Large Scale Matrix Factorization With Guarantees
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Large Scale Matrix Factorization with Guarantees
Author: Venkata Sesha Pavana Srinadh Bhojanapalli
language: en
Publisher:
Release Date: 2015
Low rank matrix factorization is an important step in many high dimensional machine learning algorithms. Traditional algorithms for factorization do not scale well with the growing data sizes and there is a need for faster/scalable algorithms. In this dissertation we explore the following two major themes to design scalable factorization algorithms for the problems: matrix completion, low rank approximation (PCA) and semi-definite optimization. (a) Sampling: We develop the optimal way to sample entries of any matrix while preserving its spectral properties. Using this sparse sketch (set of sampled entries) instead of the entire matrix, gives rise to scalable algorithms with good approximation guarantees. (b) Bi-linear factorization structure: We design algorithms that operate explicitly on the factor space instead on the matrix. While bi-linear structure of the factorization, in general, leads to a non-convex optimization problem, we show that under appropriate conditions they indeed recover the solution for the above problems. Both these techniques (individually or in combination) lead to algorithms with lower computational complexity and memory usage. Finally we extend these ideas of sampling and explicit factorization to design algorithms for higher order tensors.
Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) in its modern form has become a standard tool in the analysis of high-dimensional data sets. This book provides a comprehensive and up-to-date account of the most important aspects of the NMF problem and is the first to detail its theoretical aspects, including geometric interpretation, nonnegative rank, complexity, and uniqueness. It explains why understanding these theoretical insights is key to using this computational tool effectively and meaningfully. Nonnegative Matrix Factorization is accessible to a wide audience and is ideal for anyone interested in the workings of NMF. It discusses some new results on the nonnegative rank and the identifiability of NMF and makes available MATLAB codes for readers to run the numerical examples presented in the book. Graduate students starting to work on NMF and researchers interested in better understanding the NMF problem and how they can use it will find this book useful. It can be used in advanced undergraduate and graduate-level courses on numerical linear algebra and on advanced topics in numerical linear algebra and requires only a basic knowledge of linear algebra and optimization.
Handbook of Robust Low-Rank and Sparse Matrix Decomposition
Handbook of Robust Low-Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing shows you how robust subspace learning and tracking by decomposition into low-rank and sparse matrices provide a suitable framework for computer vision applications. Incorporating both existing and new ideas, the book conveniently gives you one-stop access to a number of different decompositions, algorithms, implementations, and benchmarking techniques. Divided into five parts, the book begins with an overall introduction to robust principal component analysis (PCA) via decomposition into low-rank and sparse matrices. The second part addresses robust matrix factorization/completion problems while the third part focuses on robust online subspace estimation, learning, and tracking. Covering applications in image and video processing, the fourth part discusses image analysis, image denoising, motion saliency detection, video coding, key frame extraction, and hyperspectral video processing. The final part presents resources and applications in background/foreground separation for video surveillance. With contributions from leading teams around the world, this handbook provides a complete overview of the concepts, theories, algorithms, and applications related to robust low-rank and sparse matrix decompositions. It is designed for researchers, developers, and graduate students in computer vision, image and video processing, real-time architecture, machine learning, and data mining.