北京理工大学：Data science from a topological viewpoint（曹越琦）

Data science from a topological viewpoint 曹越琦( Yueqi cao Beijing institute of technology

Data science from a topological viewpoint 曹越琦（Yueqi Cao） Beijing institute of technology

05 Unrolled manifold 0.5 公∷ -1.0 1.5-1.0-0.50.00.5101.5-1.5-10-0.50.00.5101.5 Recovery by LLE Recovery by HLLe 0.10 PCA 0.04 神一 43. 008100010 LLE/Hessian LLE Sourcehttps://en.wikipedia.org/wiki/nonli near dimensionality reduction#Hessian L ocally-Linear Embedding( Hessian LLE Isomap

PCA Isomap LLE/Hessian LLE Source:https://en.wikipedia.org/wiki/Nonli near_dimensionality_reduction#Hessian_L ocally-Linear_Embedding_(Hessian_LLE)

opology Connected components Holes Tunnels

Topology Connected components Holes Tunnels

Topological data analysis Subcritical Critical Supercritical nrd→>0 nr2→入∈(0,∞) Source: Hamid R. Eghbalnia, the practice of topological estimation

Topological data analysis • …… Source: Hamid R. Eghbalnia, the practice of topological estimation

Chaos topology nosehoover y=-x+ yz

Chaos & Topology 2 x y y x yz z c y  =   = − +   = −

nosehoover(dimension O) nosehoover(dimension 1) nosehoover(dimension 2) 00.020.040.060.08 00.020.040.060.08 00.020.040.060.08

Outlines Conceptions about persistent homology Bayesian inference Kunneth formula Gauss map and its generalization

Outlines • Conceptions about persistent homology; • Bayesian inference & Kuሷnneth formula; • Gauss map and its generalization;

Outlines Conceptions about persistent homology: Bayesian inference Kunneth formula Gauss map and its generalization

Outlines • Conceptions about persistent homology; • Bayesian inference & Kuሷnneth formula; • Gauss map and its generalization;

Simplical complexes simplicial homology Simplices: convex hulls of points in general positions Point Line segment triangle Tetrahedron

Simplical complexes & simplicial homology Simplices : convex hulls of points in general positions Point Line segment triangle Tetrahedron

Simplicial complexes simplicial homology Simplicial complexes: Sets consisting of simplices such that 1.| f o andτ are simplices in a complex k,a∩τis the common face of o and t; 2. f g is in K. all faces of g are in k K=A, B, C,AB, AC, BC,ABC]

Simplicial complexes & simplicial homology Simplicial complexes : Sets consisting of simplices such that 1. If 𝜎 and 𝜏 are simplices in a complex 𝐾, 𝜎 ∩ 𝜏 is the common face of 𝜎 and 𝜏; 2. If 𝜎 is in K, all faces of 𝜎 are in 𝐾 𝐾={A,B,C,AB,AC,BC,ABC} A B C