Novel Monte Carlo Methods for Large-Scale Linear Algebra Operations

Linear algebra operations play an important role in scientific computing and data analysis. With increasing data volume and complexity in the Big Data era, linear algebra operations are important tools to process massive datasets. On one hand, the advent of modern high-performance computing architectures with increasing computing power has greatly enhanced our capability to deal with a large volume of data. One the other hand, many classical, deterministic numerical linear algebra algorithms have difficulty to scale to handle large data sets. Monte Carlo methods, which are based on statistical sampling, exhibit many attractive properties in dealing with large volume of datasets, including fast approximated results, memory efficiency, reduced data accesses, natural parallelism, and inherent fault tolerance. In this dissertation, we present new Monte Carlo methods to accommodate a set of fundamental and ubiquitous large-scale linear algebra operations, including solving large-scale linear systems, constructing low-rank matrix approximation, and approximating the extreme eigenvalues/ eigenvectors, across modern distributed and parallel computing architectures. First of all, we revisit the classical Ulam-von Neumann Monte Carlo algorithm and derive the necessary and sufficient condition for its convergence. To support a broad family of linear systems, we develop Krylov subspace Monte Carlo solvers that go beyond the use of Neumann series. New algorithms used in the Krylov subspace Monte Carlo solvers include (1) a Breakdown-Free Block Conjugate Gradient algorithm to address the potential rank deficiency problem occurred in block Krylov subspace methods; (2) a Block Conjugate Gradient for Least Squares algorithm to stably approximate the least squares solutions of general linear systems; (3) a BCGLS algorithm with deflation to gain convergence acceleration; and (4) a Monte Carlo Generalized Minimal Residual algorithm based on sampling matrix-vector products to provide fast approximation of solutions. Secondly, we design a rank-revealing randomized Singular Value Decomposition (R3SVD) algorithm for adaptively constructing low-rank matrix approximations to satisfy application-specific accuracy. Thirdly, we study the block power method on Markov Chain Monte Carlo transition matrices and find that the convergence is actually depending on the number of independent vectors in the block. Correspondingly, we develop a sliding window power method to find stationary distribution, which has demonstrated success in modeling stochastic luminal Calcium release site. Fourthly, we take advantage of hybrid CPU-GPU computing platforms to accelerate the performance of the Breakdown-Free Block Conjugate Gradient algorithm and the randomized Singular Value Decomposition algorithm. Finally, we design a Gaussian variant of Freivalds’ algorithm to efficiently verify the correctness of matrix-matrix multiplication while avoiding undetectable fault patterns encountered in deterministic algorithms

Highly Accurate Fragment Library for Protein Fold Recognition

Proteins play a crucial role in living organisms as they perform many vital tasks in every living cell. Knowledge of protein folding has a deep impact on understanding the heterogeneity and molecular functions of proteins. Such information leads to crucial advances in drug design and disease understanding. Fold recognition is a key step in the protein structure discovery process, especially when traditional computational methods fail to yield convincing structural homologies. In this work, we present a new protein fold recognition approach using machine learning and data mining methodologies. First, we identify a protein structural fragment library (Frag-K) composed of a set of backbone fragments ranging from 4 to 20 residues as the structural “keywords” that can effectively distinguish between major protein folds. We firstly apply randomized spectral clustering and random forest algorithms to construct representative and sensitive protein fragment libraries from a large-scale of high-quality, non-homologous protein structures available in PDB. We analyze the impacts of clustering cut-offs on the performance of the fragment libraries. Then, the Frag-K fragments are employed as structural features to classify protein structures in major protein folds defined by SCOP (Structural Classification of Proteins). Our results show that a structural dictionary with ~400 4- to 20-residue Frag-K fragments is capable of classifying major SCOP folds with high accuracy. Then, based on Frag-k, we design a novel deep learning architecture, so-called DeepFrag-k, which identifies fold discriminative features to improve the accuracy of protein fold recognition. DeepFrag-k is composed of two stages: the first stage employs a multimodal Deep Belief Network (DBN) to predict the potential structural fragments given a sequence, represented as a fragment vector, and then the second stage uses a deep convolution neural network (CNN) to classify the fragment vectors into the corresponding folds. Our results show that DeepFrag-k yields 92.98% accuracy in predicting the top-100 most popular fragments, which can be used to generate discriminative fragment feature vectors to improve protein fold recognition

A Survey of Matrix Completion Methods for Recommendation Systems

In recent years, the recommendation systems have become increasingly popular and have been used in a broad variety of applications. Here, we investigate the matrix completion techniques for the recommendation systems that are based on collaborative filtering. The collaborative filtering problem can be viewed as predicting the favorability of a user with respect to new items of commodities. When a rating matrix is constructed with users as rows, items as columns, and entries as ratings, the collaborative filtering problem can then be modeled as a matrix completion problem by filling out the unknown elements in the rating matrix. This article presents a comprehensive survey of the matrix completion methods used in recommendation systems. We focus on the mathematical models for matrix completion and the corresponding computational algorithms as well as their characteristics and potential issues. Several applications other than the traditional user-item association prediction are also discussed

Prediction of lncRNA-Disease Associations Based on Inductive Matrix Completion

Motivation: Accumulating evidences indicate that long non-coding RNAs (lncRNAs) play pivotal roles in various biological processes. Mutations and dysregulations of lncRNAs are implicated in miscellaneous human diseases. Predicting lncRNA–disease associations is beneficial to disease diagnosis as well as treatment. Although many computational methods have been developed, precisely identifying lncRNA–disease associations, especially for novel lncRNAs, remains challenging. Results: In this study, we propose a method (named SIMCLDA) for predicting potential lncRNA– disease associations based on inductive matrix completion. We compute Gaussian interaction profile kernel of lncRNAs from known lncRNA–disease interactions and functional similarity of diseases based on disease–gene and gene–gene onotology associations. Then, we extract primary feature vectors from Gaussian interaction profile kernel of lncRNAs and functional similarity of diseases by principal component analysis, respectively. For a new lncRNA, we calculate the interaction profile according to the interaction profiles of its neighbors. At last, we complete the association matrix based on the inductive matrix completion framework using the primary feature vectors from the constructed feature matrices. Computational results show that SIMCLDA can effectively predict lncRNA–disease associations with higher accuracy compared with previous methods. Furthermore, case studies show that SIMCLDA can effectively predict candidate lncRNAs for renal cancer, gastric cancer and prostate cancer