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Lee  RB  s *DjJ8t8  <l " q  j"University of California, Berkeley## #  ^A(txp_figl | SOURCEj\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$ \psi_t(x) = \bigoplus_{m \in \mathbb Z} \rho_{m,t}(x) \varphi_m(x) $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH418(BOXHEIGHT334"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2538PICTUREFILESIZE 18396H  0޽h ? 33___PPT10i.a{G0+D=' = @B +F  F(     <Ⱦe " V5 ,$  0 RDistortion: Smallest number C such that& T*   *x  c $t 0p t  RB  s *DjJ0  Be `Z <   Hre  , < S  <$ n kIn general, we will consider euclidean embeddings and the geometry of high dimensional euclidean spaces.Nl) l  <' "  ( ,$ 0 p(Why study this kind of geometry (in CS)?)) )  < " 6  ,$ 0 @ Applicability of low-distortion Euclidean embeddings Understanding semi-definite programs Optimization, harmonic analysis, hardness of approximation, cuts and flows, Markov chains, expansion, randomness& n-:-5%v    <̱+" (,$ 0 'GEuclidean embedding: A map f : X ! 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FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1476PICTUREFILESIZE8490 " ^Atxp_fig" fN~v SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\log n \to \log^{\frac{3}{4}} n$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1366PICTUREFILESIZE74616 $ ^Atxp_fig" [   SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $\log^{\frac{5}{6}} n \to \log^{\frac{3}{4}} n$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1476PICTUREFILESIZE8627H  0޽h ? 33Q I ___PPT10) .a{G0+"D ' = @B D ' = @BA?%,( < +O%,( < +D ' =%(%(Dw ' 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6;FDjJ1Y Yf [ <ȱ "  Every metric space (X,d) is composed of many scales (resolutions)& :C+ CB \ <DjJ* ,,$@ 0 d@ 6>"`NPx ,$@ 0b e ^Atxp_fig" f$D 0 SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$k\in \mathbb Z$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH506PICTUREFILESIZE2690: f < "  wH ,$ 0 4Often easy to construct an embedding for one scale: *54 5 g ^A txp_fig"  % $D 06 SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$ \phi_k : X \to \mathbb R^d $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1176PICTUREFILESIZE6262 r ^Atxp_fig" tg$@  0\ SOURCE@\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$ \textrm{whenever $d(x,y) \in [2^k, 2^{k+1})$} $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2868PICTUREFILESIZE 15649 s ^A txp_fig"  -r $$D  0p SOURCET\documentclass{slides}\pagestyle{empty} \begin{document} $$ (1)\,\,\,\,\,||\phi_k(x) - \phi_k(y)||_{_2} \leq d(x,y) \textrm{ for all $x,y \in X$} $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH4558PICTUREFILESIZE 23486 t ^A txp_fig" 9 );S@$D  0  SOURCEp\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$ (2)\,\,\,\,\,||\phi_k(x) - \phi_k(y)||_{_2} \geq \tfrac{1}{C} d(x,y) $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH3188PICTUREFILESIZE 19151H  0޽h ? 33& &___PPT10%.a{G0+saDM%' = @B D%' = @BA?%,( < +O%,( < +D ' =%(D+ ' =%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*^%(D' =4@BB*BB%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*e%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*f%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*g%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*s%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*r%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D ' =%(D ' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*^D' =1:Bhidden*o3>+B#style.visibility<*^%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*[D' =1:Bhidden*o3>+B#style.visibility<*[%(D)' =4@BB BB%(D' =-g6B fade*<3<*\D' =1:Bhidden*o3>+B#style.visibility<*\%(D)' =4@BB BB%(D' =-g6B fade*<3<*dD' =1:Bhidden*o3>+B#style.visibility<*d%(D)' =4@BB BB%(D' =-g6B fade*<3<*eD' =1:Bhidden*o3>+B#style.visibility<*e%(+p+0+[  ++0+f  +wI --ne-(  x  c $ 0p   RB  s *DjJ0 ^ < "  wH  4Often easy to construct an embedding for one scale: *54 5X _ ^A txp_fig"  % 6 SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$ \phi_k : X \to \mathbb R^d $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1176PICTUREFILESIZE6262 ` ^Atxp_fig" tg\ SOURCE@\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$ \textrm{whenever $d(x,y) \in [2^k, 2^{k+1})$} $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2868PICTUREFILESIZE 15649 a ^A txp_fig"  -r p SOURCET\documentclass{slides}\pagestyle{empty} \begin{document} $$ (1)\,\,\,\,\,||\phi_k(x) - \phi_k(y)||_{_2} \leq d(x,y) \textrm{ for all $x,y \in X$} $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH4558PICTUREFILESIZE 23486 b ^A txp_fig" 9 );S  SOURCEp\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$ (2)\,\,\,\,\,||\phi_k(x) - \phi_k(y)||_{_2} \geq \tfrac{1}{C} d(x,y) $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH3188PICTUREFILESIZE 19151 c <U> " Ai Idea of measured descent: Combine the component embeddings in a novel way to arrive at a final embedding which has low distortion.Xv j d ^A txp_fig" D ^hJ SOURCE.\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$ \big\{\phi_k\big\}_{k \in \mathbb Z} $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH776PICTUREFILESIZE6086! e <d> " x ,$@ 0 sProblem: The component maps have to be of a very special form (Frechet embeddings) because the proof is non-black-box; it breaks the embeddings apart and puts them back together in complicated ways.~ q1 V f ^A txp_fig" )$@ 0  SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$ \phi_k$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH226PICTUREFILESIZE18762 g 633>"`8,$@ 02 h s BCDEF jJlLHn< @"`BN,$@ 02 i s B`CDEF jJT6^H&`x @"`R,$@ 0" j s BC<DEF8cT< @"`hNP,$@ 0B k s *Dmf,$@ 0B l s *Dmd,$@  0B m s *Dm,$@  0B n s *Dm,$@  0H  0޽h ? 33JB___PPT10".a{G0+Uq^D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*f%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*g%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*i%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*j%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*k%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*m%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*n%(Dj' =4@BBBB%(/%,( < +)D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*kD' =0l9 BBBB*<3<*k)?Dj' =4@BBBB%(/%,( < +)D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*lD' =0l9 BBBB*<3<*l)?Dj' =4@BBBB%(/%,( < +)D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*mD' =0l9 BBBB*<3<*m)?Dj' =4@BBBB%(/%,( < +)D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*nD' =0l9 BBBB*<3<*n)?+% }u (  x  c $ĉ> 0p >  RB  s *DjJ0 ^ <h> "  S  >xGLUING LEMMA: We can construct an embedding of the same quality using the component maps as a black box.yi y r <> " Ai Idea of measured descent: Combine the component embeddings in a novel way to arrive at a final embedding which has low distortion.Xv j s ^A txp_fig" D ^hJ SOURCE.