Persistence diagrams (PD)s play a central role in topological data analysis. This analysis requires computing distances among such diagrams such as the 1-Wasserstein distance. Accurate computation of these PD distances for large data sets that render large diagrams may not scale appropriately with the existing methods. The main source of difficulty ensues from the size of the bipartite graph on which a matching needs to be computed for determining these PD distances. We address this problem by making several algorithmic and computational observations in order to obtain an approximation. First, taking advantage of the proximity of PD points, we condense them thereby decreasing the number of nodes in the graph for computation. The increase in point multiplicities is addressed by reducing the matching problem to a min-cost flow problem on a transshipment network. Second, we use Well Separated Pair Decomposition to sparsify the graph to a size that is linear in the number of points. Both node and arc sparsifications contribute to the approximation factor where we leverage a lower bound given by the Relaxed Word Mover's distance. Third, we eliminate bottlenecks during the sparsification procedure by introducing parallelism. Fourth, we develop an open source software called1 PDoptFlow based on our algorithm, exploiting parallelism by GPU and multicore. We perform extensive experiments and show that the actual empirical error is very low. We also show that we can achieve high performance at low guaranteed relative errors, improving upon the state of the arts.

The computation of Vietoris-Rips persistence barcodes is both execution-intensive and memory-intensive. In this paper, we study the computational structure of Vietoris-Rips persistence barcodes, and identify several unique mathematical properties and algorithmic opportunities with connections to the GPU. Mathematically and empirically, we look into the properties of apparent pairs, which are independently identifiable persistence pairs comprising up to 99% of persistence pairs. We give theoretical upper and lower bounds of the apparent pair rate and model the average case. We also design massively parallel algorithms to take advantage of the very large number of simplices that can be processed independently of each other. Having identified these opportunities, we develop a GPU-accelerated software for computing Vietoris-Rips persistence barcodes, called Ripser++. The software achieves up to 30x speedup over the total execution time of the original Ripser and also reduces CPU-memory usage by up to 2.0x. We believe our GPU-acceleration based efforts open a new chapter for the advancement of topological data analysis in the post-Moore’s Law era.

Persistent homology (PH) matrix reduction is an important tool for data analytics in many application areas. Due to its highly irregular execution patterns in computation, it is challenging to gain high efficiency in parallel processing for increasingly large data sets.

Users in online social networks play a variety of social roles and statuses. For example, users in Twitter can be represented as advertiser, content contributor, information receiver, etc; users in Linkedin can be in different professional roles, such as engineer, salesperson and recruiter. Previous research work mainly focuses on using categorical and textual information to predict the attributes of users. However, it cannot be applied to a large number of users in real social networks, since much of such information is missing, outdated and non-standard. In this paper, we investigate the social roles and statuses that people act in online social networks in the perspective of network structures, since the uniqueness of social networks is connecting people. We quantitatively analyze a number of key social principles and theories that correlate with social roles and statuses. We systematically study how the network characteristics reflect the social situations of users in an online society. We discover patterns of homophily, the tendency of users to connect with users with similar social roles and statuses. In addition, we observe that different factors in social theories influence the social role/status of an individual user to various extent, since these social principles represent different aspects of the network. We then introduce an optimization framework based on Factor Conditioning Symmetry, and we propose a probabilistic model to integrate the optimization framework on local structural information as well as network influence to infer the unknown social roles and statuses of online users. We will present experiment results to show the effectiveness of the inference.

Traditional parallel processing models, such as BSP, are "scale up" based, aiming to achieve high performance by increasing computing power, interconnection network bandwidth, and memory/storage capacity within dedicated systems, while big data analytics tasks aiming for high throughput demand that large distributed systems "scale out" by continuously adding computing and storage resources through networks. Each one of the "scale up" model and "scale out" model has a different set of performance requirements and system bottlenecks. In this paper, we develop a general model that abstracts critical computation and communication behavior and computation-communication interactions for big data analytics in a scalable and fault-tolerant manner. Our model is called DOT, represented by three matrices for data sets (D), concurrent data processing operations (O), and data transformations (T), respectively. With the DOT model, any big data analytics job execution in various software frameworks can be represented by a specific or non-specific number of elementary/composite DOT blocks, each of which performs operations on the data sets, stores intermediate results, makes necessary data transfers, and performs data transformations in the end. The DOT model achieves the goals of scalability and fault-tolerance by enforcing a data-dependency-free relationship among concurrent tasks. Under the DOT model, we provide a set of optimization guidelines, which are framework and implementation independent, and applicable to a wide variety of big data analytics jobs. Finally, we demonstrate the effectiveness of the DOT model through several case studies.