4,524 research outputs found

    A dual framework for low-rank tensor completion

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    One of the popular approaches for low-rank tensor completion is to use the latent trace norm regularization. However, most existing works in this direction learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm that helps in learning a non-sparse combination of tensors. We develop a dual framework for solving the low-rank tensor completion problem. We first show a novel characterization of the dual solution space with an interesting factorization of the optimal solution. Overall, the optimal solution is shown to lie on a Cartesian product of Riemannian manifolds. Furthermore, we exploit the versatile Riemannian optimization framework for proposing computationally efficient trust region algorithm. The experiments illustrate the efficacy of the proposed algorithm on several real-world datasets across applications.Comment: Aceepted to appear in Advances of Nueral Information Processing Systems (NIPS), 2018. A shorter version appeared in the NIPS workshop on Synergies in Geometric Data Analysis 201

    Population-level Balance in Signed Networks

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    Statistical network models are useful for understanding the underlying formation mechanism and characteristics of complex networks. However, statistical models for \textit{signed networks} have been largely unexplored. In signed networks, there exist both positive (e.g., like, trust) and negative (e.g., dislike, distrust) edges, which are commonly seen in real-world scenarios. The positive and negative edges in signed networks lead to unique structural patterns, which pose challenges for statistical modeling. In this paper, we introduce a statistically principled latent space approach for modeling signed networks and accommodating the well-known \textit{balance theory}, i.e., ``the enemy of my enemy is my friend'' and ``the friend of my friend is my friend''. The proposed approach treats both edges and their signs as random variables, and characterizes the balance theory with a novel and natural notion of population-level balance. This approach guides us towards building a class of balanced inner-product models, and towards developing scalable algorithms via projected gradient descent to estimate the latent variables. We also establish non-asymptotic error rates for the estimates, which are further verified through simulation studies. In addition, we apply the proposed approach to an international relation network, which provides an informative and interpretable model-based visualization of countries during World War II

    International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

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    The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

    Generalized Low Rank Models

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    Principal components analysis (PCA) is a well-known technique for approximating a tabular data set by a low rank matrix. Here, we extend the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix completion, sparse and robust PCA, kk-means, kk-SVD, and maximum margin matrix factorization. The method handles heterogeneous data sets, and leads to coherent schemes for compressing, denoising, and imputing missing entries across all data types simultaneously. It also admits a number of interesting interpretations of the low rank factors, which allow clustering of examples or of features. We propose several parallel algorithms for fitting generalized low rank models, and describe implementations and numerical results.Comment: 84 pages, 19 figure

    First-order Convex Optimization Methods for Signal and Image Processing

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    In this thesis we investigate the use of first-order convex optimization methods applied to problems in signal and image processing. First we make a general introduction to convex optimization, first-order methods and their iteration com-plexity. Then we look at different techniques, which can be used with first-order methods such as smoothing, Lagrange multipliers and proximal gradient meth-ods. We continue by presenting different applications of convex optimization and notable convex formulations with an emphasis on inverse problems and sparse signal processing. We also describe the multiple-description problem. We finally present the contributions of the thesis. The remaining parts of the thesis consist of five research papers. The first paper addresses non-smooth first-order convex optimization and the trade-off between accuracy and smoothness of the approximating smooth function. The second and third papers concern discrete linear inverse problems and reliable numerical reconstruction software. The last two papers present a convex opti-mization formulation of the multiple-description problem and a method to solve it in the case of large-scale instances. i i

    Similarity learning in the era of big data

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    This dissertation studies the problem of similarity learning in the era of big data with heavy emphasis on real-world applications in social media. As in the saying “birds of a feather flock together,” in similarity learning, we aim to identify the notion of being similar in a data-driven and task-specific way, which is a central problem for maximizing the value of big data. Despite many successes of similarity learning from past decades, social media networks as one of the most typical big data media contain large-volume, various and high-velocity data, which makes conventional learning paradigms and off- the-shelf algorithms insufficient. Thus, we focus on addressing the emerging challenges brought by the inherent “three-Vs” characteristics of big data by answering the following questions: 1) Similarity is characterized by both links and node contents in networks; how to identify the contribution of each network component to seamlessly construct an application orientated similarity function? 2) Social media data are massive and contain much noise; how to efficiently learn the similarity between node pairs in large and noisy environments? 3) Node contents in social media networks are multi-modal; how to effectively measure cross-modal similarity by bridging the so-called “semantic gap”? 4) User wants and needs, and item characteristics, are continuously evolving, which generates data at an unprecedented rate; how to model the nature of temporal dynamics in principle and provide timely decision makings? The goal of this dissertation is to provide solutions to these questions via innovative research and novel methods. We hope this dissertation sheds more light on similarity learning in the big data era and broadens its applications in social media
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