Mathematical Analysis for Machine Learning and Data Mining 1st edition by Dan Simovici – Ebook PDF Instant Download/Delivery: 9813229683 , 978-9813229686
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ISBN 10: 9813229683
ISBN 13: 978-9813229686
Author: Dan Simovici
This compendium provides a self-contained introduction to mathematical analysis in the field of machine learning and data mining. The mathematical analysis component of the typical mathematical curriculum for computer science students omits these very important ideas and techniques which are indispensable for approaching specialized area of machine learning centered around optimization such as support vector machines, neural networks, various types of regression, feature selection, and clustering. The book is of special interest to researchers and graduate students who will benefit from these application areas discussed in the book. Related Link(s)
Mathematical Analysis for Machine Learning and Data Mining 1st Table of contents:
Part I. Set-Theoretical and Algebraic Preliminaries
1. Preliminaries
1.1 Introduction
1.2 Sets and Collections
1.3 Relations and Functions
1.4 Sequences and Collections of Sets
1.5 Partially Ordered Sets
1.6 Closure and Interior Systems
1.7 Algebras and σ-Algebras of Sets
1.8 Dissimilarity and Metrics
1.9 Elementary Combinatorics
Exercises and Supplements
Bibliographical Comments
2. Linear Spaces
2.1 Introduction
2.2 Linear Spaces and Linear Independence
2.3 Linear Operators and Functionals
2.4 Linear Spaces with Inner Products
2.5 Seminorms and Norms
2.6 Linear Functionals in Inner Product Spaces
2.7 Hyperplanes
Exercises and Supplements
Bibliographical Comments
3. Algebra of Convex Sets
3.1 Introduction
3.2 Convex Sets and Affine Subspaces
3.3 Operations on Convex Sets
3.4 Cones
3.5 Extreme Points
3.6 Balanced and Absorbing Sets
3.7 Polytopes and Polyhedra
Exercises and Supplements
Bibliographical Comments
Part II. Topology
4. Topology
4.1 Introduction
4.2 Topologies
4.3 Closure and Interior Operators in Topological Spaces
4.4 Neighborhoods
4.5 Bases
4.6 Compactness
4.7 Separation Hierarchy
4.8 Locally Compact Spaces
4.9 Limits of Functions
4.10 Nets
4.11 Continuous Functions
4.12 Homeomorphisms
4.13 Connected Topological Spaces
4.14 Products of Topological Spaces
4.15 Semicontinuous Functions
4.16 The Epigraph and the Hypograph of a Function
Exercises and Supplements
Bibliographical Comments
5. Metric Space Topologies
5.1 Introduction
5.2 Sequences in Metric Spaces
5.3 Limits of Functions on Metric Spaces
5.4 Continuity of Functions between Metric Spaces
5.5 Separation Properties of Metric Spaces
5.6 Completeness of Metric Spaces
5.7 Pointwise and Uniform Convergence
5.8 The Stone-Weierstrass Theorem
5.9 Totally Bounded Metric Spaces
5.10 Contractions and Fixed Points
5.11 The Hausdorff Metric Hyperspace of Compact Subsets
5.12 The Topological Space (R, O)
5.13 Series and Schauder Bases
5.14 Equicontinuity
Exercises and Supplements
Bibliographical Comments
6. Topological Linear Spaces
6.1 Introduction
6.2 Topologies of Linear Spaces
6.3 Topologies on Inner Product Spaces
6.4 Locally Convex Linear Spaces
6.5 Continuous Linear Operators
6.6 Linear Operators on Normed Linear Spaces
6.7 Topological Aspects of Convex Sets
6.8 The Relative Interior
6.9 Separation of Convex Sets
6.10 Theorems of Alternatives
6.11 The Contingent Cone
6.12 Extreme Points and Krein-Milman Theorem
Exercises and Supplements
Bibliographical Comments
Part III. Measure and Integration
7. Measurable Spaces and Measures
7.1 Introduction
7.2 Measurable Spaces
7.3 Borel Sets
7.4 Measurable Functions
7.5 Measures and Measure Spaces
7.6 Outer Measures
7.7 The Lebesgue Measure on Rn
