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Mathematics of Machine Learning

(MATHS-ML.AJ1) / ISBN : 979-8-90059-011-0
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1

Introduction

  • What is this course about?
  • How to read this course
  • Conventions used
  • What this course covers
2

Vectors and Vector Spaces

  • What is a vector space?
  • The basis
  • Vectors in practice
  • Summary
  • Problems
3

The Geometric Structure of Vector Spaces

  • Norms and distances
  • Inner products, angles, and lots of reasons to care about them
  • Summary
  • Problems
4

Linear Algebra in Practice

  • Vectors in NumPy
  • Matrices, the workhorses of linear algebra
  • Summary
  • Problems
5

Linear Transformations

  • What is a linear transformation?
  • Change of basis
  • Linear transformations in the Euclidean plane
  • Determinants, or how linear transformations affect volume
  • Summary
  • Problems
6

Matrices and Equations

  • Linear equations
  • The LU decomposition
  • Determinants in practice
  • Summary
  • Problems
7

Eigenvalues and Eigenvectors

  • Eigenvalues of matrices
  • Finding eigenvalue-eigenvector pairs
  • Eigenvectors, eigenspaces, and their bases
  • Summary
  • Problems
8

Matrix Factorizations

  • Special transformations
  • Self-adjoint transformations and the spectral decomposition theorem
  • The singular value decomposition
  • Orthogonal projections
  • Computing eigenvalues
  • The QR algorithm
  • Summary
  • Problems
9

Matrices and Graphs

  • The directed graph of a nonnegative matrix
  • Benefits of the graph representation
  • The Frobenius normal form
  • Summary
  • Problems
10

Functions

  • Functions in theory
  • Functions in practice
  • Summary
  • Problems
11

Numbers, Sequences, and Series

  • Numbers
  • Sequences
  • Series
  • Summary
  • Problems
12

Topology, Limits, and Continuity

  • Topology
  • Limits
  • Continuity
  • Summary
  • Problems
13

Differentiation

  • Differentiation in theory
  • Differentiation in practice
  • Summary
  • Problems
14

Optimization

  • Minima, maxima, and derivatives
  • The basics of gradient descent
  • Why does gradient descent work?
  • Summary
  • Problems
15

Integration

  • Integration in theory
  • Integration in practice
  • Summary
  • Problems
16

Multivariable Functions

  • What is a multivariable function?
  • Linear functions in multiple variables
  • The curse of dimensionality
  • Summary
17

Derivatives and Gradients

  • Partial and total derivatives
  • Derivatives of vector-valued functions
  • Summary
  • Problems
18

Optimization in Multiple Variables

  • Multivariable functions in code
  • Minima and maxima, revisited
  • Gradient descent in its full form
  • Summary
  • Problems
19

What is Probability?

  • The language of thinking
  • The axioms of probability
  • Conditional probability
  • Summary
  • Problems
20

Random Variables and Distributions

  • Random variables
  • Discrete distributions
  • Real-valued distributions
  • Density functions
  • Summary
  • Problems
21

The Expected Value

  • Discrete random variables
  • Continuous random variables
  • Properties of the expected value
  • Variance
  • The law of large numbers
  • Information theory
  • The Maximum Likelihood Estimation
  • Summary
  • Problems
A

Appendix A: It’s Just Logic

  • Mathematical logic 101
  • Logical connectives
  • The propositional calculus
  • Variables and predicates
  • Existential and universal quantification
  • Problems
B

Appendix B: The Structure of Mathematics

  • What is a definition?
  • What is a theorem?
  • What is a proof?
  • Equivalences
  • Proof techniques
C

Appendix C: Basics of Set Theory

  • What is a set?
  • Operations on sets
  • The Cartesian product
  • The cardinality of sets
  • The Russell paradox (optional)
D

Appendix D: Complex Numbers

  • The definition of complex numbers
  • The geometric representation
  • The fundamental theorem of algebra
  • Why are complex numbers important?

1

Vectors and Vector Spaces

  • Implementing Tuple and List Operations
  • Performing NumPy Array and Vector Operations
2

The Geometric Structure of Vector Spaces

  • Analyzing Vectors and Distances
3

Linear Algebra in Practice

  • Evaluating Vector Norms and Operations
  • Applying Matrix Computations Using NumPy
  • Representing Images and Text Using Vectors and Matrices
4

Matrices and Equations

  • Solving Linear Equations Using Gaussian Elimination
  • Solving Linear Equations Using Determinants and Inverses
  • Solving Linear Models in Machine Learning
  • Performing LU Decomposition
  • Computing the Determinant Using LU Decomposition
5

Eigenvalues and Eigenvectors

  • Finding Eigenvalues and Eigenvectors of Matrices
  • Analyzing Matrices Using Characteristic Polynomials
  • Visualizing Eigenvectors and Eigenspaces in Linear Algebra
6

Matrix Factorizations

  • Implementing Spectral Decomposition and PCA
  • Performing Feature Extraction and Dimensionality Reduction
  • Performing Singular Value Decomposition
  • Implementing QR Decomposition
7

Matrices and Graphs

8

Functions

  • Implementing Callable Functions
9

Numbers, Sequences, and Series

  • Visualizing Mathematical Sequences and Approximations
  • Visualizing the Harmonic Series
10

Topology, Limits, and Continuity

  • Analyzing Openness, Closedness, and Compactness of Sets
11

Differentiation

  • Applying the Chain Rule
12

Optimization

  • Implementing Gradient Descent for Model Training
13

Integration

  • Approximating Integrals Using the Trapezoidal Rule
14

Multivariable Functions

15

Derivatives and Gradients

16

Optimization in Multiple Variables

  • Training Machine Learning Models with Gradient Descent
17

What is Probability?

18

Random Variables and Distributions

19

The Expected Value

Mathematics of Machine Learning

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