finding yet why an arena first - university of...
TRANSCRIPT
Finding the average is one of the key procedures in deteranalysis Yet littleattention is paid to why an arena
exists at all in the firstplace
Definition let X be a topological space Huu averaging is a ingpig associatingto a tupleits averageM l nuRequirements w se sc se
w is SnequivariantMkii Rin beta au foranypointsonhe is continuous
Topology of the Space Hi the ambo slackTate X S Then for fixed X sets walk aheaps 8 S There is whomof degreeof any cupping between
Compact oriented manifolds countingpreinases of a genniepants withorientation essentially looking at thereduced cuppingofHIX Hand IL the degree is an integer
So if the degree of mzCn S is D then
toy symmetry so is the degreeof C a S
However N 1 3 mix a se is
a cupping of degree 1 yet Xthe diagonal D flax C Sks
Data aggregation
is homologous to the sum of a parallel a meridian
on the torus six S so by funchialitythe degree of service a shouldbe equal to the sums of the degrees
of Rl wzCx 2 Khadr Ra I
i e QD Which cannot be L x
So i we 2 averaging on 8 for any sphereTurns out existence of h means for all u z means
that X is contractible
The B Eckman154 If X is homology equivalentto finite shephrideouplex or finitely way Suphies ineach dimension and there exist a averaging forany U2 2 then X is contractible
The proof is essentially taking the handgiesand establishing why the reasoning as above thatHulk 2 is h divisible for each h This is Kossthe onlyif Hulk 2 is like ie needs manyempties to
generate or is 0
Now if all homologies are zero then Hurumall hearehepie grepps one 0 which ihphies Whiteheadthat X is contractible
We should let that one can construct quiteearlySome pathological non contractible spaces wife wereprySuplest example is 2 solenoid defined as infinitesequences y ya Yu of points in Slsuch Thet ya 2yu for all u 4,2 Thenwz 41,42 n 41,42 1 42142,4343 1 is
a 2 averaging
Gaversely if X is www.acfible it admits n annoying
for all 2 This is proven by homotopy extension
easy but outside our scopeSeine examples are of course easy R has a whirl
averaging etcthere broad and igortarf class of examples is givenby
We war will be ceastruehry means require ware
from the space namely that X is a path metricspace will be assuming couplefeness I.e we havea dislaeae function dfa y and we can always
find a halfway point o
and therefore reconstruct a
geodesic the shortest timeconnecting a pair of points
Spaces of non-positive curvature.
Standard examples include of course Riemannian wfdsfar Ruder reefds
Among path metric spaces Nen Positive Curvature NPCspaces for any geoderie A
triangle ABC and ft c
D on the geodesic Be bB e
the distance e is E Dawhat it should love been C
1h Euclidean space a.k.a CATCo ar HadamardspearDowerfulcookeries as a distances in linear
beuotopies along
geodesics are convex
Distance is convex function in each argument
thetic balls are convex
i.e round spheres are not NPC
ds2 deedy7
Example Hyperbolic spaces
Iketric trees
Hilbert spaces duh but no further Banachspacesproduct of NPC spaces with d.sk do dsf
Gluing along convex subsets
Ihperlant properties of NPC spaces0
any two paints are connected by unique geodesichuegeodesic as a weaponry I 7 X depends continuouslyon its endpoints
BaycentersFor NPC spaces dYa is convex restriction to anygeodesic segment is convex
For any collection of points or better for anyprobability measure q with compact support defineits barycenter as the unique point bCq whichminimizes
a Jatta dqe
Banycenters define averaging on NPC spaces nMula Ru f Ta e easy to see it sahih
the axioms
Easy to see that for q a8a ta Ja PG is onthe geodesic sensed splitting it a ratio a i a Wewill call this point M no 2 ft t affaRemarkably simple algorithm to find beepFor any sequence of pants see au C Xwe define the doreded mean as
Mink Mz M 74,22142 MexFM Kc Mkl
I 1
Remark i then depends on order of pointsSampleHowever repeating
µwe have Ceuvergeue
The Sturm If m au isan iid Sample then 9 thenfun defined as above converges to b g
Application phylogenetic trees
Phylogenetic trees are metric trees with roots labeled
leaves same distance from roof to leaves
µ fiftystandard application deferreddistance
through mutationfrequencies
9 can be made into a metric1 3 2456
Space
HK 4
i z
XIf I473 Positions of the branches
parameterize cells cubes of cells is1 3 5 X X a 3
Theorem Biwa Holmes Westman spaceof phylogenetictrees is NPC
Droz Gromov if the link of any cube in a
cubical couplex is a flag complex then thespace is NPC Links in the spaceof treescorrespond to compatible famlies ofedges deputybested bisections they all can be variedsimultaneously a flag eenylex I