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An Alternating Projection Framework for Elementwise Masked Nonlinear Matrix Decomposition
Kulathil, Muhammed Noshin Poovan
Kulathil, Muhammed Noshin Poovan
Date
2025-11
Advisor
Type
Thesis
Degree
Citations
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35.232-2025.63a Muhammed Noshin Poovan Kulathil_COMPRESSED.pdf
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Description
A Master of Science thesis in Machine Learning by Muhammed Noshin Poovan Kulathil entitled, “An Alternating Projection Framework for Elementwise Masked Nonlinear Matrix Decomposition”, submitted in November 2025. Thesis advisor is Dr. Mohamed Alhajri. Soft copy is available (Thesis, Completion Certificate, Approval Signatures, and AUS Archives Consent Form).
Abstract
Achieving extremely low-rank representations without loss of data fidelity remains a central challenge for conventional matrix approximation techniques. This thesis presents a novel masked alternating projection algorithm for nonlinear matrix
decomposition (NMD) that addresses this limitation by selectively injecting both positive and negative values into the zero entries of data matrices while preserving all nonzero data. By exploiting the structure of zero entries, the proposed method
introduces additional degrees of freedom on the low-rank manifold, enabling aggressive rank reduction without compromising reconstruction accuracy. Four variants of the algorithm were developed to compare two low-rank projectors: truncated singular value decomposition (TSVD) and randomized QR, each tested with and without momentum acceleration. All variants achieve high reconstruction accuracy, with the randomized QR variant with momentum (AP-QRm) matching this accuracy while delivering significantly faster runtimes, achieving efficiency gains of 70% to 99.9% over the SVD-based approach. Extensive benchmarking across 25 diverse datasets demonstrates that AP-QRm consistently delivers state-of-the-art performance, surpassing both established nonlinear NMD and linear low-rank methods in terms of attainable rank and reconstruction error. The method enables aggressive rank reductions by 90% to 99.9%, frequently attaining an effective rank-1 while maintaining low numerical error and high perceptual quality for image data. Furthermore, AP-QRm offers substantial memory savings by reducing storage requirements by up to 680× compared to dense matrix representations and outperforming traditional sparse formats such as CSR and CSC across a wide range of sparsity levels. This ability to compress data into extremely low-rank representations translates directly into exceptional storage efficiency, making AP-QRm a scalable and robust solution for data compression, large-scale machine learning, and scientific computing, where efficient and reliable matrix representations are essential.