Mathematics for Machine Learning Specialization

Introduction

• Machine learning uses mathematical tools like matrix notation & calculus
• Specialization aims to build intuition, not focus on heavy details
• Ideal for getting confidence & motivation to explore applied ML courses
• Designed for accessibility beyond computer scientists

Course 1: Linear Algebra

Module 1: Vectors and Spaces

• Vectors: Objects that move us around space, can be data attributes
  • Vector Addition: v + u = w
  • Scalar Multiplication: c * v = w
• Dot Product: Measures angle similarity; commutative, distributive over addition
• Magnitudes: Length of a vector via the dot product

Module 2: Matrices

• Matrix Transformation: Rotations, scalings, reflections
• Matrix Compositions: Successive transformations
  • Associative but not commutative
• Inverse Matrices: Matrix A where A * A^-1 = I (identity matrix)
• Determinants: Scalar value representing matrix’s scaling effect
  • Zero determinant means matrix has no inverse

Module 3: Linear Equations

• Gaussian Elimination: Solve linear systems by transforming to row echelon form
• Inverse Calculation: Utilize row operations for matrix inversion
• Special Cases: Handling of non-invertible matrices

Module 4: Eigenvalues and Eigenvectors

• Eigenvalues/Vectors: Vectors unchanged by certain transformations
• Diagonalization: Simplifies raising matrices to powers, T^n = C D^n C^-1
  • Useful for computational optimizations

Module 5: Applications

• Google PageRank: Ranks web pages using link probabilities & eigenvalue problems

Course 2: Multivariate Calculus

Module 1: Fundamentals of Calculus

• Derivatives: Measure of how function changes, computed via rise/run
• Rules: Sum rule, Power rule, Product rule, Chain rule
  • Example: f(x) = x^2, f'(x) = 2x
• Special Cases: 1/x, e^x, trigonometric functions

Module 2: Multivariate Calculus

• Partial Derivatives: Deals with multi-variable functions
• Jacobian Matrix: Vector of first-order partial derivatives
• Hessian: Matrix of second-order partial derivatives, helps find extrema
  • Critical Points: Min/Max points where gradient is zero

Module 3: Optimization

• Gradient Descent: Method to find local minima, uses gradient vectors
• Lagrange Multipliers: Optimize functions with constraints

Module 4: Power Series

• Taylor Series: Approximates functions using polynomials
  • f(x) ≈ f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! +...
• Linearization: Simplified approximation, considering higher-order terms as error

Module 5: Numerical Methods

• Finite Differences: Numerical differentiation using discrete points
• Data Fitting: Using least squares for linear & non-linear models

Course 3: Principal Component Analysis (PCA)

Module 1: Statistical Concepts

• Mean & Variance: Describes data spread
• Covariance: Measures dependency between variables
  • Covariance Matrix: Represents variances & covariances in multiple dimensions

Module 2: Inner Products & Geometry

• Inner Product: Generalizes dot product to compute lengths & angles
  • Basis for defining orthogonality
• Orthogonal Projections: Projects data onto lower-dimensional spaces

Module 3: PCA Derivation

• Objectives: Minimize squared reconstruction error
  • Find principal subspace retaining maximum variance
• Eigen Decomposition: Uses data covariance matrix
  • Principal subspace basis from eigenvectors of largest eigenvalues

Module 4: Practical PCA

• Preprocessing: Centering data, scaling by standard deviation
• Algorithm Steps: Compute covariance, eigen decomposition, project data