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Abstract

Non-negative matrix factorization (NMF) is a fundamental technique in modern data analysis. Its distinguishing characteristic—requiring non-negative factors—ensures that it models only additive components, producing interpretable parts-based representations. When augmented with sum-to-one constraints, yielding stochastic matrix factorization (SMF), this interpretability becomes mathematically precise through an exact correspondence with latent variable models. This thesis develops a comprehensive framework extending these ideas to continuous multivariate probability densities via non-negative simplex-constrained tensor factorizations. We demonstrate the framework’s versatility through two novel applications: sediment provenance tracing in geology and spatial transcriptomics denoising in biology. Algorithmically, we develop a novel rescaled block coordinate descent method that efficiently handles simplex constraints while maintaining iterate bound- edness—a key challenge in constrained NMF. By exploiting the semialgebraic structure of our objectives through the Kurdyka-Łojasiewicz (KL) property, we establish subsequential and full-sequence convergence guarantees. This analysis framework proves remarkably versatile: we successfully apply it to both our probabilistic tensor factorization models and to an adaptive graph-regularized denoising method for spatial transcriptomics, where the regularization evolves with the solution. Theoretically, we investigate why NMF methods work well despite non-convexity. Drawing inspiration from the benign geometry of low-rank matri- ces, we analyze the structure of SMF feasible sets through novel reparameter- izations and boundary characterizations. Our analysis reveals that spurious stationary points primarily occur at boundaries, suggesting new algorithmic strategies for avoiding local minima. These geometric insights, combined with our convergence guarantees and practical applications, provide a unified view of constrained factorization methods that bridges theory and practice.