Real World: Data Visualization

Linear Algebra for Machine Learning · Axiom Academy

Using dimensionality reduction to explore high-dimensional datasets Humans are visual creatures. A 2D or 3D scatter plot reveals patterns that statistics alone might miss. But how do you visualize data with thousands of features? MNIST: 70,000 images of handwritten digits (0-9). Each image is 28×28 = 784 dimensions. Strong clustering by digit (all 3s cluster together, all 8s cluster together) Some overlap (e.g., 3s and 8s are similar in shape) Clear visual separation of most digit classes Variance Retained: Just 2 components capture ~20% of variance, yet digit separation is clear! Using t-SNE on MNIST produces even more striking visualizations: Very tight, well-separated clusters for each digit Clear manifold structure (digits form a learned embedding) Computational cost: ~1-2 hours on a standard machine Word2Vec learns 300-dimensional embeddings for ~100,000 English words. Each word is a 300D vector capturing semantic meaning. Stage 1: PCA reduces 300D to 50D (preserves meaning, removes noise) Stage 2: t-SNE reduces 50D to 2D (produces visualization) Semantically related words cluster together (colors: red, blue, green) Analogies visible geometrically (king - man + woman ≈ queen) Syntactic relationships (verbs in one region, nouns in another) RNA-seq: Measure expression of ~20,000 genes across multiple samples/conditions. Each gene is a feature, each sample is a point. Biological Insight from Visualization PCA applied to gene expression data reveals:

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