Introduction to Lattice Algebra: With Applications in AI, Pattern Recognition, Image Analysis, and Biomimetic Neural Networks
- Length: 432 pages
- Edition: 1
- Language: English
- Publisher: Chapman and Hall/CRC
- Publication Date: 2021-08-24
- ISBN-10: 0367720299
- ISBN-13: 9780367720292
- Sales Rank: #2672170 (See Top 100 Books)
Lattice theory extends into virtually every branch of mathematics, ranging from measure theory and convex geometry to probability theory and topology. A more recent development has been the rapid escalation of employing lattice theory for various applications outside the domain of pure mathematics. These applications range from electronic communication theory and gate array devices that implement Boolean logic to artificial intelligence and computer science in general.
Introduction to Lattice Algebra: With Applications in AI, Pattern Recognition, Image Analysis, and Biomimetic Neural Networks lays emphasis on two subjects, the first being lattice algebra and the second the practical applications of that algebra. This textbook is intended to be used for a special topics course in artificial intelligence with a focus on pattern recognition, multispectral image analysis, and biomimetic artificial neural networks. The book is self-contained and – depending on the student’s major – can be used for a senior undergraduate level or first-year graduate level course. The book is also an ideal self-study guide for researchers and professionals in the above-mentioned disciplines.
Features
- Filled with instructive examples and exercises to help build understanding
- Suitable for researchers, professionals and students, both in mathematics and computer science
- Every chapter consists of exercises with solution provided online at www.Routledge.com/9780367720292
Cover Half Title Title Page Copyright Page Dedication Contents Preface CHAPTER 1: Elements of Algebra 1.1. SETS, FUNCTIONS, AND NOTATION 1.1.1. Special Sets and Families of Sets 1.1.2. Functions 1.1.3. Finite, Countable, and Uncountable Sets 1.2. ALGEBRAIC STRUCTURES 1.2.1. Operations on Sets 1.2.2. Semigroups and Groups 1.2.3. Rings and Fields 1.2.4. Vector Spaces 1.2.5. Homomorphisms and Linear Transforms CHAPTER 2: Pertinent Properties of Euclidean Space 2.1. ELEMENTARY PROPERTIES OF R 2.1.1. Foundations 2.1.2. Topological Properties of R 2.2. ELEMENTARY PROPERTIES OF EUCLIDEAN SPACES 2.2.1. Metrics on Rn 2.2.2. Topological Spaces 2.2.3. Topological Properties of Rn 2.2.4. Aspects of Rn, Artificial Intelligence, Pattern Recognition, and Artificial Neural Networks CHAPTER 3: Lattice Theory 3.1. HISTORICAL BACKGROUND 3.2. PARTIAL ORDERS AND LATTICES 3.2.1. Order Relations on Sets 3.2.2. Lattices 3.3. RELATIONS WITH OTHER BRANCHES OF MATHEMATICS 3.3.1. Topology and Lattice Theory 3.3.2. Elements of Measure Theory 3.3.3. Lattices and Probability 3.3.4. Fuzzy Lattices and Similarity Measures CHAPTER 4: Lattice Algebra 4.1. LATTICE SEMIGROUPS AND LATTICE GROUPS 4.2. MINIMAX ALGEBRA 4.2.1. Valuations, Metrics, and Measures 4.3. MINIMAX MATRIX THEORY 4.3.1. Lattice Vector Spaces 4.3.2. Lattice Independence 4.3.3. Bases and Dual Bases of l-Vector Spaces 4.4. THE GEOMETRY OF S (X) 4.4.1. Affine Structures in Rn 4.4.2. The Shape of S(X) CHAPTER 5: Matrix-Based Lattice Associative Memories 5.1. HISTORICAL BACKGROUND 5.1.1. The Classical ANN Model 5.2. LATTICE ASSOCIATIVE MEMORIES 5.2.1. Basic Properties of Matrix-Based LAMs 5.2.2. Lattice Auto-Associative Memories 5.2.3. Pattern Recall in the Presence of Noise 5.2.4. Kernels and Random Noise 5.2.5. Bidirectional Associative Memories 5.2.6. Computation of Kernels 5.2.7. Addendum CHAPTER 6: Extreme Points of Data Sets 6.1. RELEVANT CONCEPTS OF CONVEX SET THEORY 6.1.1. Convex Hulls and Extremal Points 6.1.2. Lattice Polytopes 6.2. AFFINE SUBSETS OF EXT(P(X)) 6.2.1. Simplexes and Affine Subspaces of Rn 6.2.2. Analysis of ext(P(X)) ˆ Rn 6.2.2.1. The case n = 6.2.2.2. The case n = 6.2.2.3. The case n >= CHAPTER 7: Image Unmixing and Segmentation 7.1. SPECTRAL ENDMEMBERS AND LINEAR UNMIXING 7.1.1. The Mathematical Basis of the WM-Method 7.1.2. A Validation Test of the WM-Method 7.1.3. Candidate and Final Endmembers 7.2. AVIRIS HYPERSPECTRAL IMAGE EXAMPLES 7.3. ENDMEMBERS AND CLUSTERING VALIDATION INDEXES 7.4. COLOR IMAGE SEGMENTATION 7.4.1. About Segmentation and Clustering 7.4.2. Segmentation Results and Comparisons CHAPTER 8: Lattice-Based Biomimetic Neural Networks 8.1. BIOMIMETIC ARTIFICIAL NEURAL NETWORKS 8.1.1. Biological Neurons and Their Processes 8.1.2. Biomimetic Neurons and Dendrites 8.2. LATTICE BIOMIMETIC NEURAL NETWORKS 8.2.1. Simple Examples of Lattice Biomimetic Neural Networks CHAPTER 9: Learning in Biomimetic Neural Networks 9.1. LEARNING IN SINGLE-LAYER LBNNS 9.1.1. Training Based on Elimination 9.1.2. Training Based on Merging 9.1.3. Training for Multi-Class Recognition 9.1.4. Training Based on Dual Lattice Metrics 9.2. MULTI-LAYER LATTICE BIOMIMETIC NEURAL NETWORKS 9.2.1. Constructing a Multi-Layer DLAM 9.2.2. Learning for Pattern Recognition 9.2.3. Learning Based on Similarity Measures Epilogue Bibliography Index
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