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Basis (linear algebra)

About: Basis (linear algebra) is a research topic. Over the lifetime, 14069 publications have been published within this topic receiving 278522 citations. The topic is also known as: Hamel basis & algebraic basis.


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TL;DR: A new online optimization algorithm is proposed, based on stochastic approximations, which scales up gracefully to large data sets with millions of training samples, and extends naturally to various matrix factorization formulations, making it suitable for a wide range of learning problems.
Abstract: Sparse coding--that is, modelling data vectors as sparse linear combinations of basis elements--is widely used in machine learning, neuroscience, signal processing, and statistics. This paper focuses on the large-scale matrix factorization problem that consists of learning the basis set, adapting it to specific data. Variations of this problem include dictionary learning in signal processing, non-negative matrix factorization and sparse principal component analysis. In this paper, we propose to address these tasks with a new online optimization algorithm, based on stochastic approximations, which scales up gracefully to large datasets with millions of training samples, and extends naturally to various matrix factorization formulations, making it suitable for a wide range of learning problems. A proof of convergence is presented, along with experiments with natural images and genomic data demonstrating that it leads to state-of-the-art performance in terms of speed and optimization for both small and large datasets.

2,256 citations

Journal ArticleDOI
TL;DR: The construction of transferable, hierarchical basis sets are demonstrated, allowing the calculation to range from qualitative tight-binding like accuracy to meV-level total energy convergence with the basis set, since all basis functions are strictly localized.

2,178 citations

Journal ArticleDOI
25 Jan 2010-Analyst
TL;DR: The increasing interest in Support Vector Machines (SVMs) over the past 15 years is described, including its application to multivariate calibration, and why it is useful when there are outliers and non-linearities.
Abstract: The increasing interest in Support Vector Machines (SVMs) over the past 15 years is described. Methods are illustrated using simulated case studies, and 4 experimental case studies, namely mass spectrometry for studying pollution, near infrared analysis of food, thermal analysis of polymers and UV/visible spectroscopy of polyaromatic hydrocarbons. The basis of SVMs as two-class classifiers is shown with extensive visualisation, including learning machines, kernels and penalty functions. The influence of the penalty error and radial basis function radius on the model is illustrated. Multiclass implementations including one vs. all, one vs. one, fuzzy rules and Directed Acyclic Graph (DAG) trees are described. One-class Support Vector Domain Description (SVDD) is described and contrasted to conventional two- or multi-class classifiers. The use of Support Vector Regression (SVR) is illustrated including its application to multivariate calibration, and why it is useful when there are outliers and non-linearities.

1,899 citations

Book
02 Dec 2012
TL;DR: Huzinaga et al. as mentioned in this paper provided information pertinent to the Gaussian basis sets, with emphasis on lithium, radon, and important ions, and discussed the polarization functions prepared for lithium through radon for further improvement of the basis sets.
Abstract: Gaussian Basis Sets for Molecular Calculations-S. Huzinaga 2012-12-02 Physical Sciences Data, Volume 16: Gaussian Basis Sets for Molecular Calculations provides information pertinent to the Gaussian basis sets, with emphasis on lithium, radon, and important ions. This book discusses the polarization functions prepared for lithium through radon for further improvement of the basis sets. Organized into three chapters, this volume begins with an overview of the basis set for the most stable negative and positive ions. This text then explores the total atomic energies given by the basis sets. Other chapters consider the distinction between diffuse functions and polarization function. This book presents as well the exponents of polarization function. The final chapter deals with the Gaussian basis sets. This book is a valuable resource for chemists, scientists, and research workers.

1,798 citations

Book
01 Jan 2000
TL;DR: Second Quantization Spin in Second Quantization Orbital Rotations Exact and Approximate Wave Functions The Standard Models Atomic Basis Functions Short-range Interactions and Orbital Expansions Gaussian Basis Sets Molecular Integral Evaluation Hartree-Fock Theory Configuration-Interaction Theory Multiconfigurational Self-Consistent Field Theory Coupled-Cluster Theory Perturbation Theory Calibration of the Electronic-Structure Models List of Acronyms Index
Abstract: Second Quantization Spin in Second Quantization Orbital Rotations Exact and Approximate Wave Functions The Standard Models Atomic Basis Functions Short-Range Interactions and Orbital Expansions Gaussian Basis Sets Molecular Integral Evaluation Hartree-Fock Theory Configuration-Interaction Theory Multiconfigurational Self-Consistent Field Theory Coupled-Cluster Theory Perturbation Theory Calibration of the Electronic-Structure Models List of Acronyms Index

1,740 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20244
20231,321
20222,849
2021714
2020693
2019657