\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$ \big\{\phi_k\big\}_{k \in \mathbb Z} $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH776PICTUREFILESIZE6086 t <> " x  sProblem: The component maps have to be of a very special form (Frechet embeddings) because the proof is non-black-box; it breaks the embeddings apart and puts them back together in complicated ways.~ q1 * u ^A txp_fig" )  SOURCE\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$ \phi_k$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH226PICTUREFILESIZE1876l2 v 633>"`8 w s BCDEF jJlLHn< @"`BN x s B`CDEF jJT6^H&`x @"`R y s BC<DEF8cT< @"`hNPRB z s *DmfRB { s *DmdRB | s *DmRB } s *Dm ~ <> "  D  HQuantitatively, the distortion of the final map is .II I  ^Atxp_fig"    h` SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $$O(\sqrt{C \log n})$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1176PICTUREFILESIZE7731RB  s *D> "   C 0A'bumps-title- 8 ,$D 0"  <> "  L,$ 0 tPROOF:@ H  0޽h ? 33___PPT10.a{G0+CJD9' = @B D' = @BA?%,( < +O%,( < +D+' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+8+0+>  +[S )'!'.&(  x  c $\> 0p >  RB  s *DjJ0  <> " F  *Metrics of negative type arise as follows:4+   +V  <X> " H 0Let X Rk be endowed with the distance function~1' 1  ^Atxp_fig" 6v Ztl SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $d(x,y) = ||x-y||_{_2}^{^2}$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1736PICTUREFILESIZE8915f  0jJ"`P U  <> " =" ?If (X,d) is a metric space, then X is a space of negative type.n@  @2  6>"` Bz ,$@ 02  6>"` 0F ,$@ 0B  0D> 4Td ,$@ 0B  0D>d .0 ,$@ 02  6>"`M O ,$@ 0<    BKCDEF>"$EIHlK&@  "` ! ,$@ 0; ! <$> " B Pl ,$ 0  90oZ  " <> " z/ ,$  0 Ex  # <pF> " * _T ,$  0 Ey  $ <> "   ,$  0 Ez @ % ^Atxp_fig" Z r $@  0p SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $d(x,z) \leq d(x,y) + d(y,z)$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2408PICTUREFILESIZE 13261 & 0jJ"` P ,$@  0B ' s *DjJ v ,$@ 0B ( s *DjJ pp ,$@ 0j ) <> " . &,$ 0 XThe family of negative type metrics is much bigger than the family of Euclidean metrics!6Y' ' Y^ * ^Atxp_fig"   $@ 0 SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $$ ||x-z||^2 \leq ||x-y||^2 + ||y-z||^2 $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH3018PICTUREFILESIZE 10966B + s *DjJ qq ,$D 0T . ^A%txp_fig" q b%$D 0 SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $ = x \cdot x - 2 x \cdot y + y \cdot y$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH2066PICTUREFILESIZE7516H  0޽h ? 33++___PPT10+.a{G0+ԻDn*' = @B D)*' = @BA?%,( < +O%,( < +D$'' =%(D4$' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' 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=1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bdissolve*<3<*(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-o6Bdissolve*<3<**D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-o6Bdissolve*<3<*+D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-g6B fade*<3<*)D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(++0+! ++0+" ++0+# ++0+$ ++0+) +:U &&0,BU&(   B <T" ,$  0 4Improved version: Possible to choose the sets A and B  at random. D  Dx  c $X0p  RB  s *DjJ0  <Q" X THE ARV THEOREMJ RB  s *D>  < Q" <O rLet X Sn-1 be an n-point subset of the unit sphere endowed with the squared Euclidean metric d(x,y) = ||x-y||2, and suppose thatL    ^Atxp_fig" %^ &` SOURCED\documentclass{slides}\pagestyle{empty} \usepackage{amsmath,amssymb} \begin{document} $$ \tfrac{1}{n^2} \sum_{x,y \in X} d(x,y) \geq 0.1 $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1998PICTUREFILESIZE 14605 z AR   (  ,$D 0n2  0jJ"`f ZB  s *DjJj ZB  s *DjJp0 0 ZB   s *DRBZB ! s *D<* * ZB "B s *DA+ + ZB # s *D`  n2 $ 0jJ"`R  n2 % 0jJ"`b" V n2 & 0jJ"` ! n2 ' 0jJ"`b  n2 ( 0jJ"`y  n2 ) 0jJ"`  n2 * 0jJ"`  ? n2 + 0jJ"`h*n2 , 0jJ"`fn2 - 0jJ"`n2 . 