Geometric Approximation Using Coresets

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1 Geometic Appoximation Using Coesets Pankaj K. Agawal epatment of Compute Science uke Univesity

2 Kinetic Geomety : Set of moving points in Maintain the diamete (width, smallest enclosing disk) of. [A., Guibas, Heshbege, Veach] iametal pai can change Kinetic data stuctue with times events Can we maintain the appoximate diamete of Is thee a small coeset s.t.? moe efficiently? Kinetic bounding box hieachies? Geometic Appoximation Using Coesets 1

3 &' $. # %! Shape Fitting : Set of points in Fit a cylinde though Find a cylinde O I " Optimal solution: Shai] (*) -, -appoximation: [A., Aonov, Can we compute an -appoximation of in linea time? Is thee is a small coeset so that δ appoximates /? Geometic Appoximation Using Coesets 2

4 3 4 9 Geomety in Steaming Model t=1 t=2 t=3 t=4 An incoming steam of points in 1 Maintain cetain geometic/statistical measues of the input steam iamete, smallest enclosing disk, 2-clusteing Use "*5 687 space and pocessing time Much wok done on maintaining a summay of 1 data Little wok on multidimensional geometic data [A., Kishnan, Mustafa, Venkatasubamanian], [Heshbege, Sui], [Bagchi, Chaudhay, Eppstein, Goodich] How much stoage and pocessing time (pe point) needed to maintain -appox of smallest disk enclosing? Maintain a coe set! Geometic Appoximation Using Coesets 3

5 C C : ; C A A B; ( <;, >= : Set system (ange space)?: VC-dimension of -appoximation if fo all C C C " 3 4 H C A C L-Appoximation and Random Sampling A andom subset Gof size is an with high pobability [Vapnik-Chevonenkis] HJI K I -appoximation of Efficient deteministic algoithms fo computing an [Matoušek, Chazelle] -appoximation Geometic Appoximation Using Coesets 4

6 E A M N A E ; L-Appoximations in a combi- An -appoximation is a coeset of natoial sense : Set of points in Ais a an not a coeset <; -appoximation of in a metic/geometic sense Abest-fit cicle not appoximate the best-fit cicle fo What about othe sampling schemes? does not appoximate Geometic Appoximation Using Coesets 5

7 Unified Famewok fo Coesets Notion of coeset is poblem specicific Is thee a unified famewok that constucts coesets fo a wide class of poblems? Random subset is an spaces! -appoximation fo a lage class of ange Easy to compute efine the notion of -kenel Coe set fo a wide class of poblems Geometic Appoximation Using Coesets 6

8 Y 7 X V V 7 X P S T Q V Y V \ V BW Y V % &' ( Z [ &' ( Z [ # R P OP : Set of points in Uω (u,q) ω (u,s) iectional width: Fo V BW, u -kenel: is an -kenel of if L-Kenels Geometic Appoximation Using Coesets 7

9 b 6 7 X baaffine tansfom e -kenel of d b ] X b Computing L-Kenels Theoem A: [AHV, YAPV, Ch] -kenel of size of in time 9`, ^. We can compute an. ` Lemma 1: s.t. Unit hypecube c is the smallest box enclosing is fat is an is an -kenel of Geometic Appoximation Using Coesets 8

10 g f X 7 X 7 X X ] d Computing L-Kenels Lemma 2: : Set of fat points in ^. We can compute an c, -kenel of of size 9` in time `. Sketch: Algoithm in two phases ε Compute -size appoximation B aw a sphee =fof adius centeed at oigin q aw a gid of size 9` on Fo each gid point, select its neaest neigh- CH(Q) bo in (Suffices to compute appoximate NN.) Geometic Appoximation Using Coesets 9

11 q mq q mq q p q o mq uq q o p w u t n v q o q qsphee q m n m m t m w u u o q o p m n s s nq q s w w n vq s u p q s vm w u q p m s w m s s n w q s u mq s v u p w s v o u v s pv n s v o m s s q q s t p m v p s pv o vq s v t n s q s p s v s m s q t t n u s u vu o s t vp Computing -kenels fo h ^i j^on vaious synthetic inputs Input Input Kenel Size Type Size cylinde clusteed s v m o q s n vp u s o nq s p w t s p s m kl kl kl kl Kenel Computation Geometic Appoximation Using Coesets 1

