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Reprentation discovery using Harmonic analysis 1st Edition by Sridhar Mahadevan ISBN 3031004183 9783031004186
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Authors:Representation discovery using harmonic analysis-Morgan; Claypool Publishers (2008) , Author sort:analysis-Morgan, Representation discovery using harmonic & Publishers, Claypool , Published:Published:Jul 2008
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ISBN 10: 3031004183
ISBN 13: 9783031004186
Author: Sridhar Mahadevan
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
Reprentation discovery using Harmonic analysis 1st Edition Table of contents:
- WHAT IS A REPRESENTATION?
- PRINCIPLES OF REPRESENTATION DISCOVERY
- OVERVIEW OF THE BOOK
- Road Map to the Book
- Theory of Basis Construction: Vector Spaces
- Generalizing Fourier and Wavelet Analysis
- Algorithms and Computational Tractability
- Case Studies
- BIBLIOGRAPHICAL REMARKS
- Vector Spaces
- ANALYSIS–SYNTHESIS FRAMEWORK
- Approximating 3D Objects in Computer Graphics
- Abstract Fourier Expansion
- Issues in Basis Construction and Selection
- DUAL BASES
- LINEAR MAPPINGS AND MATRIX REPRESENTATIONS
- INVARIANT SUBSPACES
- Dual Bases and Direct Sum Decompositions
- QR Decomposition and Gram–Schmidt Orthogonalization
- Eigenspace Decomposition
- Singular Value Decomposition
- BASES IN INFINITE-DIMENSIONAL SPACES
- Normed spaces
- Inner Product Spaces
- Banach and Hilbert Spaces
- PROJECTIONS
- Projections onto Finite-Dimensional Spaces
- Projections in Infinite-Dimensional Hilbert Spaces
- Reproducing Kernel Hilbert Spaces
- BIBLIOGRAPHICAL REMARKS
- Fourier Bases on Graphs
- ANALYSIS–SYNTHESIS PERSPECTIVE REVISITED
- Function Approximation Using Laplacian Eigenfunctions
- Analysis–Synthesis Example
- RANDOM WALKS AND THE LAPLACIAN
- Variational Analysis of Laplacian Eigenfunctions
- DIRECTED GRAPH LAPLACIAN
- GRAPH PARTITIONING AND CHEEGER CONSTANTS
- BIBLIOGRAPHICAL REMARKS
- Multiscale Bases on Graphs
- INTRODUCTION
- MULTI-RESOLUTION ANALYSIS AND SYNTHESIS
- Multi-resolution Analysis on Graphs
- Examples of Diffusion Wavelets
- DIFFUSION ANALYSIS
- Basic Setup and Notation
- Multiscale Analysis of Functions and Stochastic Processes
- DIFFUSION WAVELETS
- Construction of Diffusion Wavelets
- Multiscale Compression of a Simple Markov Chain
- Comparison of Eigenfunction and Diffusion Wavelet Bases
- BIBLIOGRAPHICAL REMARKS
- Scaling to Large Spaces
- KRONECKER SUM DECOMPOSITION
- Product Spaces: Complex Graphs from Simple Ones
- SCALING TO LARGE GRAPHS USING APPROXIMATION METHODS
- Kronecker Product Approximation
- Graph Partitioning and Fast Laplacian Solvers
- SCALING TO CONTINUOUS SPACES
- RIEMANNIAN MANIFOLDS
- Manifolds
- Hodge Theorem
- THE NYSTRÖM INTERPOLATION OF EIGENFUNCTIONS
- SAMPLING TECHNIQUES
- EXPLOITING DOMAIN KNOWLEDGE
- BIBLIOGRAPHICAL REMARKS
- Case Study: State-Space Planning
- INTRODUCTION
- MARKOV DECISION PROCESSES
- Hilbert Space Formulation of Value Function Approximation
- Least-Squares Approximation of Action-Value Functions
- REPRESENTATION POLICY ITERATION
- Sample Run of RPI on the Two-Room Environment
- Comparison with Hand-coded Parametric Bases
- SCALING PROTO-VALUE FUNCTIONS: PRODUCT SPACES
- Factored Representation Policy Iteration for Structured Domains
- Experimental Results
- RPI IN CONTINUOUS DOMAINS
- Three Control Tasks
- RPI with On-Policy Sampling
- Comparing PVFs with RBFs on Continuous MDPs
- Policy and Reward-Sensitive PVFs
- Extensions of Proto-Value Functions
- MULTISCALE BASIS CONSTRUCTION FOR MARKOV DECISION PROCESSES
- Preliminaries
- Direct Solution of Bellman’s Equation
- Experiments
- BIBLIOGRAPHICAL REMARKS
- Case Study: Computer Graphics
- INTRODUCTION
- SPECTRAL MESH COMPRESSION: FOURIER VERSUS WAVELET BASES
- APPROXIMATION OF MESH GEOMETRY USING OBJECT-SPECIFIC BASES
- Global Laplacian Eigenfunctions
- Diffusion Wavelet Bases
- SCALING TO LARGE GRAPHS USING GRAPH PARTITIONING
- Computing Local Basis Functions
- MESH COMPRESSION USING FOURIER AND WAVELET BASES
- Compression of Small Objects
- Partition Size Versus Error
- Compression of Large Objects
- SUMMARY
- BIBLIOGRAPHICAL REMARKS
- Case Study: Natural Language
- INTRODUCTION
- FOURIER ANALYSIS OF TEXT
- Term–Document Matrix Representation
- Latent Semantic Indexing
- MULTISCALE ANALYSIS OF TEXT USING DIFFUSION WAVELETS
- Fourier Versus Wavelet Analysis of Text
- Main Algorithm
- Advantages of the Diffusion Wavelet Approach
- EXPERIMENTAL RESULTS
- NIPS Paper Data Set
- Diffusion Model: Test 1
- Diffusion Model: Test 2
- Running Times for Various Approaches
- eBay Discussion Forum Data Set
- CONCLUSIONS
- BIBLIOGRAPHICAL REMARKS
- Future Directions
- COMPRESSED SENSING
- HARMONIC ANALYSIS AND LOGIC
- GROUP REPRESENTATION THEORY
- BIBLIOGRAPHICAL REMARKS
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