Dimensionality Reduction
High-dimensional data has structure. You find it by looking from the right angle. Language: Python Implement PCA from scratch: center data, compute the covariance matrix, eigendecompose, and project. Use explained variance ratio and the elbow method to choose the number of principal components. Compare PCA, t-SNE, and UMAP for visualizing MNIST digits in 2D and explain their tradeoffs. Apply kernel PCA with an RBF kernel to separate nonlinear data structures that standard PCA cannot handle. You have a dataset with 784 features per sample. Maybe it is pixel values of handwritten digits. Maybe it is gene expression levels. Maybe it is user behavior signals. You cannot visualize 784 dimensions. You cannot plot them. You cannot even think about them. But most of those 784 features are redundant. The actual information lives on a much smaller surface. A handwritten "7" does not need 784 independent numbers to describe it. It needs a few: the angle of the stroke, the length of the crossbar, how much it leans. The rest is noise. Dimensionality reduction finds that smaller surface. It takes your 784-dimensional data and compresses it to 2, 10, or 50 dimensions while keeping the structure that matters. High-dimensional spaces are unintuitive. Three things break as dimensions grow. Distance becomes meaningless. In high dimensions, the distance between any two random points converges…
Dimensionality Reduction: High-dimensional data has structure. You find it by looking from the right angle. Language: Python Implement PCA from scratch:…
This free lesson is part of the AI Engineering from Scratch curriculum. Read the full explanation, run the lesson code, and verify the result in the interactive reader or from the repository source.
Browse the complete course catalog or open this lesson on GitHub.