12 ƒ~ z {cylinde {clusteed { ~ ~I zƒ~i I z y { } { z { ~I y z { } } { ~ { ~{ ~ { I z II ƒ~i { } { z { ~ y Input Input Type Size Pep Quey Pep Quey Pep Quey sphee Pep: Time (in secs.) in conveting input into a fat set and pepocessing fo neaest-neighbo (NN) queies Quey: Time (in secs.) in answeing NN queies Expeiments un on Pentium IV 3.6GHz pocesso with 2GB RAM { } } ~ {y { I ~y{ ƒ ~{ } { { z I ~{ ƒ ~{ y { } { z { { { } } ~ { ~ { { y ƒ~i } z { { y ~{ { } y { ~{ Px I Px Px Kenel Computation: Running Time Geometic Appoximation Using Coesets 11

13 w vn v w vn u w m n w n u -kenels fo h ^i m n n q u m p u q q q o t o p o u n q q Kenel Computation Computing j^on 3 models Input Input Running Time Kenel iamete Type Size Pep Quey Size Eo bunny dagon buddha s o s t s o Geometic Appoximation Using Coesets 12

14 Expeimental Results: Appoximation Eo Tadeoff between kenel size and appoximation eo Appoximation Eo Appoximation Eo Bunny: 35,947 vetices agon: 437,645 vetices Kenel Size Kenel Size Geometic Appoximation Using Coesets 13

15 F F ˆ a ˆ ] ^ Faithful Extent Measues : Function defined ove point sets in is faithful if ^fo all fo any -kenel of faithful measue unfaithful measue Faithful measues: iamete, width, adius of smallest enclosing ball, volume of the smallest enclosing box (simplex) Nonfaithful measues: width of the thinnest spheical shell containing Geometic Appoximation Using Coesets 14

16 ] ^ g ] ^ ˆ X g ) ] ^ Computing Faithful Measues : Set of points, : A faithful measue, Compute an -kenel of Compute using a known algoithm Retun By definition,, Can compute a pai in time ` Bs.t., Can compute an -kenel of the smallest simplex enclosing in time ` - How does one handle unfaithful measues? Geometic Appoximation Using Coesets 15

17 Ž Ž Š: Lowe envelope of ŠExtent of Š Ž N Š Ž F \ B Ž Ž Ž # Ž U F : Uppe envelope of % M 7 i i i Œ Extents of Functions )-vaiate : functions E (x) F U F E (x) G E (x) F L F x L F x : -kenel: Šis an-kenel ofšif Geometic Appoximation Using Coesets 16

18 X ] L-Kenel of Linea Functions U F E F (x) E F (x) E (x) G LF x x Many functions can be mapped to linea functions using lineaization Uppe and lowe envelopes of linea functions ae convex polyheda Relationship between linea functions and points Theoeam A uality: B: Theoem an -kenel of : set of ) -vaiate linea functions, of size ` in time ` 7 ^. W e can compute. Geometic Appoximation Using Coesets 17

19 Š M 7 i i i Œ N )-vaiate : polynomials Lineaization [Yao-Yao, A.-Matoušek] Map 1, 7 i i i ] e^ š2-vaiate linea function Each maps to a ; 6 X ] š B ] ^ M š e N L-Kenels of Polynomials 2: imension of lineaization Lemma: is an -kenel of is an -kenel of Š. Theoem C: Š: a family of 2) -vaiate polynomials, : dimension of lineaization, ^. We can compute an -kenel of Šof size ` in time ` 7. Geometic Appoximation Using Coesets 18

20 N Š Ž V [ " = [ Y F F œ N, \-kenel if 7 X &' ( Z &' ( Z Y V F V M B : Set of moving points in Y V % V # V [ an V BW B ω (u,q) ω (u,s) efine Claim: -kenel of Š M 7 i i i Œ, -kenel of. Y V V V Z polynomial ; is a u Apply Theoem C! Application I: Kinetic Geomety Geometic Appoximation Using Coesets 19

