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Joint Nonnegative Matrix Factorization (JointNMF) is a hybrid method for mining information from datasets that contain both feature and connection information. We propose distributed-memory parallelizations of three algorithms for solving the JointNMF problem based on Alternating Nonnegative Least Squares, Projected Gradient Descent, and Projected Gauss-Newton. We extend well-known communication-avoiding algorithms using a single processor grid case to our coupled case on two processor grids. We demonstrate the scalability of the algorithms on up to 960 cores (40 nodes) with 60\% parallel efficiency. The more sophisticated Alternating Nonnegative Least Squares (ANLS) and Gauss-Newton variants outperform the first-order gradient descent method in reducing the objective on large-scale problems. We perform a topic modelling task on a large corpus of academic papers that consists of over 37 million paper abstracts and nearly a billion citation relationships, demonstrating the utility and scalability of the methods. 

Advances in single cell transcriptomics have allowed us to study the identity of single cells. This has led to the discovery of new cell types and high resolution tissue maps of them. Technologies that measure multiple modalities of such data add more detail, but they also complicate data integration. We offer an integrated analysis of the spatial location and gene expression profiles of cells to determine their identity. We propose scHybridNMF (single-cell Hybrid Nonnegative Matrix Factorization), which performs cell type identification by combining sparse nonnegative matrix factorization (sparse NMF) with k-means clustering to cluster high-dimensional gene expression and low-dimensional location data. We show that, under multiple scenarios, including the cases where there is a small number of genes profiled and the location data is noisy, scHybridNMF outperforms sparse NMF, k-means, and an existing method that uses a hidden Markov random field to encode cell location and gene expression data for cell type identification. 

We consider the problem of low-rank approximation of massive dense nonnegative tensor data, for example, to discover latent patterns in video and imaging applications. As the size of data sets grows, single workstations are hitting bottlenecks in both computation time and available memory. We propose a distributed-memory parallel computing solution to handle massive data sets, loading the input data across the memories of multiple nodes, and performing efficient and scalable parallel algorithms to compute the low-rank approximation. We present a software package called Parallel Low-rank Approximation with Nonnegativity Constraints, which implements our solution and allows for extension in terms of data (dense or sparse, matrices or tensors of any order), algorithm (e.g., from multiplicative updating techniques to alternating direction method of multipliers), and architecture (we exploit GPUs to accelerate the computation in this work). We describe our parallel distributions and algorithms, which are careful to avoid unnecessary communication and computation, show how to extend the software to include new algorithms and/or constraints, and report efficiency and scalability results for both synthetic and real-world data sets.