Simovici D. Mathematical Analysis for Machine Learning and Data Mining

World Scientific Publishing, 2018. — 968 p. — ISBN 978-981-3229-68-6.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.True PDFSet-Theoretical and Algebraic Preliminaries
PreliminariesSets and Collections
Relations and Functions
Sequences and Collections of Sets
Partially Ordered Sets
Closure and Interior Systems
Algebras and σ-Algebras of Sets
Dissimilarity and Metrics
Elementary Combinatorics
Exercises and Supplements
Bibliographical Comments
Linear SpacesLinear Spaces and Linear Independence
Linear Operators and Functionals
Linear Spaces with Inner Products
Seminorms and Norms
Linear Functionals in Inner Product Spaces
Hyperplanes
Exercises and Supplements
Bibliographical Comments
Algebra of Convex SetsConvex Sets and Affine Subspaces
Operations on Convex Sets
Cones
Extreme Points
Balanced and Absorbing Sets
Polytopes and Polyhedra
Exercises and Supplements
Bibliographical Comments
Topology
TopologyTopologies
Closure and Interior Operators in Topological Spaces
Neighborhoods
Bases
Compactness
Separation Hierarchy
Locally Compact Spaces
Limits of Functions
Nets
Continuous Functions
Homeomorphisms
Connected Topological Spaces
Products of Topological Spaces
Semicontinuous Functions
The Epigraph and the Hypograph of a Function
Exercises and Supplements
Bibliographical Comments
Metric Space TopologiesSequences in Metric Spaces
Limits of Functions on Metric Spaces
Continuity of Functions between Metric Spaces
Separation Properties of Metric Spaces
Completeness of Metric Spaces
Pointwise and Uniform Convergence
The Stone-Weierstrass Theorem
Totally Bounded Metric Spaces
Contractions and Fixed Points
The Hausdorff Metric Hyperspace of Compact Subsets
The Topological Space (R, O)
Series and Schauder Bases
Equicontinuity
Exercises and Supplements
Bibliographical Comments
Topological Linear SpacesTopologies of Linear Spaces
Topologies on Inner Product Spaces
Locally Convex Linear Spaces
Continuous Linear Operators
Linear Operators on Normed Linear Spaces
Topological Aspects of Convex Sets
The Relative Interior
Separation of Convex Sets
Theorems of Alternatives
The Contingent Cone
Extreme Points and Krein-Milman Theorem
Exercises and Supplements
Bibliographical Comments
Measure and Integration
Measurable Spaces and MeasuresMeasurable Spaces
Borel Sets
Measurable Functions
Measures and Measure Spaces
Outer Measures
The Lebesgue Measure on Rn
Measures on Topological Spaces
Measures in Metric Spaces
Signed and Complex Measures
Probability Spaces
Exercises and Supplements
Bibliographical Comments
IntegrationThe Lebesgue Integral
The Dominated Convergence Theorem
Functions of Bounded Variation
Riemann Integral vs. Lebesgue Integral
The Radon-Nikodym Theorem
Integration on Products of Measure Spaces
The Riesz-Markov-Kakutani Theorem
Integration Relative to Signed Measures and Complex Measures
Indefinite Integral of a Function
Convergence in Measure
Lp and Lp Spaces
Fourier Transforms of Measures
Lebesgue-Stieltjes Measures and Integrals
Distributions of Random Variables
Random Vectors
Exercises and Supplements
Bibliographical Comments
Functional Analysis and Convexity
Banach SpacesBanach Spaces — Examples
Linear Operators on Banach Spaces
Compact Operators
Duals of Normed Linear Spaces
Spectra of Linear Operators on Banach Spaces
Exercises and Supplements
Bibliographical Comments
Differentiability of Functions Defined on Normed SpacesThe Fréchet and Gâteaux Differentiation
Taylor’s Formula
The Inverse Function Theorem in Rn
Normal and Tangent Subspaces for Surfaces in Rn
Exercises and Supplements
Bibliographical Comments
Hilbert SpacesHilbert Spaces — Examples
Classes of Linear Operators in Hilbert Spaces
Orthonormal Sets in Hilbert Spaces
The Dual Space of a Hilbert Space
Weak Convergence
Spectra of Linear Operators on Hilbert Spaces
Functions of Positive and Negative Type
Reproducing Kernel Hilbert Spaces
Positive Operators in Hilbert Spaces
Exercises and Supplements
Bibliographical Comments
Convex FunctionsConvex Functions — Basics
Constructing Convex Functions
Extrema of Convex Functions
Differentiability and Convexity
Quasi-Convex and Pseudo-Convex Functions
Convexity and Inequalities
Subgradients
Exercises and Supplements
Bibliographical Comments
Applications
OptimizationLocal Extrema, Ascent and Descent Directions
General Optimization Problems
Optimization without Differentiability
Optimization with Differentiability
Duality
Strong Duality
Exercises and Supplements
Bibliographical Comments
Iterative AlgorithmsNewton’s Method
The Secant Method
Newton’s Method in Banach Spaces
Conjugate Gradient Method
Gradient Descent Algorithm
Stochastic Gradient Descent
Exercises and Supplements
Bibliographical Comments
Neural NetworksNeurons
Neural Networks
Neural Networks as Universal Approximators
Weight Adjustment by Back Propagation
Exercises and Supplements
Bibliographical Comments
RegressionLinear Regression
A Statistical Model of Linear Regression
Logistic Regression
Ridge Regression
Lasso Regression and Regularization
Exercises and Supplements
Bibliographical Comments
Support Vector MachinesLinearly Separable Data Sets
Soft Support Vector Machines
Non-linear Support Vector Machines
Perceptrons
Exercises and Supplements
Bibliographical Comments