7.8 Measures on Topological Spaces
7.9 Measures in Metric Spaces
7.10 Signed and Complex Measures
7.11 Probability Spaces
Exercises and Supplements
Bibliographical Comments
8. Integration
8.1 Introduction
8.2 The Lebesgue Integral
8.2.1 The Integral of Simple Measurable Functions
8.2.2 The Integral of Non-negative Measurable Functions
8.2.3 The Integral of Real-Valued Measurable Functions
8.2.4 The Integral of Complex-Valued Measurable Functions
8.3 The Dominated Convergence Theorem
8.4 Functions of Bounded Variation
8.5 Riemann Integral vs. Lebesgue Integral
8.6 The Radon-Nikodym Theorem
8.7 Integration on Products of Measure Spaces
8.8 The Riesz-Markov-Kakutani Theorem
8.9 Integration Relative to Signed Measures and Complex Measures
8.10 Indefinite Integral of a Function
8.11 Convergence in Measure
8.12 Lp and Lp Spaces
8.13 Fourier Transforms of Measures
8.14 Lebesgue-Stieltjes Measures and Integrals
8.15 Distributions of Random Variables
8.16 Random Vectors
Exercises and Supplements
Bibliographical Comments
Part IV. Functional Analysis and Convexity
9. Banach Spaces
9.1 Introduction
9.2 Banach Spaces — Examples
9.3 Linear Operators on Banach Spaces
9.4 Compact Operators
9.5 Duals of Normed Linear Spaces
9.6 Spectra of Linear Operators on Banach Spaces
Exercises and Supplements
Bibliographical Comments
10. Differentiability of Functions Defined on Normed Spaces
10.1 Introduction
10.2 The Fréchet and Gâteaux Differentiation
10.3 Taylor’s Formula
10.4 The Inverse Function Theorem in Rn
10.5 Normal and Tangent Subspaces for Surfaces in Rn
Exercises and Supplements
Bibliographical Comments
11. Hilbert Spaces
11.1 Introduction
11.2 Hilbert Spaces — Examples
11.3 Classes of Linear Operators in Hilbert Spaces
11.3.1 Self-Adjoint Operators
11.3.2 Normal and Unitary Operators
11.3.3 Projection Operators
11.4 Orthonormal Sets in Hilbert Spaces
11.5 The Dual Space of a Hilbert Space
11.6 Weak Convergence
11.7 Spectra of Linear Operators on Hilbert Spaces
11.8 Functions of Positive and Negative Type
11.9 Reproducing Kernel Hilbert Spaces
11.10 Positive Operators in Hilbert Spaces
Exercises and Supplements
Bibliographical Comments
12. Convex Functions
12.1 Introduction
12.2 Convex Functions — Basics
12.3 Constructing Convex Functions
12.4 Extrema of Convex Functions
12.5 Differentiability and Convexity
12.6 Quasi-Convex and Pseudo-Convex Functions
12.7 Convexity and Inequalities
12.8 Subgradients
Exercises and Supplements
Bibliographical Comments
Part V. Applications
13. Optimization
13.1 Introduction
13.2 Local Extrema, Ascent and Descent Directions
13.3 General Optimization Problems
13.4 Optimization without Differentiability
13.5 Optimization with Differentiability
13.6 Duality
13.7 Strong Duality
Exercises and Supplements
Bibliographical Comments
14. Iterative Algorithms
14.1 Introduction
14.2 Newton’s Method
14.3 The Secant Method
14.4 Newton’s Method in Banach Spaces
14.5 Conjugate Gradient Method
14.6 Gradient Descent Algorithm
14.7 Stochastic Gradient Descent
Exercises and Supplements
Bibliographical Comments
15. Neural Networks
15.1 Introduction
15.2 Neurons
15.3 Neural Networks
15.4 Neural Networks as Universal Approximators
15.5 Weight Adjustment by Back Propagation
Exercises and Supplements
Bibliographical Comments
16. Regression
16.1 Introduction
16.2 Linear Regression
16.3 A Statistical Model of Linear Regression
16.4 Logistic Regression
16.5 Ridge Regression
16.6 Lasso Regression and Regularization
Exercises and Supplements
Bibliographical Comments
17. Support Vector Machines
17.1 Introduction
17.2 Linearly Separable Data Sets
17.3 Soft Support Vector Machines
17.4 Non-linear Support Vector Machines
17.5 Perceptrons
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