Graph Summarization

The continuous and rapid growth of highly interconnected datasets, which are both voluminous and complex, calls for the development of adequate processing and analytical techniques. One method for condensing and simplifying such datasets is graph summarization. It denotes a series of application-specific algorithms designed to transform graphs into more compact representations while preserving structural patterns, query answers, or specific property distributions. As this problem is common to several areas studying graph topologies, different approaches, such as clustering, compression, sampling, or influence detection, have been proposed, primarily based on statistical and optimization methods. The focus of our chapter is to pinpoint the main graph summarization methods, but especially to focus on the most recent approaches and novel research trends on this topic, not yet covered by previous surveys.Comment: To appear in the Encyclopedia of Big Data Technologie

Feature Grouping Using Weighted L1 Norm For High-Dimensional Data

Building effective prediction models from high-dimensional data is an important problem in several domains such as in bioinformatics, healthcare analytics and general regression analysis. Extracting feature groups automatically from such data with several correlated features is necessary, in order to use regularizers such as the group lasso which can exploit this deciphered grouping structure to build effective prediction models. Elastic net, fused-lasso and Octagonal Shrinkage Clustering Algorithm for Regression (oscar) are some of the popular feature grouping methods proposed in the literature which recover both sparsity and feature groups from the data. However, their predictive ability is affected adversely when the regression coefficients of adjacent feature groups are similar, but not exactly equal. This happens as these methods merge such adjacent feature groups erroneously, which is widely known as the misfusion problem. In order to solve this problem, in this thesis, we propose a weighted L1 norm-based approach which is effective at recovering feature groups, despite the proximity of the coefficients of adjacent feature groups, building extremely accurate prediction models. This convex optimization problem is solved using the fast iterative soft-thresholding algorithm (FISTA). We depict how our approach is more successful than competing feature grouping methods such as the elastic net, fused-lasso and oscar at solving the misfusion problem on synthetic datasets. We also compare the goodness of prediction of our algorithm against state-of-the-art non-convex feature grouping methods when applied on a real-world breast cancer dataset, the 20-Newsgroups dataset and synthetic datasets

Feature Grouping Using Weighted L1 Norm For High-Dimensional Data

Building effective prediction models from high-dimensional data is an important problem in several domains such as in bioinformatics, healthcare analytics and general regression analysis. Extracting feature groups automatically from such data with several correlated features is necessary, in order to use regularizers such as the group lasso which can exploit this deciphered grouping structure to build effective prediction models. Elastic net, fused-lasso and Octagonal Shrinkage Clustering Algorithm for Regression (oscar) are some of the popular feature grouping methods proposed in the literature which recover both sparsity and feature groups from the data. However, their predictive ability is affected adversely when the regression coefficients of adjacent feature groups are similar, but not exactly equal. This happens as these methods merge such adjacent feature groups erroneously, which is widely known as the misfusion problem. In order to solve this problem, in this thesis, we propose a weighted L1 norm-based approach which is effective at recovering feature groups, despite the proximity of the coefficients of adjacent feature groups, building extremely accurate prediction models. This convex optimization problem is solved using the fast iterative soft-thresholding algorithm (FISTA). We depict how our approach is more successful than competing feature grouping methods such as the elastic net, fused-lasso and oscar at solving the misfusion problem on synthetic datasets. We also compare the goodness of prediction of our algorithm against state-of-the-art non-convex feature grouping methods when applied on a real-world breast cancer dataset, the 20-Newsgroups dataset and synthetic datasets

DANR: Discrepancy-aware Network Regularization

Network regularization is an effective tool for incorporating structural prior knowledge to learn coherent models over networks, and has yielded provably accurate estimates in applications ranging from spatial economics to neuroimaging studies. Recently, there has been an increasing interest in extending network regularization to the spatio-temporal case to accommodate the evolution of networks. However, in both static and spatio-temporal cases, missing or corrupted edge weights can compromise the ability of network regularization to discover desired solutions. To address these gaps, we propose a novel approach---{\it discrepancy-aware network regularization} (DANR)---that is robust to inadequate regularizations and effectively captures model evolution and structural changes over spatio-temporal networks. We develop a distributed and scalable algorithm based on the alternating direction method of multipliers (ADMM) to solve the proposed problem with guaranteed convergence to global optimum solutions. Experimental results on both synthetic and real-world networks demonstrate that our approach achieves improved performance on various tasks, and enables interpretation of model changes in evolving networks

