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Fast Conical Hull Algorithms for Near-separable Non-negative Matrix Factorization [article]

Abhishek Kumar, Vikas Sindhwani, Prabhanjan Kambadur
2012 arXiv   pre-print
The separability assumption (Donoho & Stodden, 2003; Arora et al., 2012) turns non-negative matrix factorization (NMF) into a tractable problem.  ...  In this paper, we reformulate the separable NMF problem as that of finding the extreme rays of the conical hull of a finite set of vectors.  ...  Acknowledgments: We thank Victor Bittorf and Ben Recht for graciously providing their code, associated parameters and technical support.  ... 
arXiv:1210.1190v1 fatcat:pbcp77uzwzf7to364f5rqf25f4

Fast SVM Trained by Divide-and-Conquer Anchors

Meng Liu, Chang Xu, Chao Xu, Dacheng Tao
2017 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence  
In this paper, we propose to choose the representative points which are noted as anchors obtained from non-negative matrix factorization (NMF) in a divide-and-conquer framework, and then use the anchors  ...  Supporting vector machine (SVM) is the most frequently used classifier for machine learning tasks. However, its training time could become cumbersome when the size of training data is very large.  ...  Approximate Convex Hull of NMF Non-negative matrix factorization (NMF) decomposes a matrix X ∈ R n×m + which contains n non-negative mdimensional vectors {x i } n i=1 into the form of X = HW , where H  ... 
doi:10.24963/ijcai.2017/323 dblp:conf/ijcai/LiuX0T17 fatcat:jfyqcqq3lrc3bgyd6onwhf6gza

Scalable methods for nonnegative matrix factorizations of near-separable tall-and-skinny matrices [article]

Austin R. Benson, Jason D. Lee, Bartek Rajwa, David F. Gleich
2014 arXiv   pre-print
Numerous algorithms are used for nonnegative matrix factorization under the assumption that the matrix is nearly separable.  ...  One key component to these improved methods is an orthogonal matrix transformation that preserves the separability of the NMF problem.  ...  In the remainder of the introduction, we review the state of the art for computing non-negative matrix factorizations.  ... 
arXiv:1402.6964v1 fatcat:eszeisjg75fptd2ngscxijwxc4

Robust Near-Separable Nonnegative Matrix Factorization Using Linear Optimization [article]

Nicolas Gillis, Robert Luce
2013 arXiv   pre-print
Nonnegative matrix factorization (NMF) has been shown recently to be tractable under the separability assumption, under which all the columns of the input data matrix belong to the convex cone generated  ...  However, Hottopixx has two important drawbacks: (i) the input matrix has to be normalized, and (ii) the factorization rank has to be known in advance.  ...  Acknowledgments The authors would like to thank the reviewers for their feedback which helped improve the paper.  ... 
arXiv:1302.4385v2 fatcat:7sdsgaf7hffarjt4xsyofzckme

The Why and How of Nonnegative Matrix Factorization [article]

Nicolas Gillis
2014 arXiv   pre-print
Nonnegative matrix factorization (NMF) has become a widely used tool for the analysis of high-dimensional data as it automatically extracts sparse and meaningful features from a set of nonnegative data  ...  We review some standard NMF algorithms, and also present a recent subclass of NMF problems, referred to as near-separable NMF, that can be solved efficiently (that is, in polynomial time), even in the  ...  Acknowledgment The author would like to thank Rafal Zdunek, Wing-Kin Ma, Marc Pirlot and the Editors of the book 'Regularization, Optimization, Kernels, and Support Vector Machines' for insightful comments  ... 
arXiv:1401.5226v2 fatcat:qw2jsn3g2za6rev45dzjdk5f6e

GLOPTLAB: a configurable framework for the rigorous global solution of quadratic constraint satisfaction problems

Ferenc Domes
2009 Optimization Methods and Software  
The algorithms implemented in GloptLab are used to reduce the search space: scaling, constraint propagation, linear relaxations, strictly convex enclosures, conic methods, and branch and bound.  ...  GloptLab is an easy-to-use testing and development platform for solving quadratic constraint satisfaction problems, written in Matlab.  ...  Acknowledgements I would like to thank Arnold Neumaier, for his help and support and Martin Fuchs for his creative ideas and corrections.  ... 
doi:10.1080/10556780902917701 fatcat:imk4sacl6nbqna4uivjej4ei7a

Decomposition into low-rank plus additive matrices for background/foreground separation: A review for a comparative evaluation with a large-scale dataset

Thierry Bouwmans, Andrews Sobral, Sajid Javed, Soon Ki Jung, El-Hadi Zahzah
2017 Computer Science Review  
However, similar robust implicit or explicit decompositions can be made in the following problem formulations: Robust Non-negative Matrix Factorization (RNMF), Robust Matrix Completion (RMC), Robust Subspace  ...  testing and ranking existing algorithms for background/foreground separation.  ...  thank the following researchers: Zhouchen Lin (Visual Computing Group, Microsoft Research Asia) who has kindly provided the solver LADMAP [192] and the l 1 -filtering [196] , Shiqian Ma (Institute for  ... 
doi:10.1016/j.cosrev.2016.11.001 fatcat:vdh7ic4n6zfkjlccnyiq74z5wu

Conic Scan-and-Cover algorithms for nonparametric topic modeling [article]

