THE GRAPH REEB Is EDIT FLUFD BAUER ULRICH DISTANCE UNIVERSAL WORKSHOP EINSTEIN ON & TOPOLOGY DISCRETE GEOMETRY MARCH 151 2018 MODENA ) LANDI CLAUDIA ( WORK WITH JOINT U U ) MEMOLI ( AND FACUNDO STATE OHIO
on things Here two that are are close each other reasonably to , 6£ want to them and I compare . WEINBERGER S .
GRAPHS REEB f :M→R I € § 1 identify oil M Rf D - Components ' of f- level sets G) : fjRp→lR Rf M/~f = , where ( t ) ⇐ ' in TER × > of f- xiy same component some rfy . , :Rg→k :M→R j g ^ ^ ✓ 1€ M → Rg
FORMAL SETTING function F We consider domains ) locally Haxsdorff Reeb compact ( . spaces ( Reeb quotient maps ) with connected quotient maps fibers proper • closed stable pullbacks composition and undo under are maps , . Define Reeb graph a as These g=f°q Hoff Reeb with domain Rp . a function ) discrete fibers R with Reeb → ( : Rf a . A Reeb Rf is graph the Rees of function Y → graph f :X R if → a fop quotient → f x-P Reeb for = X Rf some : p . map . In this ± Rf case X ^ /~f I . Moreover let : Reeb : × be quotient , , Rf a q map . y9→ , Then Rf is Reeb also the graph of . R Reeb quotient Reeb graphs maps preserve . .
GOALS ( f How to ? M R two Reeb Rep → unknown ) graphs Rg are compare , g : , glla Rgi ds ds Assign metric distance ( extended ) d ( Rp , Rg ) pseudo - . Desirable properties : X For and Stability fig R Rf : :X Reeb → yielding graphs any space , . d f- ( , Rg ) Rf E 11 . Universality For other stable distance : any , ( Rf Rg ) d 1 Rg ) ( E Rpi .
A DISTANCE CANONICAL UNIVERSAL Given Reeb Rf Rg with functions I , of define graphs , , , Rg ) imf = Has - g Pf X Pg ÷ neither dulrt Hf ⇒ I taken all Reeb Reeb quotient over × and Pg maps pfi . domains This distance is ) pullback triangle inequality consider a ( : * µ ↳ ⇒ Rg Rn Rg ~ ~ ~ nd flu gd R R R stability immediate definition and universality from • with arbitrary working is X unfeasible • spaces
PREVIOUS WORK FUNCTIONAL DISTORTION DISTANCE [ B.) Ge , Wang ] 20^4 : £ with On Reeb Rf R metric graph Rf consider . : → a , " ] } a / 40,4 the f- { Laib df ( int b- in component of , y ) H y same : × x. . I¥I¥÷I : * Given : Rf ¢ Rg consider • maps → y Rg Rf : → , , { G( ) } Rg } 4,4 lxiolx ) ) { ylyl xerf u ( I y ) ye = I , . . distortion Define the ) of as D F) / ( ¢ y ) tfldflxix ) dglyi snp , = - . GUN ) lxiy ) E. F) e , distortion Define the functional distance as . ( - g°dH• { did , Rg ) ( Rf inf Dully Hf . ) foul max , 11g = - , 414
FUNCTIONAL EXAMPLE DISTANCE Distortion : . 'Q¥i¥# ¥ - @ Rg Rp Rf snptfldlxit dly ,yY) 21 ) = , 4) - DH = E Rg y|y~ I Rer where E xi , 441 with YCH or × = , - y - YCFI E yCE1=F or
&al PREVIOUS DISTANCE [ Silva WORK INTERLEAVING Bnbeuik & de 2016 ] al 2015 : . . ; to ( " F ( ) ) ^ Reeb F Set I I Interpret Rf fmctor tntk I → graph a : → as , YI ( the intervals wrt Inta E) poset as are open a , . is J pair natural A between F of interleaving and G - a transformations . G ( Bo ( I ) ) ) ( with components FCI ) that such if y → : ... , , ( I ) ) I ) ( F ( ( It ) F ( - F Bo → Bu yytfee # * eat , Bolt ) ) ( I ) ( Bu ( I ) ) ( all G G G commutes for Iektr → - unlabeled induced inclusion ) ( by maps The distance is interleaving . A } between ( Rp dt ( Rg ) imf { o I F F into leaving F and : = problem ; Open ] FDFD Edt DFD Thur [ = Munch 2015 B. , Wang ? , ' the lower bound is tight
ABSTRACT AND TOPOLOGICAL INTERLEAVING S ' II. ± to / # yE# b / h÷\÷ - " | |s.r S÷Rf Snrf I OEI ' x , / - Rg , H H
LEVEL SET PERSISTENT HOMOLOGY ] de Silva 2005 Thru Morotov C , , Given X R ( f PL with → X compact ) : : , Carlsson # ) Ct ) level sets of ( and generally , Homology more ; " " ( j ) IEJ ) ( I ) intervals of inclusions f ↳ f for H*lf" : . is encoded ( up Isomorphism ) by unique to a collection barcode ( level set persistence of intervals ) . graphs Reeb for Example . • ~ Rf Bare ( f )
