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Topology for ML

Topological spaces, manifolds, and persistent homology for understanding data shape.

5 concepts

Intermediate1

Topological Data Analysis (TDA)

Topological Data Analysis (TDA) studies the shape of data using tools from algebraic topology, producing summaries like Betti numbers, barcodes, and persistence diagrams.

#topological data analysis#persistent homology#vietoris–rips complex+12

Advanced4

∑MathAdvanced

Topological Spaces & Continuity

A topological space abstracts the idea of “closeness” using open sets instead of distances, allowing geometry without measuring lengths.

#topological space#open set#continuity+12

∑MathAdvanced

Manifolds & Manifold Hypothesis

A manifold is a space that locally looks like Euclidean space, stitched together by coordinate charts and smooth transition maps.

#manifold#topological manifold#smooth manifold+12

∑MathAdvanced

Persistent Homology

Persistent homology tracks how topological features (components, loops, voids) appear and disappear as you grow a scale parameter over a filtered simplicial complex.

#persistent homology#filtration#vietoris-rips+12

∑MathAdvanced

Betti Numbers

Betti numbers count independent k-dimensional holes: β₀ counts connected components, β₁ counts independent loops/tunnels, and β₂ counts voids.

#betti numbers#homology#simplicial complex+12