Mikhail Yurochkin, Aritra Guha, XuanLong Nguyen
2017 arXiv   pre-print
We propose new algorithms for topic modeling when the number of topics is unknown.  ...  Our approach relies on an analysis of the concentration of mass and angular geometry of the topic simplex, a convex polytope constructed by taking the convex hull of vertices representing the latent topics  ...  A notable recent algorithm coming from the matrix factorization perspective is RecoverKL (Arora et al., 2012) , which solves non-negative matrix factorization (NMF) efficiently under assumptions on the  ... 
arXiv:1710.02952v1 fatcat:aw5amcszszgh7l3s66eshps7nu

Conic Linear Programming [chapter]

David G. Luenberger, Yinyu Ye
2016 International Series in Operations Research and Management Science  
i ii Preface This monograph is developed for MS&E 314, "Conic Linear Programming", which I am teaching at Stanford.  ...  Then came the primal-dual algorithm of many authors, the iii iv PREFACE SDP approximation algorithm for Max-Cut, ... -SDP eventually established its full popularity.  ...  Example 1.6 The non-negative orthant is a polyhedral cone, and neither the positive semi-definite matrix cone nor the second-order cone is polyhedral.  ... 
doi:10.1007/978-3-319-18842-3_6 fatcat:dsna7n4nprfv3jq6wjwnf5lzda

Rank-One NMF-Based Initialization for NMF and Relative Error Bounds Under a Geometric Assumption

Zhaoqiang Liu, Vincent Y. F. Tan
2017 IEEE Transactions on Signal Processing  
factorization (NMF).  ...  On face image and hyperspectral imaging datasets, we demonstrate that our algorithm provides an excellent initialization for applying other NMF algorithms at a low computational cost.  ...  The algorithm and the theorem for near-separable case is also presented in [16] .  ... 
doi:10.1109/tsp.2017.2713761 fatcat:nm2grw7sfzfmdhnorsjdmmcfpq

k-Space Deep Learning for Parallel MRI: Application to Time-Resolved MR Angiography [article]

Eunju Cha, Eung Yeop Kim, Jong Chul Ye
2018 arXiv   pre-print
Moreover, the k-space sampling patterns have been specially designed for a specific generalized autocalibrating partial parallel acquisition (GRAPPA) factor so that it is not possible to reduce the number  ...  To address these issues, this paper proposes a novel k-space deep learning approach for parallel MRI.  ...  Interestingly, this conic (nonnegative) coding scheme is known to allow part-by representation, which is the key idea of non-negative matrix factorization (NMF) [28] , [29] , [29] , [30] .  ... 
arXiv:1806.00806v2 fatcat:aj46e3cvbzeojp7aomjo5qtpr4

Convex Hull and Voronoi Diagram of Additively Weighted Points [chapter]

Jean-Daniel Boissonnat, Christophe Delage
2005 Lecture Notes in Computer Science  
We provide a complete description of dynamic algorithms for constructing convex hulls and Voronoi diagrams of additively weighted points of R d .  ...  We present simple algorithms and provide a complete description of all the predicates. The algorithms have been implemented in R 3 and experimental results are reported.  ...  On this non-degenerate input, there is, roughly, a factor 5 slow-down compared to double-precision arithmetic. The input here comes from a direct application of our algorithm.  ... 
doi:10.1007/11561071_34 fatcat:wm2cnl7jnbeztbvw5g7a4wbdna

Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints [article]

Dimitris Bertsimas, Ryan Cory-Wright, Jean Pauphilet
2021 arXiv   pre-print
Our algorithms also supply certifiably near-optimal solutions for larger problem sizes and outperform existing heuristics, by deriving an alternative to the popular nuclear norm relaxation which generalizes  ...  Using currently available spatial branch-and-bound codes, not tailored to projection matrices, we can scale our exact (resp. near-exact) algorithms to matrices with up to 30 (resp. 600) rows/columns.  ...  Acknowledgments: We thank the two anonymous referees and the associate editor for many valuable comments which improved the paper substantially.  ... 
arXiv:2009.10395v2 fatcat:rq6zwnp5vfezbbhcfsegbtkzwu

Construction and validation of a rigorous surface hopping algorithm for conical crossings

Caroline Lasser, Torben Swart, Stefan Teufel
2007 Communications in Mathematical Sciences  
This article presents and evaluates a surface hopping algorithm for time-dependent two-level Schrödinger systems with conically intersecting eigenvalues.  ...  The algorithm implements an asymptotic semigroup for approximating the solution's Wigner function, which was rigorously defined and derived from the Schrödinger equation by two of the authors in previous  ...  Near Γ × R d an effective non-adiabatic transfer of weight between w + t and w − t occurs.  ... 
doi:10.4310/cms.2007.v5.n4.a3 fatcat:tx277gmgpvfjjkr222s7tgba6u

Adaptive cutaways for comprehensible rendering of polygonal scenes

Michael Burns, Adam Finkelstein
2008 ACM SIGGRAPH Asia 2008 papers on - SIGGRAPH Asia '08  
This paper introduces a method for generating cutaway renderings of polygonal scenes at interactive frame rates, using illustrative and non-photorealistic rendering cues to expose objects of interest in  ...  Figure 1 : From left to right, the hull of a galleon is cut away to reveal cannons on the interior. Changing camera updates the cutaway shape, exemplified by the leftmost exposed cannon.  ...  Acknowledgements We would like to thank the paper reviewers, the Tiggraph gang, and Wilmot Li for many helpful suggestions. Many of the models shown herein are from the Google 3D Warehouse.  ... 
doi:10.1145/1457515.1409107 fatcat:co2276mc2vgjzmv2kkahzspqle
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