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How much structure do we see in noise (a topological perspective)?
Published on Mar 10, 20164139 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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Chapter list
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
Regimes13:43
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
Bound on Volume38:16
Isoperimetric Inequality39:03
Putting it together40:12
General Result41:22
Experiments - 242:53
Experiments - 343:10
Future Work43:26
Questions?45:59