📐 MemoLearning Dimensionality Reduction

Reduce feature complexity while preserving essential information and patterns

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Dimensionality Reduction Curriculum

11
Core Units
~70
DR Techniques
12+
Algorithms
20+
Applications
1

Introduction to Dimensionality Reduction

Understand the need for dimensionality reduction and its role in machine learning and data analysis.

  • Curse of dimensionality
  • High-dimensional data challenges
  • Linear vs non-linear methods
  • Feature selection vs feature extraction
  • Supervised vs unsupervised DR
  • Visualization applications
  • Computational benefits
  • Information preservation
2

Principal Component Analysis (PCA)

Master the most fundamental linear dimensionality reduction technique using eigenvalue decomposition.

  • PCA mathematical foundation
  • Covariance matrix
  • Eigenvalues and eigenvectors
  • Principal components
  • Explained variance ratio
  • Choosing number of components
  • Data standardization
  • PCA interpretation
3

Linear Discriminant Analysis (LDA)

Learn supervised dimensionality reduction that maximizes class separability.

  • LDA vs PCA comparison
  • Between-class and within-class scatter
  • Fisher's linear discriminant
  • Maximizing class separation
  • Multiclass LDA
  • Assumptions and limitations
  • Classification applications
  • Regularized LDA
4

t-SNE and UMAP

Explore advanced non-linear techniques for visualization and manifold learning.

  • t-SNE algorithm overview
  • Probabilistic approach
  • Perplexity parameter
  • UMAP principles
  • Topological data analysis
  • Hyperparameter tuning
  • Visualization best practices
  • Computational considerations
5

Manifold Learning

Understand how to discover low-dimensional manifolds embedded in high-dimensional spaces.

  • Manifold hypothesis
  • Locally Linear Embedding (LLE)
  • Isomap algorithm
  • Multidimensional Scaling (MDS)
  • Laplacian eigenmaps
  • Neighborhood preservation
  • Geodesic distances
  • Non-linear dimensionality
6

Feature Selection Methods

Learn techniques to select the most relevant features rather than transforming them.

  • Filter methods
  • Wrapper methods
  • Embedded methods
  • Univariate feature selection
  • Recursive feature elimination
  • L1 regularization
  • Mutual information
  • Feature importance ranking
7

Matrix Factorization

Explore matrix decomposition techniques for dimensionality reduction and data compression.

  • Singular Value Decomposition (SVD)
  • Non-negative Matrix Factorization
  • Truncated SVD
  • Independent Component Analysis
  • Factor analysis
  • Matrix completion
  • Latent factor models
  • Recommender systems
8

Autoencoders

Learn neural network-based approaches for non-linear dimensionality reduction.

  • Autoencoder architecture
  • Encoder-decoder structure
  • Bottleneck layer
  • Variational autoencoders
  • Denoising autoencoders
  • Sparse autoencoders
  • Deep autoencoders
  • Generative applications
9

Evaluation and Validation

Learn methods to evaluate the quality of dimensionality reduction and choose optimal parameters.

  • Reconstruction error
  • Preservation of distances
  • Neighborhood preservation
  • Silhouette analysis
  • Trustworthiness metrics
  • Cross-validation strategies
  • Downstream task performance
  • Visual assessment
10

Text and Image Applications

Apply dimensionality reduction techniques to specific domains like text processing and image analysis.

  • Text preprocessing for DR
  • TF-IDF and word embeddings
  • Latent Semantic Analysis
  • Topic modeling
  • Image dimensionality reduction
  • Face recognition applications
  • Computer vision preprocessing
  • Feature maps visualization
11

Practical Implementation

Implement dimensionality reduction in real-world projects with best practices and optimization.

  • Scikit-learn implementation
  • Preprocessing pipelines
  • Scalability considerations
  • Memory optimization
  • Online and incremental methods
  • Integration with ML pipelines
  • Performance monitoring
  • Production deployment

Unit 1: Introduction to Dimensionality Reduction

Understand the need for dimensionality reduction and its role in machine learning and data analysis.

