Topological Data Analysis
Visual presentation of multidimensional data sets
Current vs New SQL Topological Data Analysis
Topology
The Seven Bridges of Königsberg, a problem solved by Leonard Euler (1736).
The study of qualitative properties of certain objects (topological spaces) that are invariant under a certain kind of transformation (continuous map), especially those properties that are invariant under a certain kind of equivalence (homeomorphism).
Topology Data Analysis Pipeline
a b
a. First approximate the unknown space X in a combinatorial structure K
b. Then compute topological invariants of K
Combinatorial Representations The Čech Complex
Combinatorial Representations Alpha Complex Vietoris-‐‑Rips Complex
Cubical Complex Witness Complex
Topological Invariants A topological invariant is a map f that assigns the same object to homeomorphic spaces, that is:
Homology: is a machine that converts local data about a space into global algebraic structure
Reference: Wikipedia, 2010.
Morse Theory and Reeb Graph Theorem: Suppose h : X g is a discrete Morse function. Then X is homotopy equivalent to a CW-‐‑complex with exactly one cell of dimension p for each critical simplex of dimension p.
Reference: Teng Ma ; Zhuangzhi Wu ; Pei Luo ; Lu Feng. Reeb graph computation through spectral clustering, 2011.
Case study: Demographics
Data shape: [220:45]
Case study: YT channel stats
Data shape: [1500:12]
Case study: Netflix dataset
Data shape: [17770:480189] 8.5 billions of elements
Case study: Netflix dataset
Music
Indian
Anime
French
Honk Kong
US Cartoons
Kids Movie
German
US Retro
Horror
Case study: Netflix comparison
PCA Isomap
LLE
Spectral Embedding
LTSA Hessian LLE
Case study: Netflix (music)
Case study: Netflix (kids movie)
Case study: Netflix (horror)
Questions?
