Variable sparsity non-negative matrix and tensor factorization algorithms and their applications in cancer genomics

The oncology field has seen major progress in personalized cancer treatments through novel therapies and molecular diagnostics. However, cancer's evolving and resistant nature poses a significant challenge due to its dynamic behavior. This complexity at molecular, cellular, and clinical levels underscores the necessity for innovative modeling and classification approaches.

Matrix factorization algorithms, such as non-negative matrix factorization (NMF), have been utilized to create transcriptional signatures that encapsulate the most significant patterns in cancer datasets. NMF generates dimensionality-reduced representations encoded with positive components, summarizing input data. However, the sparseness of the resulting NMF factor matrices, W and H, is often incidental, providing limited control over sparseness and the balance between local and global structures.

To address this, we introduce a novel variable-sparsity Non-negative Matrix Factorization (vsNMF) algorithm. Unlike other sparse versions of NMF, vsNMF explicitly enforces sparsity through a projective step for each column of W and row of H after a standard NMF update. The user inputs the input matrix along with two one-dimensional vectors to specify sparsity functions.

This fine-grained control over sparsity allows for NMF solutions tailored to generate parts-based encodings of overlapping features or soft-clustering models, enhancing biological interpretability. These low-dimensionality representations can integrate and merge datasets, define cancer subtypes, and define tumor states.

We extend this approach to tensors, developing a variable-sparsity Non-Negative Tensor Factorization (vsNTF) algorithm. This algorithm accepts an input tensor and sparsity functions for each tensor dimension. By enabling more flexible representations of biological data, vsNMF and vsNTF can significantly advance our understanding of cancer biology, provide interpretable models of cancer, and improve functional cancer taxonomies and precision oncology paradigms.