Curse of Dimensionality

Learn how high-dimensional data creates challenges for machine learning algorithms and data analysis.

Distance Concentration Sparse Data Overfitting
As dimensions increase, data points become increasingly sparse and equidistant, making pattern recognition and clustering more difficult.

High-Dimensional Data Challenges

Understand computational and statistical problems that arise with many features.

import numpy as np
# Demonstrate distance concentration
# In high dimensions, distances become similar
dims = [2, 10, 50, 100]
for d in dims:
  data = np.random.randn(1000, d)
  distances = np.linalg.norm(data, axis=1)
  print(f"Dim {d}: std/mean = {distances.std()/distances.mean():.3f}")

Linear vs Non-linear Methods

Compare linear transformations like PCA with non-linear methods like t-SNE.

Linear methods: PCA, LDA, Factor Analysis
Non-linear methods: t-SNE, UMAP, Autoencoders
Choose based on data structure and goals
from sklearn.decomposition import PCA
from sklearn.manifold import TSNE

# Linear reduction
pca = PCA(n_components=2)
X_pca = pca.fit_transform(X)

# Non-linear reduction
tsne = TSNE(n_components=2)
X_tsne = tsne.fit_transform(X)

Feature Selection vs Feature Extraction

Distinguish between selecting existing features and creating new transformed features.

Feature Selection: Choose subset of original features
Feature Extraction: Create new features as combinations
from sklearn.feature_selection import SelectKBest
from sklearn.decomposition import PCA

# Feature selection - keeps original features
selector = SelectKBest(k=5)
X_selected = selector.fit_transform(X, y)

# Feature extraction - creates new features
pca = PCA(n_components=5)
X_extracted = pca.fit_transform(X)

Supervised vs Unsupervised DR

Learn when to use methods that consider target variables versus those that don't.

Supervised: LDA Unsupervised: PCA
# Unsupervised - doesn't use labels
pca = PCA(n_components=2)
X_pca = pca.fit_transform(X) # Only X, no y

# Supervised - uses labels for better separation
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
lda = LinearDiscriminantAnalysis(n_components=2)
X_lda = lda.fit_transform(X, y) # Uses both X and y

Visualization Applications

Use dimensionality reduction to create 2D and 3D visualizations of high-dimensional data.

import matplotlib.pyplot as plt

# Visualize high-dimensional data in 2D
plt.figure(figsize=(12, 5))

plt.subplot(1, 2, 1)
plt.scatter(X_pca[:, 0], X_pca[:, 1], c=y, cmap='viridis')
plt.title('PCA Visualization')

plt.subplot(1, 2, 2)
plt.scatter(X_tsne[:, 0], X_tsne[:, 1], c=y, cmap='viridis')
plt.title('t-SNE Visualization')

Computational Benefits

Understand how dimensionality reduction improves computational efficiency and storage requirements.

• Faster training and prediction
• Reduced memory usage
• Lower storage requirements
• Improved numerical stability
• Noise reduction

Information Preservation

Learn to balance dimensionality reduction with preserving important information in the data.

# Check how much variance is preserved
pca = PCA()
pca.fit(X)

# Cumulative explained variance
cumsum = np.cumsum(pca.explained_variance_ratio_)
n_components = np.argmax(cumsum >= 0.95) + 1
print(f"Need {n_components} components for 95% variance")

Unit 2: Principal Component Analysis (PCA)

Master the most fundamental linear dimensionality reduction technique using eigenvalue decomposition.

PCA Mathematical Foundation

Understand the mathematical principles behind PCA and how it finds principal components.

PCA finds directions of maximum variance:
C = (1/n)X^T X (covariance matrix)
Cv = λv (eigenvalue equation)
import numpy as np
from sklearn.decomposition import PCA

# Manual PCA computation
X_centered = X - np.mean(X, axis=0)
cov_matrix = np.cov(X_centered.T)
eigenvals, eigenvecs = np.linalg.eig(cov_matrix)