Imaging Mass Spectrometry (IMS) data calls for computational methods to perform factorizations efficiently that facilitate interpretable components. Despite the widespread use of Nonnegative Matrix Factorization (NMF) for IMS-related dimensionality reduction, conventional NMF algorithms are not up to par to scale for extremely large datasets. To remedy this problem, this thesis explores Randomized Nonnegative Matrix Factorization as an efficient alternative to compute IMS data on whole-body mouse pup tissue data.

In this thesis, the author demonstrates how Randomized NMF inherits its two-stage framework for computation from Randomized SVD. The main bottleneck between Randomized NMF replicating the success of its predecessor is the lower-dimensional basis used to compress the data matrix. Although the basis has been conventionally effective for Randomized SVD with provable guarantees, the lack of entry-wise nonnegativity constraint requires existing methods to compromise the projection quality and interpretability and rely on the relaxation of nonnegativity constraints within the update rules. Therefore, this work motivates and explores the use of alternative bases that directly satisfy entry-wise nonnegativity.

In medical research, Imaging Mass Spectrometry (IMS) is a powerful tool that facilitates the spatial mapping of biomolecules in tissue samples, contributing to the identification of disease biomarkers and the analysis of drug effects. While offering valuable insights, IMS data is often large and complex, presenting challenges in data interpretation. Traditional techniques such as principal component analysis (PCA) and non-negative matrix factorization (NMF) have been employed to address these issues, but they fall short due to their inability to fully capture the complex, mixed nature of IMS data. Deep non-negative matrix factorization (deep NMF) offers a promising solution by employing a multi-layer architecture, allowing for dimensionality reduction while improving interpretability.

Deep NMF can decompose IMS data into multiple structures, helping to uncover complex biochemical interactions. This thesis explores the application of deep NMF to IMS data, deriving update algorithms for both the Frobenius norm and generalized Kullback-Leibler (gKL) divergence, and testing these methods on MALDI-TOF IMS data. Results show that the deep multiplicative update rule (MUR) outperforms non-deep methods in terms of interpretability and surpasses multi-layer structures in both reconstruction error and interpretability. Computational challenges remain in tuning algorithm parameters and achieving convergence in deep Alternating Direction Method of Multipliers (ADMM) NMF.