PLANC

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.

A cubic nonlinear population growth model for single species: theory, an explicit–implicit solution algorithm and applications

In this paper, we extend existing population growth models and propose a model based on a nonlinear cubic differential equation that reveals itself as a special subclass of Abel differential equations of first kind. We first summarize properties of the time-continuous problem formulation. We state the boundedness, global existence, and uniqueness of solutions for all times. Proofs of these properties are thoroughly given in the Appendix to this paper. Subsequently, we develop an explicit–implicit time-discrete numerical solution algorithm for our time-continuous population growth model and show that many properties of the time-continuous case transfer to our numerical explicit–implicit time-discrete solution scheme. We provide numerical examples to illustrate different behaviors of our proposed model. Furthermore, we compare our explicit–implicit discretization scheme to the classical Eulerian discretization. The latter violates the nonnegativity constraints on population sizes, whereas we prove and illustrate that our explicit–implicit discretization algorithm preserves this constraint. Finally, we describe a parameter estimation approach to apply our algorithm to two different real-world data sets.

Matrix Factorization-based Improved Classification of Gene Expression Data

Background: The medical data, in the form of prescriptions and test reports, is very extensive which needs a comprehensive analysis. Objective: The gene expression data set is formulated using a very large number of genes associated to thousands of samples. Identifying the relevant biological information from these complex associations is a difficult task. Methods: For this purpose, a variety of classification algorithms are available which can be used to automatically detect the desired information. K-Nearest Neighbour Algorithm, Latent Dirichlet Allocation, Gaussian Naïve Bayes and support Vector Classifier are some of the well known algorithms used for the classification task. Nonnegative Matrix Factorization is a technique which has gained a lot of popularity because of its nonnegativity constraints. This technique can be used for better interpretability of data. Results: In this paper, we applied NMF as a pre-processing step for better results. We also evaluated the given classifiers on the basis of four criteria: accuracy, precision, specificity and Recall. Conclusion: The experimental results shows that these classifiers give better performance when NMF is applied at pre-processing of data before giving it to the said classifiers. Gaussian Naïve Bias algorithm showed a significant improvement in classification after the application of NMF at preprocessing.

Multiplicative Updates for Optimization Problems with Dynamics

We consider the problem of optimizing general convex objective functions with nonnegativity constraints. Using the Karush-Kuhn-Tucker (KKT) conditions for the nonnegativity constraints we will derive fast multiplicative update rules for several problems of interest in signal processing, including non-negative deconvolution, point-process smoothing, ML estimation for Poisson Observations, nonnegative least squares and nonnegative matrix factorization (NMF). Our algorithm can also account for temporal and spatial structure and regularization. We will analyze the performance of our algorithm on simultaneously recorded neuronal calcium imaging and electrophysiology data.