CenetBiplot: a new proposal of sparse and orthogonal biplots methods by means of elastic net CSVD

[EN[ In this work, a new mathematical algorithm for sparse and orthogonal constrained biplots, called CenetBiplots, is proposed. Biplots provide a joint representation of observations and variables of a multidimensional matrix in the same reference system. In this subspace the relationships between them can be interpreted in terms of geometric elements. CenetBiplots projects a matrix onto a low-dimensional space generated simultaneously by sparse and orthogonal principal components. Sparsity is desired to select variables automatically, and orthogonality is necessary to keep the geometrical properties that ensure the biplots graphical interpretation. To this purpose, the present study focuses on two different objectives: 1) the extension of constrained singular value decomposition to incorporate an elastic net sparse constraint (CenetSVD), and 2) the implementation of CenetBiplots using CenetSVD. The usefulness of the proposed methodologies for analysing high-dimensional and low-dimensional matrices is shown. Our method is implemented in R software and available for download from https://github.com/ananieto/SparseCenetMA.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work was not supported by any grant.Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

Optimization with Sparsity-Inducing Penalties

Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many of the related estimation problems can be cast as convex optimization problems by regularizing the empirical risk with appropriate non-smooth norms. The goal of this paper is to present from a general perspective optimization tools and techniques dedicated to such sparsity-inducing penalties. We cover proximal methods, block-coordinate descent, reweighted $\ell_2$-penalized techniques, working-set and homotopy methods, as well as non-convex formulations and extensions, and provide an extensive set of experiments to compare various algorithms from a computational point of view

Model estimation, identification and inference for next-generation functional data and spatial data

This dissertation is composed of three research projects focused on model estimation, identification, and inference for next-generation functional data and spatial data. The first project deals with data that are collected on a count or binary response with spatial covariate information. In this project, we introduce a new class of generalized geoadditive models (GGAMs) for spatial data distributed over complex domains. Through a link function, the proposed GGAM assumes that the mean of the discrete response variable depends on additive univariate functions of explanatory variables and a bivariate function to adjust for the spatial effect. We propose a two-stage approach for estimating and making inferences of the components in the GGAM. In the first stage, the univariate components and the geographical component in the model are approximated via univariate polynomial splines and bivariate penalized splines over triangulation, respectively. In the second stage, local polynomial smoothing is applied to the cleaned univariate data to average out the variation of the first-stage estimators. We investigate the consistency of the proposed estimators and the asymptotic normality of the univariate components. We also establish the simultaneous confidence band for each of the univariate components. The performance of the proposed method is evaluated by two simulation studies and the crash counts data in the Tampa-St. Petersburg urbanized area in Florida. In the second project, motivated by recent work of analyzing data in the biomedical imaging studies, we consider a class of image-on-scalar regression models for imaging responses and scalar predictors. We propose to use flexible multivariate splines over triangulations to handle the irregular domain of the objects of interest on the images and other characteristics of images. The proposed estimators of the coefficient functions are proved to be root-$n$ consistent and asymptotically normal under some regularity conditions. We also provide a consistent and computationally efficient estimator of the covariance function. Asymptotic pointwise confidence intervals (PCIs) and data-driven simultaneous confidence corridors (SCCs) for the coefficient functions are constructed. A highly efficient and scalable estimation algorithm is developed. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed method. The proposed method is applied to the spatially normalized Positron Emission Tomography (PET) data of Alzheimer\u27s Disease Neuroimaging Initiative (ADNI). In the third project, we propose a heterogeneous functional linear model to simultaneously estimate multiple coefficient functions and identify groups, such that coefficient functions are identical within groups and distinct across groups. By borrowing information from relevant subgroups, our method enhances estimation efficiency while preserving heterogeneity. We use an adaptive fused lasso penalty to shrink subgroup coefficients to shared common values within each group. We also establish the theoretical properties of our adaptive fused lasso estimators. To enhance the computation efficiency and incorporate neighborhood information, we propose to use a graph-constrained adaptive lasso. A highly efficient and scalable estimation algorithm is developed. Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed method. The proposed method is applied to a dataset of hybrid maize grain yields from the Genomes to Fields consortium