo, Youdong, "Adaptive Component Expansion in Nonnegative Matrix Factorization: Theory and Applications" (2026). McKelvey School of Engineering Theses & Dissertations. 1331.
The definitive version is available at https://doi.org/10.7936/mj8q-c977