fa25-cs540 cs540
Lecture 3: Linear Algebra and PCA Lecture 3: Linear Algebra and PCA
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Regularized Estimate
Motivation
Estimating 1D Gaussian Parameters
Gaussian Distribution
Mean Estimate
Variance Estimates
Estimation Theory
Quality of Estimates
Estimation Errors
Mean Squared Error
Bias-Variance Decomposition
Bias-Variance Tradeoff
Correlation vs. Causation
Important Distinction
Example: Yellow Fingers and Lung Cancer
Famous Example: Chocolate and Nobel Prizes
Linear Algebra
Why Linear Algebra in AI/ML?
Vectors
Definition and Interpretations
High-Dimensional Vectors in AI/ML
Matrices
Definition
Interpretations
Indexing
Transposition
Definition
Vector Operations
Addition
Scalar Multiplication
Inner Product (Dot Product)
Outer Product
Vector Norms and Orthogonality
Vector Norm (Length)
Orthogonality
Orthonormal Vectors
Matrix Operations
Matrix Addition
Scalar Multiplication
Matrix-Vector Multiplication
Definition
Application: Neural Networks
Matrix Multiplication
Definition
Important Properties
Identity Matrix
Definition
Matrix Inverses
Definition
Example
Eigenvalues and Eigenvectors
Definition
Intuition
Properties
Dimensionality Reduction
Motivation
Examples
Principal Component Analysis (PCA)
Overview
Goal
PCA Procedure
Projection Intuition
Applications of PCA
Image Compression
Exploratory Data Analysis
Visualization
Other Uses
Variations and Extensions
Practice Problems
Key Takeaways
Coming Up
Further Reading