Algebraic Topology Simplical Complex Tsau Young (‘T. Y.’) Lin Institute of Data Science and Computing and Computer Science Department, San Jose State University Ty.lin@sjsu.edu ; prof.tylin@gmail.com

Formal Theory of Simplicial Complex The mathematical Structure Ssimplex similar to R3 are used in algebraic topology (U, Ssimplex)

Open Simplexes  0- Simplex ( 1 point)

Open Simplexes -    1-simplex (no end points)

Open Simplexes     2-simplex (no boundary)

Open Simplexes 3-simplex(no boundary) Open tetrahedron

Data Mining Theorem 1: Isomorphic Information Tables  Isomorphic Patterns PAKDD 1999

BiT IT Tuple : What did a customer buy?) Diaper Beer Milk Pen …. 1

BiT Geometric IT Instead of item names, we may use name of the “unit vector” First item is A=(1, 0, 0, …) The second item is B=(0, 1, 0, 0, ….)

BiT IT (IT=A Relation in Relational Database) C D …. 1

Closed 3-simplex : It summarizes all information of First tuple B A C

An Example of Bit IT A B C D E F G 1 #1 #2 #3 #4 #5 #6 #7 Pen Beer #1 #2 #3 #4 #5 #6 #7 Pen Beer Milk Diaper Paper Tape Knife

Its Simplicial Complex View Open tetrahedron 2 d #3 d Open tetrahedron 1 b #2 b c #1 c a #4 e #5 g # 7 f #6