Publications

The aim of this project is to develop low dimension parametric (deterministic) models of complex networks, to use compressive sensing (CS) and multiscale analysis to do so and to exploit the structure of complex networks (some are self-similar under coarsening). CS provides a new way of sampling and reconstructing networks. The approach is based on multiresolution decomposition of the adjacency matrix and its efficient sampling. It requires preprocessing of the adjacency matrix to make it 'blocky' which is the biggest (combinatorial) algorithm challenge. Current CS reconstruction algorithm makes no use of the structure of a graph, its very general (and so not very efficient/customized). Other model-based CS techniques exist, but not yet adapted to networks. Obvious starting point for future work is to increase the efficiency of reconstruction.

The problem of incomplete data - i.e., data with missing or unknown values - in multi-way arrays is ubiquitous in biomedical signal processing, network traffic analysis, bibliometrics, social network analysis, chemometrics, computer vision, communication networks, etc. We consider the problem of how to factorize data sets with missing values with the goal of capturing the underlying latent structure of the data and possibly reconstructing missing values (i.e., tensor completion). We focus on one of the most well-known tensor factorizations that captures multi-linear structure, CANDECOMP/PARAFAC (CP). In the presence of missing data, CP can be formulated as a weighted least squares problem that models only the known entries. We develop an algorithm called CP-WOPT (CP Weighted OPTimization) that uses a first-order optimization approach to solve the weighted least squares problem. Based on extensive numerical experiments, our algorithm is shown to successfully factorize tensors with noise and up to 99% missing data. A unique aspect of our approach is that it scales to sparse large-scale data, e.g., 1000 x 1000 x 1000 with five million known entries (0.5% dense). We further demonstrate the usefulness of CP-WOPT on two real-world applications: a novel EEG (electroencephalogram) application where missing data is frequently encountered due to disconnections of electrodes and the problem of modeling computer network traffic where data may be absent due to the expense of the data collection process.

Automated processing, modeling, and analysis of unstructured text (news documents, web content, journal articles, etc.) is a key task in many data analysis and decision making applications. As data sizes grow, scalability is essential for deep analysis. In many cases, documents are modeled as term or feature vectors and latent semantic analysis (LSA) is used to model latent, or hidden, relationships between documents and terms appearing in those documents. LSA supplies conceptual organization and analysis of document collections by modeling high-dimension feature vectors in many fewer dimensions. While past work on the scalability of LSA modeling has focused on the SVD, the goal of our work is to investigate the use of distributed memory architectures for the entire text analysis process, from data ingestion to semantic modeling and analysis. ParaText is a set of software components for distributed processing, modeling, and analysis of unstructured text. The ParaText source code is available under a BSD license, as an integral part of the Titan toolkit. ParaText components are chained-together into data-parallel pipelines that are replicated across processes on distributed-memory architectures. Individual components can be replaced or rewired to explore different computational strategies and implement new functionality. ParaText functionality can be embedded in applications on any platform using the native C++ API, Python, or Java. The ParaText MPI Process provides a 'generic' text analysis pipeline in a command-line executable that can be used for many serial and parallel analysis tasks. ParaText can also be deployed as a web service accessible via a RESTful (HTTP) API. In the web service configuration, any client can access the functionality provided by ParaText using commodity protocols ... from standard web browsers to custom clients written in any language.

The data in many disciplines such as social networks, web analysis, etc. is link-based, and the link structure can be exploited for many different data mining tasks. In this paper, we consider the problem of temporal link prediction: Given link data for time periods 1 through T, can we predict the links in time period T + 1? Specifically, we look at bipartite graphs changing over time and consider matrix- and tensor-based methods for predicting links. We present a weight-based method for collapsing multi-year data into a single matrix. We show how the well-known Katz method for link prediction can be extended to bipartite graphs and, moreover, approximated in a scalable way using a truncated singular value decomposition. Using a CANDECOMP/PARAFAC tensor decomposition of the data, we illustrate the usefulness of exploiting the natural three-dimensional structure of temporal link data. Through several numerical experiments, we demonstrate that both matrix- and tensor-based techniques are effective for temporal link prediction despite the inherent difficulty of the problem.

