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Mathematics of Machine Learning

(MATHS-ML.AJ1)
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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
  • 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

7

Matrices and Graphs

8

Functions

9

Numbers, Sequences, and Series

10

Topology, Limits, and Continuity

11

Differentiation

12

Optimization

13

Integration

14

Multivariable Functions

15

Derivatives and Gradients

16

Optimization in Multiple Variables

17

What is Probability?

18

Random Variables and Distributions

19

The Expected Value

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