Machine Learning/Learning Center/Dimensionality Reduction

Dimensionality Reduction & Metric Learning

Master techniques to solve the curse of dimensionality. Learn how to map high-dimensional data to low-dimensional spaces while preserving essential structure, from linear methods (PCA) to nonlinear manifold learning (Isomap, LLE) and metric learning.

k-Nearest Neighbors (kNN)

Module 1

Learn the lazy learning algorithm that serves as a key application scenario for dimensionality reduction. Understand kNN classification and regression, error rate analysis, and how dimensionality affects neighbor finding performance.

Topics Covered:

kNN Algorithm & Lazy Learning

Classification & Regression

Error Rate Analysis

Nearest Neighbor Theory

Connection to Dimensionality Reduction

Low-Dimensional Embedding & Curse of Dimensionality

Module 2

Understand the fundamental problem of high-dimensional data. Learn about the curse of dimensionality, why sample density decreases exponentially, and the core logic of finding low-dimensional embeddings that preserve essential structure.

Topics Covered:

Curse of Dimensionality Definition

Sample Density in High Dimensions

Distance Computation Challenges

Low-Dimensional Embedding Goals

Structure Preservation Principles

Multidimensional Scaling (MDS)

Module 3

Master the foundational linear dimensionality reduction method. Learn how MDS preserves pairwise distances, derives inner product matrices from distance matrices, and finds optimal low-dimensional representations through eigenvalue decomposition.

Topics Covered:

MDS Core Assumptions

Distance Matrix to Inner Product

Centering & Global Statistics

Eigenvalue Decomposition

Low-Dimensional Mapping

Principal Component Analysis (PCA)

Module 4

Master the most classic linear dimensionality reduction method. Learn PCA's dual criteria (minimum reconstruction error and maximum separability), optimization derivation, algorithm steps, and methods for selecting the optimal dimensionality.

Topics Covered:

PCA Core Criteria

Minimum Reconstruction Error

Maximum Separability

Covariance Matrix & Eigenvectors

Dimensionality Selection Methods

Kernel PCA

Module 5

Extend PCA to handle nonlinear data through the kernel trick. Learn how to map data to high-dimensional feature spaces implicitly, perform PCA in feature space, and apply kernel functions without explicit high-dimensional computation.

Topics Covered:

Kernel Trick for Nonlinear Data

Feature Space Mapping

Kernel Matrix Construction

Implicit High-Dimensional PCA

New Sample Projection

Manifold Learning Overview

Module 6

Introduction to nonlinear dimensionality reduction through manifold learning. Understand the core assumption that high-dimensional data lies on low-dimensional manifolds, and learn when to use manifold learning versus linear methods.

Topics Covered:

Manifold Learning Assumptions

Local vs Global Structure

Nonlinear vs Linear Methods

Isomap vs LLE Comparison

When to Use Manifold Learning

Isomap

Module 7

Master isometric mapping for preserving global geodesic distances. Learn how Isomap constructs neighbor graphs, computes shortest paths as geodesic distances, and applies MDS to find low-dimensional embeddings that preserve manifold structure.

Topics Covered:

Geodesic Distance Concept

Neighbor Graph Construction

Shortest Path Algorithms

Global Structure Preservation

Swiss Roll Example

Locally Linear Embedding (LLE)

Module 8

Learn how LLE preserves local linear reconstruction relationships. Master the algorithm that finds local neighborhoods, computes reconstruction weights, and solves for low-dimensional coordinates that maintain local geometry through eigenvalue decomposition.

Topics Covered:

Local Linear Reconstruction

Neighborhood Selection

Weight Matrix Computation

Eigenvalue Decomposition

Local Structure Preservation

Metric Learning

Module 9

Elevate from dimensionality reduction to learning optimal distance metrics. Understand the evolution from Euclidean to weighted Euclidean to Mahalanobis distance, learn NCA and constrained metric learning, and discover how metric learning optimizes for task-specific objectives.

Topics Covered:

Distance Metric Evolution

Mahalanobis Distance

Metric Matrix Constraints

Neighborhood Component Analysis (NCA)

Supervised Metric Learning

Suggested Learning Paths

Fundamentals Path

Start with kNN and the curse of dimensionality

k-Nearest Neighbors
Low-Dimensional Embedding
MDS

Linear Reduction Path

Master linear dimensionality reduction

MDS
PCA
Kernel PCA

Nonlinear Reduction Path

Explore manifold learning methods

Manifold Overview
Isomap
LLE
Metric Learning

Why Learn Dimensionality Reduction?

Solve the Curse of Dimensionality

High-dimensional data suffers from sparse samples, meaningless distances, and poor generalization. Dimensionality reduction addresses these fundamental challenges.

Improve Model Performance

Reducing dimensions can improve kNN accuracy, reduce overfitting, speed up training, and enhance visualization for better insights.

Industry Applications

Essential for image processing, natural language processing, recommendation systems, and any domain with high-dimensional feature spaces.

Foundation for Advanced ML

Understanding dimensionality reduction is crucial for deep learning, feature engineering, and modern machine learning pipelines.

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