PERSCSTENCE BARCODES THE BETWEEN BOTTLENECK DISTANCE =) . • I tttiti . I o.O . . 0 intervals o , unmarked I I • • 8- matching A between (f) two barcodes Bare satisfies Barccg ) : , , J ) matched ( I , ] ) have distance dtd to I . intervals have 28 length E . bottleneck distance The ( ) is do fig - mauling inf J between ( f ) Bare J Bare (g) F : ,
ZOO AND INEQUALITIES A OF DISTANCES [ Carlsson Morozov 2009 de Silva ] , , dp ( f- , Rg ) H glto Re a- ] B Ye 20^4 [ Wang . , , ¥ dB ( dtp , Rg ) ( Rp ± E , Rg ) f- Rf H gH• E alt ] ¥ Munch ] 10^5 Wang [ B. , , dt , Rg ) , Rg ) ( Rp , Rg ) ( Rp ( Rp E dtp dtp ± 13 Botnam Lesnick [ 2016 , ( Rf , Rg ) dB ( , Rg ) Rf k n' ] [ Bjerke 2016 12dB dt ( Rp , Rg ) E ( , Rg ) Rf
& UNIVERSAL FUNCTIONAL ARE NOT Consider with two a functions f. cylinder g e- ⇐ : , i '¥¥t of Food DISTANCES DISTORTION INTERLEAVING " Rf du( Rp , Rg ) he Hf ^ . - = glla EF du( , Rg ) ( Rpi < " At Rg ) , Rg ) Edt ( Rg : Rep , ± % ± Rgo / Rg ' Rg / Rf ,
FROM TO CLOSE REED GRAPHS CLOSE FUNCTIONS problem Open with Given two , Rg d± Reeb graphs Rf ( Rg , Rg ) or = . X with He Is there f R C. or :X Hf space → a - g g , , , constant ( ? fixed Reeb graphs for yielding Rg Rg some , , is ⇒ 0 | the previous example By : then Rg if Rf C 2 yes = ,
THE EDIT DISTANCE TOPOLOGICAL Consider tig Z Reeb quotient diagrams of . zag maps - R Rn Re= Rn Rn Rg = } , * y ' y y in " Sz / ' ' x. ' limit Lz rift Lz ↳ . " : ↳ the all maps ) and take ( note Reels quotient ↳ : maps are . fi Ri R fi Ri Each to → R → → : : composes . R Define fn ... , fn → the the functions : • spread of as , R fjlx → filx ) ) × I → may - ngin . , ( topological ) the edit distance Define . as detop ( Rp Hstlls Rg ) = inzf . , stable Prop detap is and universal . .
THE GRAPH DISTANCE EDIT Consider Z of Reeb quotient tig diagrams . - maps zag Gn Rn R Re= Rn Rg yRn= } T Y T " i. / . REEB . G G . . . . in be finite before but Ri as restrict Gj Z to graphs , , . edi@ Interpretation IR need R : { µ - Ttitn fit y Ritny Ri Run Y T Gi Gi Gin fi → IR modifn Gitn modify fit Gi Gi to to : , , , the Reeb domain the maintaining graph Ri+ Gi maintaining , the Define Reeb graph edit distance analogously . as dearapn ( Rg ) Rf Halls = inzf , .
1 MAIN RESULT Turn [ B Mehdi ] , Landi The Reeb edit distance is stable & universal graph . , . an edit hard part Rtox restrict the PL We to here category . . stability ,yR×iR8 between The is • : X X=lk1 ) , ( for triangulation → IR PL construct given fig how to : , - glls ? spread with c- Hf Rp and Rg zigzag : • Idea fx t ) Consider straight line tf homotopy = + ( o - r - g The of Rf often structure Rx changes only finitely ° = , Rp n ) ( say at du ti ) parameters c. to < Choose E o= ( tin = , , . . pi . . titna Rt a Mix ... ¥ Construct " ° tigtag . . . . . . . . . How quotient in this ? Reeb ° to the zigzag get maps
CRITICAL A PL STRAIGHT LINE HOMOTOPY INSTANTS OF - ... . t¥¥I¥÷€fI¥÷% ← ( regular ) ( critical ) # µ , Xof order presenting snrjection - X ( P4 imf in : → g k ) v ) ) glu ( ( F revert have • We = . ! But X of = g Reeb greph the have X°f and same However . . . : g .
LIFTING REPARAMETRIZATIONS Then to h=X° f X lifts www.a Let a Rf . Gµ⇒¥ } Rn Reeb quotient : → map IN . x ' Ti in h imf - ? f Rf
THE REEB WORKHORSE QUOTIENT MAPS INTERPOLATION : FROM PL X ( order ) preserving in f → : - , g( Vert K f ( y If 4 : re . = IKI |K| :b :L ii. :* ing maps sf±¥n¥÷ k=qn°¢ relation Lemma The ing imf - Tf snrjeckon , af |qf nstu ) to g) . . ' . 9u ' and , , to X Reeb quotient lifts a map Rf → Rg . Rx Rtitn ; " r a the This provides Rpi interpolation for zigzag our .
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