21 ] i i X X Application I: Kinetic Geomety Coollay: : linealy moving points in, ^. An -kenel of size ` 7 can be computed in ` time. Maintaining the -appoximate bounding box of : Compute an -kenel of Use a kinetic data stuctue to maintain the extent of in each diection Bounding box is defined by the left-, ight-, top-, and bottom-most points Events: When one of these fou points change Appoach woks fo maintaining width, smallest enclosing ball/ectangle/simplex,i Geometic Appoximation Using Coesets 2

22 h ^i ^i Bounding Box: Quality of Kenels 1, moving points; thei tajectoies chosen to ensue lage numbe of events Tajectoies linea o quadatic Eo ^ž(98% time) and Ÿ^(1% time) fo kenel Quality Quality Kenel Size = 16 Kenel Size = Time Linea Motion.94 Kenel Size = 16 Kenel Size = Time Quadatic Motion Geometic Appoximation Using Coesets 21

23 Bounding Box: Numbe of Events Linea Motion, Input Size = 1, Quadatic Motion, Input Size = 1, Linea Motion, Kenel Size = 24 Quadatic Motion, Kenel Size = # Events Time Exact algoithm # Events Time Appoximation Geometic Appoximation Using Coesets 22

24 ^ ^ ^ iamete & Width: Quality of Kenel Maintaining the kenel of ^moving points Slow implementation of kinetic convex hull algoithm Quality Kenel Size = 46 Kenel Size = 128 Kenel Size = Time Quality.95.9 Kenel Size = 46 Kenel Size = 128 Kenel Size = Time Quality.95.9 Kenel Size = 46 Kenel Size = 128 Kenel Size = Time Convex hull iamete Width Geometic Appoximation Using Coesets 23

25 Convex Hull: Numbe of Events # Events # Events Kenel Size = 46 Kenel Size = 128 Kenel Size = Time Exact algoithms Time -kenel Geometic Appoximation Using Coesets 24

26 B š Š M Functions ae not polynomials in many applications istance between point cicle of adius Aand centeed at N M 7 i i i š F Œ š F œ N N A ˆ A)-vaiate polynomial, ˆ ] B M L-Kenels of Factional Polynomials )-vaiate : functions ` š 7, : Theoem : is an is an -kenel of Š. -kenel of, ^a constant, then Coollay: If admits a lineaization of dimension 2, then we can compute an -kenel of size Šof in time. ` X ` 7 Geometic Appoximation Using Coesets 25

27 ) and -kenel Š ) Š % &' ( ) # &' ( ) : Set of points in Find the best-fit cicle though (minimize the max distance between : Min width of annulus containing : istance between and..) centeed at O I x Compute Compute an # Ž, of Compute Retun Time: Ž ; Ž F! " # ; M 7 i i i Œ N Ž Application II: Shape Fitting Geometic Appoximation Using Coesets 26

28 M ž «³ ³ ³ ) % Shape Fitting: Incemental Algoithm : Set of points in Find the best-fit cicle though a c : &'ª A simple iteative while : Initially, : Best fit cicle fo B: Point fathest ²± ³ ³8 ²µ ²± ²± b Claim: The algoithm teminates in steps. Woks fo othe shape-fitting poblems as well. Geometic Appoximation Using Coesets 27

29 o u u o m q p q m ¼ q q À» t ¼ 5.1 ½l w q ¾ mq» t ¼ 5.18 q º¹» t ¼ 5.51 q À» m ¼.51 ½l w q Á¾ v» m ¼ q Â¹» m ¼.514 q ÁÀ» m ¼.5 ½l w q ¾» m q º¹» m ¼.51 Input Input Running Output Type Size Time Width.5.54 Points chosen andomly inside an annulus of width Numbe of iteations is h ^, inne adius i ^ Minimum Width Annulus Geometic Appoximation Using Coesets 28

