Mathematical Analysis for Machine Learning and Data Mining

The following sections are included:

• Preface
• Contents

The following sections are included:

• Introduction
• Sets 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

The following sections are included:

• Introduction
• Linear 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

The following sections are included:

• Introduction
• Convex Sets and Affine Subspaces
• Operations on Convex Sets
• Cones
• Extreme Points
• Balanced and Absorbing Sets
• Polytopes and Polyhedra
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Topologies
• 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

The following sections are included:

• Introduction
• Sequences 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, $\mathcal{O}$)
• Series and Schauder Bases
• Equicontinuity
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Topologies 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

The following sections are included:

• Introduction
• Measurable 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

The following sections are included:

• Introduction
• The Lebesgue Integral
  • The Integral of Simple Measurable Functions
  • The Integral of Non-negative Measurable Functions
  • The Integral of Real-Valued Measurable Functions
  • The Integral of Complex-Valued Measurable Functions
• 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
• $\mathcal{L}$^p and L^p Spaces
• Fourier Transforms of Measures
• Lebesgue-Stieltjes Measures and Integrals
• Distributions of Random Variables
• Random Vectors
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Banach 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

The following sections are included:

• Introduction
• The 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

The following sections are included:

• Introduction
• Hilbert Spaces — Examples
• Classes of Linear Operators in Hilbert Spaces
  • Self-Adjoint Operators
  • Normal and Unitary Operators
  • Projection Operators
• 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

The following sections are included:

• Introduction
• Convex 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

The following sections are included:

• Introduction
• Local Extrema, Ascent and Descent Directions
• General Optimization Problems
• Optimization without Differentiability
• Optimization with Differentiability
• Duality
• Strong Duality
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Newton’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

The following sections are included:

• Introduction
• Neurons
• Neural Networks
• Neural Networks as Universal Approximators
• Weight Adjustment by Back Propagation
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Linear Regression
• A Statistical Model of Linear Regression
• Logistic Regression
• Ridge Regression
• Lasso Regression and Regularization
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Introduction
• Linearly Separable Data Sets
• Soft Support Vector Machines
• Non-linear Support Vector Machines
• Perceptrons
• Exercises and Supplements
• Bibliographical Comments

The following sections are included:

• Bibliography
• Index