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How much structure do we see in noise (a topological perspective)?
Published on 2016-03-104145 Views
Exploratory data analysis is the search for patterns in sampled data. In this talk, I will concentrate on geometric data, where structure often takes the form of certain shapes representing constraint
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Presentation
How much structure do we see in noise?00:00
Joint Work with00:48
Structure in Spatial Data - 101:18
Structure in Spatial Data - 202:19
Dynamic Data03:11
Structure and Randomness04:00
Completely Random Points05:18
Motivation06:16
Topology06:32
Random Spaces07:30
Geometric Random Graphs07:51
Definition08:18
Classical Result09:01
Simplicial Complexes10:38
Geometric Random Complexes11:15
Graphs to Complexes11:42
Known Results12:51
Poisson Processes: Two Views15:13
Persistence - 116:06
Persistence - 216:52
Persistence - 317:02
Persistence - 417:10
Persistence - 517:16
Persistence - 617:21
Persistence - 717:23
Persistence - 817:33
Persistence - 917:36
Persistence - 1017:43
Persistence - 1117:56
Persistence - 1218:01
Persistence - 1318:13
Persistence - 1418:20
Persistence - 1518:27
Persistence - 1618:39
Persistence - 1718:42
Persistence - 1819:09
Distance Functions19:32
Random Persistent Homology20:40
Lifetime21:12
Relative Lifetime21:35
Different Cycles22:29
Geometric Intuition - 123:08
Geometric Intuition - 223:55
Definitions24:47
Question25:17
Main Result25:35
Experiments - 125:59
Regimes26:51
Lower Bound - 127:19
Lower Bound - 228:39
Higher Codimension29:32
Upper Bound - 130:13
Upper Bound - 231:33
Upper Bound - 331:46
Early-Born Cycles32:26
Size of Components32:35
Bound on Volume33:28
Bound Number of Simplices - 134:13
Bound Number of Simplices - 235:06
Bound Number of Simplices - 335:15
Bound Number of Simplices - 435:19
Proof: Step I - 135:36
Proof: Step I - 236:01
Proof: Step I - 336:04
Proof: Step I - 436:06
Proof: Step I - 536:10
Proof: Step I - 636:12
Proof: Step I - 736:16
Proof: Step I - 836:39
Proof: Step II37:16
Isoperimetric Inequality39:03
Putting it together40:12
General Result41:22
Experiments - 242:53
Experiments - 343:10
Future Work43:26
Questions?45:59