30 y ~ ydagon ƒƒ ~ I ~I ƒ Ä Å I zy { I Ã { } } I z ƒ~ { ~{{ ~I Ä { Å Æ x I { } { z~ I ~{ y { { } z~ { y {{ Äƒ Å { } } I I z~ y ~I{ ƒy Æ x I { } { ƒ~i { ~{ z I Äƒ Å { } z~ { I {{ Äƒ Å { } } ~I{ { ~{ z Ä Å Æ x ~I { } II ƒz~ ~{ y{ { } I { {{ Points chosen andomly on a cylindical suface of adius µ, height Algoithms APPR: Appoximation, CORE: Kenel based INCR: Incemental Input Input Running Time Output Radius Type Size APPR CORE INCR APPR CORE INCR ÅÄz ÅÄz bunny buddha }z I I{{ ƒz~i ~{ { Äƒ Å ƒ }zy ~I y Äƒ Å }y ƒ z ~ y z { Minimum Enclosing Cylinde ÅÄy ÅÄz Geometic Appoximation Using Coesets 29

31 Ç " ÊF Ê Ç 3 4! # Ë = É 7 Ç È i i i : Steam of points in Maintain the -kenel using " space Patition fo some ÇÇinto subsets, s ae not maintained explicitly Í Ê Ì = Ç Ç Maintain an -kenel of is an Ì = Í Application III: Handling ata Steam P 1 P 2 Q 3 P 3 -kenel of Ç. Maintain an ²Î-kenel of 4 Q P 4 Geometic Appoximation Using Coesets 3

32 Ì Ì 4 Ñ = Ð Ê Ñ Ç Ç Ç ;! # Ë Ð Ñ Ê Ð Ê Ç Ç Ï M N Ñ Ç Ð Ñ of Ç Ï Ç Ï Inseting a Point Ceate a new set If thee ae two sets Compute an elete and add ; of ank -kenel ; P z Q z is an ÑÇ -kenel of Qx Px Q U Q x y Qy Py 3Space: "., Pocessing time: (amotized) Impovement (Chan). Space:., time: (amotized) Extend to highe diemnsions. Coollay: using appoximation of width, smalelst enclosing box,... 9space and time in the steaming model. Geometic Appoximation Using Coesets 31

33 9 Ó.) Extensions Computing -kenels in high dimensions [Bădoiu, Ha-Peled, Indyk], [Bădoiu, Clakson], [Ha-Peled, Vaadaajan], [Kuma, Mitchell, Yildiim], [Kuma, Yildiim] Smallest enclosing ball Ò Smallest enclosing ellipsoid -median Computing -kenels in pesence of outlies [Ha-Peled, Wang] Computing -kenels fo 2-clustes [Ha-Peled], [A., Pocopiuc, Vaadaajan] 2-centes 2-medians 2-line-centes Geometic Appoximation Using Coesets 32

34 ), compute a cylinde Conclusions -kenels in high dimensions Polynomial dependence on Geneal technique fo computing coe sets fo clusteing Coe sets fo shape fitting if we want to minimize the ms distance Given and is minimum so that the ms distance between Coe sets and ange spaces with finite VC dimensions Geometic Appoximation Using Coesets 33

35 Refeences Review aticle. A., S. Ha-Peled, K. Vaadaajan, Geometic appoximation via coesets, Cuent Tends in Combinatoial and Computational Geomety (E. Welzl, eds.), to appea. Technical atciles. A., S. Ha-Peled, K. Vaadaajan, Appoximating extent measues of points, J. ACM, 51(24), A., C. M. Pocopiuc, and K. Vaadaajan, Appoximation algoithms fo k-line cente, 1th Annual Euopean Sympos. Algoithms, 22. T. M. Chan, Faste coe-set constuctions and data steam algoithms in fixed dimensions, in Poc. 2th Annu. ACM Sympos. Comput. Geom., 24, H. Yu, A., R. Poeddy, K. Vaadaajan, Pactical methods fo shape fitting and kinetic data stuctues using coe sets, in Poc. 2th Annu. Sympos. Comput. Geom., 24, Geometic Appoximation Using Coesets 34

Fall 2014 Randomized Algorithms Oct 8, Lecture 3

Fall 2014 Randomized Algorithms Oct 8, Lecture 3 Fall 204 Randomized Algoithms Oct 8, 204 Lectue 3 Pof. Fiedich Eisenband Scibes: Floian Tamè In this lectue we will be concened with linea pogamming, in paticula Clakson s Las Vegas algoithm []. The main