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Machine Learning and Deep Learning

381The Curse of Dimensionality
382Dimensionality Reduction Goals
383Linear vs Nonlinear Methods
384Variance as Information
385PCA Problem Formulation
386Covariance Matrix Construction
387Eigendecomposition for PCA
388Principal Components as New Basis
389Choosing Number of Components
390PCA Transformation and Reconstruction
391Standardization Before PCA
392PCA Implementation in scikit-learn
393Interpreting Principal Components
394PCA Limitations and Assumptions
395Limitations of Linear Methods and the Manifold Hypothesis
396t-SNE: Probabilistic Neighbor Embedding
397t-SNE: The Cost Function and Optimization
398t-SNE: Perplexity and Hyperparameter Tuning
399t-SNE: Practical Considerations and Common Pitfalls
400UMAP: Uniform Manifold Approximation and Projection
401UMAP: Algorithm Components and Construction
402UMAP: Hyperparameters and Their Effects
403UMAP vs t-SNE: Comparative Analysis
404Isomap: Geodesic Distance Preservation
405Locally Linear Embedding (LLE)
406Autoencoders for Dimensionality Reduction

Machine Learning and Deep Learning

381The Curse of Dimensionality
382Dimensionality Reduction Goals
383Linear vs Nonlinear Methods
384Variance as Information
385PCA Problem Formulation
386Covariance Matrix Construction
387Eigendecomposition for PCA
388Principal Components as New Basis
389Choosing Number of Components
390PCA Transformation and Reconstruction
391Standardization Before PCA
392PCA Implementation in scikit-learn
393Interpreting Principal Components
394PCA Limitations and Assumptions
395Limitations of Linear Methods and the Manifold Hypothesis
396t-SNE: Probabilistic Neighbor Embedding
397t-SNE: The Cost Function and Optimization
398t-SNE: Perplexity and Hyperparameter Tuning
399t-SNE: Practical Considerations and Common Pitfalls
400UMAP: Uniform Manifold Approximation and Projection
401UMAP: Algorithm Components and Construction
402UMAP: Hyperparameters and Their Effects
403UMAP vs t-SNE: Comparative Analysis
404Isomap: Geodesic Distance Preservation
405Locally Linear Embedding (LLE)
406Autoencoders for Dimensionality Reduction

← Machine Learning and Deep Learning

Lesson 395 of 3,538·9. Dimensionality ReductionPro lesson

Limitations of Linear Methods and the Manifold Hypothesis

Why PCA fails on nonlinear structures and how data often lies on lower-dimensional manifolds in high-dimensional space.

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