The problem of missing data is ubiquitous in domains such as biomedical signal processing, network traffic analysis, bibliometrics, social network analysis, chemometrics, computer vision, and communication networks|all domains in which data collection is subject to occasional errors. Moreover, these data sets can be quite large and have more than two axes of variation, e.g., sender, receiver, time. Many applications in those domains aim to capture the underlying latent structure of the data; in other words, they need to factorize data sets with missing entries. If we cannot address the problem of missing data, many important data sets will be discarded or improperly analyzed. Therefore, we need a robust and scalable approach for factorizing multi-way arrays (i.e., tensors) in the presence of missing data. We focus on one of the most well-known tensor factorizations, CANDECOMP/PARAFAC (CP), and formulate the CP model as a weighted least squares problem that models only the known entries. We develop an algorithm called CP-WOPT (CP Weighted OPTimization) using a first-order optimization approach to solve the weighted least squares problem. Based on extensive numerical experiments, our algorithm is shown to successfully factor tensors with noise and up to 70% missing data. Moreover, our approach is significantly faster than the leading alternative and scales to larger problems. To show the real-world usefulness of CP-WOPT, we illustrate its applicability on a novel EEG (electroencephalogram) application where missing data is frequently encountered due to disconnections of electrodes.

We present Poblano v1.0, a Matlab toolbox for solving gradient-based unconstrained optimization problems. Poblano implements three optimization methods (nonlinear conjugate gradients, limited-memory BFGS, and truncated Newton) that require only first order derivative information. In this paper, we describe the Poblano methods, provide numerous examples on how to use Poblano, and present results of Poblano used in solving problems from a standard test collection of unconstrained optimization problems.

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Tensor decompositions are higher-order analogues of matrix decompositions and have proven to be powerful tools for data analysis. In particular, we are interested in the canonical tensor decomposition, otherwise known as the CANDECOMP/PARAFAC decomposition (CPD), which expresses a tensor as the sum of component rank-one tensors and is used in a multitude of applications such as chemometrics, signal processing, neuroscience, and web analysis. The task of computing the CPD, however, can be difficult. The typical approach is based on alternating least squares (ALS) optimization, which can be remarkably fast but is not very accurate. Previously, nonlinear least squares (NLS) methods have also been recommended; existing NLS methods are accurate but slow. In this paper, we propose the use of gradient-based optimization methods. We discuss the mathematical calculation of the derivatives and further show that they can be computed efficiently, at the same cost as one iteration of ALS. Computational experiments demonstrate that the gradient-based optimization methods are much more accurate than ALS and orders of magnitude faster than NLS.

Link analysis typically focuses on a single type of connection, e.g., two journal papers are linked because they are written by the same author. However, often we want to analyze data that has multiple linkages between objects, e.g., two papers may have the same keywords and one may cite the other. The goal of this paper is to show that multilinear algebra provides a tool for multilink analysis. We analyze five years of publication data from journals published by the Society for Industrial and Applied Mathematics. We explore how papers can be grouped in the context of multiple link types using a tensor to represent all the links between them. A PARAFAC decomposition on the resulting tensor yields information similar to the SVD decomposition of a standard adjacency matrix. We show how the PARAFAC decomposition can be used to understand the structure of the document space and define paper-paper similarities based on multiple linkages. Examples are presented where the decomposed tensor data is used to find papers similar to a body of work (e.g., related by topic or similar to a particular author's papers), find related authors using linkages other than explicit co-authorship or citations, distinguish between papers written by different authors with the same name, and predict the journal in which a paper was published.

Tensor decompositions (e.g., higher-order analogues of matrix decompositions) are powerful tools for data analysis. In particular, the CANDECOMP/PARAFAC (CP) model has proved useful in many applications such chemometrics, signal processing, and web analysis; see for details. The problem of computing the CP decomposition is typically solved using an alternating least squares (ALS) approach. We discuss the use of optimization-based algorithms for CP, including how to efficiently compute the derivatives necessary for the optimization methods. Numerical studies highlight the positive features of our CPOPT algorithms, as compared with ALS and Gauss-Newton approaches.

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This report describes the Licensing Support Network (LSN) Assistant--a set of tools for categorizing e-mail messages and documents, and investigating and correcting existing archives of categorized e-mail messages and documents. The two main tools in the LSN Assistant are the LSN Archive Assistant (LSNAA) tool for recategorizing manually labeled e-mail messages and documents and the LSN Realtime Assistant (LSNRA) tool for categorizing new e-mail messages and documents. This report focuses on the LSNAA tool. There are two main components of the LSNAA tool. The first is the Sandia Categorization Framework, which is responsible for providing categorizations for documents in an archive and storing them in an appropriate Categorization Database. The second is the actual user interface, which primarily interacts with the Categorization Database, providing a way for finding and correcting categorizations errors in the database. A procedure for applying the LSNAA tool and an example use case of the LSNAA tool applied to a set of e-mail messages are provided. Performance results of the categorization model designed for this example use case are presented.

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