1. 1 ALGEBRAIC TOPOLOGY SIMPLICAL COMPLEX 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

2. 2 Formal Theory of Simplicial Complex The mathematical Structure S simplex similar to R3 are used in algebraic topology (U, S simplex )

3. 3 Open Simplexes Open Simplexes  0- Simplex ( 1 point)

4. 4 Open Simplexes Open Simplexes -    1-simplex (no end points)

5. 5 Open Simplexes Open Simplexes     2-simplex (no boundary)

6. 6 Open Simplexes Open Simplexes 3-simplex(no boundary) Open tetrahedron

7. 7 Data Mining Theorem 1: Isomorphic Information Tables  Isomorphic Patterns PAKDD 1999

8. 8 BiT IT Tuple : What did a customer buy?) D iaper B eer M ilk P en …. 111100

9. 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, ….) 9

10. 10 BiT IT (IT= A Relation in Relational Database) A BC D…. 111100

11. 11 Closed 3-simplex : It summarizes all information of First tuple Closed 3-simplex : It summarizes all information of First tuple C A B D

12. 12 An Example of Bit IT A BC DEFG 111 1000#1 011 1100#2 10100 11#3 #4#2#1 #3#5#6#7 PenBeerMilkD iaper PaperTapeKnife

13. 13 Its Simplicial Complex View Its Simplicial Complex View a #4 b #2 c #1 d #3 c b e #5 d f #6 g # 7 Open tetrahedron 1 Open tetrahedron 2