Representation Discovery using Harmonic Analysis
Representation Discovery using Harmonic Analysis

Author: Sridhar López

Publisher: Springer Nature

Published: 2022-05-31

Total Pages: 147

ISBN-13: 3031015460

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Representations are at the heart of artificial intelligence (AI). This book is devoted to the problem of representation discovery: how can an intelligent system construct representations from its experience? Representation discovery re-parameterizes the state space - prior to the application of information retrieval, machine learning, or optimization techniques - facilitating later inference processes by constructing new task-specific bases adapted to the state space geometry. This book presents a general approach to representation discovery using the framework of harmonic analysis, in particular Fourier and wavelet analysis. Biometric compression methods, the compact disc, the computerized axial tomography (CAT) scanner in medicine, JPEG compression, and spectral analysis of time-series data are among the many applications of classical Fourier and wavelet analysis. A central goal of this book is to show that these analytical tools can be generalized from their usual setting in (infinite-dimensional) Euclidean spaces to discrete (finite-dimensional) spaces typically studied in many subfields of AI. Generalizing harmonic analysis to discrete spaces poses many challenges: a discrete representation of the space must be adaptively acquired; basis functions are not pre-defined, but rather must be constructed. Algorithms for efficiently computing and representing bases require dealing with the curse of dimensionality. However, the benefits can outweigh the costs, since the extracted basis functions outperform parametric bases as they often reflect the irregular shape of a particular state space. Case studies from computer graphics, information retrieval, machine learning, and state space planning are used to illustrate the benefits of the proposed framework, and the challenges that remain to be addressed. Representation discovery is an actively developing field, and the author hopes this book will encourage other researchers to explore this exciting area of research. Table of Contents: Overview / Vector Spaces / Fourier Bases on Graphs / Multiscale Bases on Graphs / Scaling to Large Spaces / Case Study: State-Space Planning / Case Study: Computer Graphics / Case Study: Natural Language / Future Directions

Representation Discovery using Harmonic Analysis
Representation Discovery using Harmonic Analysis

Author: Sridhar Mahadevan

Publisher: Morgan & Claypool Publishers

Published: 2008-07-08

Total Pages: 160

ISBN-13: 1598296604

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Representations are at the heart of artificial intelligence (AI). This book is devoted to the problem of representation discovery: how can an intelligent system construct representations from its experience? Representation discovery re-parameterizes the state space - prior to the application of information retrieval, machine learning, or optimization techniques - facilitating later inference processes by constructing new task-specific bases adapted to the state space geometry. This book presents a general approach to representation discovery using the framework of harmonic analysis, in particular Fourier and wavelet analysis. Biometric compression methods, the compact disc, the computerized axial tomography (CAT) scanner in medicine, JPEG compression, and spectral analysis of time-series data are among the many applications of classical Fourier and wavelet analysis. A central goal of this book is to show that these analytical tools can be generalized from their usual setting in (infinite-dimensional) Euclidean spaces to discrete (finite-dimensional) spaces typically studied in many subfields of AI. Generalizing harmonic analysis to discrete spaces poses many challenges: a discrete representation of the space must be adaptively acquired; basis functions are not pre-defined, but rather must be constructed. Algorithms for efficiently computing and representing bases require dealing with the curse of dimensionality. However, the benefits can outweigh the costs, since the extracted basis functions outperform parametric bases as they often reflect the irregular shape of a particular state space. Case studies from computer graphics, information retrieval, machine learning, and state space planning are used to illustrate the benefits of the proposed framework, and the challenges that remain to be addressed. Representation discovery is an actively developing field, and the author hopes this book will encourage other researchers to explore this exciting area of research. Table of Contents: Overview / Vector Spaces / Fourier Bases on Graphs / Multiscale Bases on Graphs / Scaling to Large Spaces / Case Study: State-Space Planning / Case Study: Computer Graphics / Case Study: Natural Language / Future Directions

Unitary Representations and Harmonic Analysis
Unitary Representations and Harmonic Analysis

Author: M. Sugiura

Publisher: Elsevier

Published: 1990-03-01

Total Pages: 469

ISBN-13: 0080887597

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The principal aim of this book is to give an introduction to harmonic analysis and the theory of unitary representations of Lie groups. The second edition has been brought up to date with a number of textual changes in each of the five chapters, a new appendix on Fatou's theorem has been added in connection with the limits of discrete series, and the bibliography has been tripled in length.

Computational Music Analysis
Computational Music Analysis

Author: David Meredith

Publisher: Springer

Published: 2015-10-27

Total Pages: 483

ISBN-13: 3319259318

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This book provides an in-depth introduction and overview of current research in computational music analysis. Its seventeen chapters, written by leading researchers, collectively represent the diversity as well as the technical and philosophical sophistication of the work being done today in this intensely interdisciplinary field. A broad range of approaches are presented, employing techniques originating in disciplines such as linguistics, information theory, information retrieval, pattern recognition, machine learning, topology, algebra and signal processing. Many of the methods described draw on well-established theories in music theory and analysis, such as Forte's pitch-class set theory, Schenkerian analysis, the methods of semiotic analysis developed by Ruwet and Nattiez, and Lerdahl and Jackendoff's Generative Theory of Tonal Music. The book is divided into six parts, covering methodological issues, harmonic and pitch-class set analysis, form and voice-separation, grammars and hierarchical reduction, motivic analysis and pattern discovery and, finally, classification and the discovery of distinctive patterns. As a detailed and up-to-date picture of current research in computational music analysis, the book provides an invaluable resource for researchers, teachers and students in music theory and analysis, computer science, music information retrieval and related disciplines. It also provides a state-of-the-art reference for practitioners in the music technology industry.

A Wavelet Tour of Signal Processing
A Wavelet Tour of Signal Processing

Author: Stephane Mallat

Publisher: Elsevier

Published: 1999-09-14

Total Pages: 663

ISBN-13: 0080520839

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This book is intended to serve as an invaluable reference for anyone concerned with the application of wavelets to signal processing. It has evolved from material used to teach "wavelet signal processing" courses in electrical engineering departments at Massachusetts Institute of Technology and Tel Aviv University, as well as applied mathematics departments at the Courant Institute of New York University and ÉcolePolytechnique in Paris. - Provides a broad perspective on the principles and applications of transient signal processing with wavelets - Emphasizes intuitive understanding, while providing the mathematical foundations and description of fast algorithms - Numerous examples of real applications to noise removal, deconvolution, audio and image compression, singularity and edge detection, multifractal analysis, and time-varying frequency measurements - Algorithms and numerical examples are implemented in Wavelab, which is a Matlab toolbox freely available over the Internet - Content is accessible on several level of complexity, depending on the individual reader's needs New to the Second Edition - Optical flow calculation and video compression algorithms - Image models with bounded variation functions - Bayes and Minimax theories for signal estimation - 200 pages rewritten and most illustrations redrawn - More problems and topics for a graduate course in wavelet signal processing, in engineering and applied mathematics

Explorations in Harmonic Analysis
Explorations in Harmonic Analysis

Author: Steven G. Krantz

Publisher: Springer Science & Business Media

Published: 2009-05-24

Total Pages: 367

ISBN-13: 0817646698

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This self-contained text provides an introduction to modern harmonic analysis in the context in which it is actually applied, in particular, through complex function theory and partial differential equations. It takes the novice mathematical reader from the rudiments of harmonic analysis (Fourier series) to the Fourier transform, pseudodifferential operators, and finally to Heisenberg analysis.