0jJ"`4n2 / 0jJ"`n2 0 0jJ"`n2n2 1 0jJ"`  n2 2 0jJ"`  n2 3 0jJ"`n2 4 0jJ"`Dn2 5 0jJ"`8,l`2 6 <kLjJ"`]8 Y ,$@ 02 7 <LjJ"` & ,$@ 0B 8 ND8c ' ,$@ 0 9 <Q" ? j,$ 0 EA  : <ҟ" h  ,$ 0 EB r ; <XpI " qV ,$ 0 Then there exist two subsets A, B X such that |A|,|B| W(n) andC  C < ^Atxp_fig" n  _ $@  0h SOURCEL\documentclass{slides}\pagestyle{empty} \usepackage{amsmath} \usepackage{amssymb} \begin{document} $$ d(A, B) \geq \tfrac{1}{\sqrt{\log n}} $$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1556PICTUREFILESIZE9721 = 0jJ"`4 U ,$D  0B ? <D>> 6 ,$@  0` A <I " ,$  0 jGood enough for lots of applications& but not others.66 6H  0޽h ? 33..___PPT10-.a{G0+/|Dy,' = @B D4,' = @BA?%,( < +O%,( < +Ds' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' D=-s6Bwipe(down)*<3<*%(D' =eB]0,0; 0.14,0.36; 0.43,0.73; 0.71,0.91; 1.0,1.0+4 8?fCB#ppt_x-0.25BCB#ppt_xB*Y3>B ppt_x<*%(D' =mBe0.0,0.0; 0.25,0.07; 0.50,0.2; 0.75,0.467; 1.0,1.0+4 8?lCB?B'#ppt_y-sin(pi*$)/3CB?B*Y3>B ppt_y<*%(D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?lCBB'#ppt_y-sin(pi*$)/9CB?B*Y3>B ppt_y<*%(D' L=B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/27CB?B*Y3>B ppt_y<*%(,D' =B0, 0; 0.125,0.2665; 0.25,0.4; 0.375,0.465; 0.5,0.5; 0.625,0.535; 0.75,0.6; 0.875,0.7335; 1,1+4 8?nCBB)#ppt_y-sin(pi*$)/81CB?B*Y3>B ppt_y<*%(xD' =0l9 BBBpB*<3<*%(D' =0l9 BBBB*<3<*%())?D' =0l9 BBBB*<3<*%( D' =0l9 BBBB*<3<*%(:))?D' =0l9 BBBB*<3<*%(jD' =0l9 BBBB*<3<*%())?D' =0l9 BBBB*<3<*%(D' =0l9 BBBB*<3<*%(*))?D' =%(Do' =%(D[' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-g6B fade*<3<*6D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D[' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-g6B fade*<3<*7D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-o6Bdissolve*<3<*<D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =-o6Bdissolve*<3<*=DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*A%(+P+0+B ++0+9 ++0+:I  ++0+;I  ++0+AI  ++0+AI  + @ DDPa oD(   | <ff"`  ,$@ 0x  c $I 0p I  RB  s *DjJ0  <~~b L AR  -#  n2 . 0jJ"`f ZB / s *DjJj ZB 0 s *DjJp0 0 ZB 1 s *DRBZB 2 s *D<* * ZB 3B s *DA+ + ZB 4 s *D`  n2 5 0jJ"`R  n2 6 0jJ"`b" V n2 7 0jJ"` ! n2 8 0jJ"`b  n2 9 0jJ"`y  n2 : 0jJ"`  n2 ; 0jJ"`  ? n2 < 0jJ"`h*n2 = 0jJ"`fn2 > 0jJ"`n2 ? 0jJ"`4n2 @ 0jJ"`n2 A 0jJ"`n2n2 B 0jJ"`  n2 C 0jJ"`  n2 D 0jJ"`n2 E 0jJ"`Dn2 F 0jJ"`8,l`x2 G BkLjJ"`e&ax2 H BLjJ"`   I BDI " GX EA  J BtI " p   EB Bl oo  x9+ ,$@  0t K 6RW"` K z L <RWff"` `B MB 0Do9 o `B N 0D0 R <I " rd& :Choosing A and B  at random. N    w <LI " tp ,$ 0 r1. Choose a random hyperplane.& 6 y <I " q @,$ 0 42. Prune the  exceptions. &  { 6"` U ,$@ 0B }@ <;DjJ $ ,$@ 0B  6;DjJ6 ,$@ 02  0jJ"`7k,$@ 02  0jJ"`v,$@ 02  0jJ"` 4 ,$@ 0   <I "  \ ,$j 0 ^Pruning ) d(A,B) is large. The hard part is showing that |A|,|B| = W(n) whp.M)  M"  <lI " F R-,$n 0 t,ARV yields .-- -2  <(I " ,$k 0 <We obtain the optimal bound: .== =@  ^Atxp_fig"  $@l 0p SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $d(A,B) \sim 1/\sqrt{\log n}$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1878PICTUREFILESIZE 10353P  ^Atxp_fig"  x$@m 0 SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $d(A,B) \sim 1/\log^{\frac{2}{3}}(n)$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH354(BOXHEIGHT426"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono(ORIGWIDTH1988PICTUREFILESIZE 122342  0jJ"` ; o G ,$@ 02  0jJ"`}W 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