Remarks on the Geometry of Coordinate Projections in R n

Transcription

1 Remarks o the Geometry of Coordiate Projectios i R S. Medelso R. Vershyi Abstract We study geometric properties of coordiate projectios. Amog other results, we show that if a body K R has a almost extremal volume ratio, the it has a projectio of proportioal dimesio which is close to the cube. We also establish a sharp estimate o the shatterig dimesio of the covex hull of a class of fuctios i terms of the shatterig dimesio of the class itself. 1 Itroductio I this article we preset several results o coordiate projectios. The majority of this article is devoted to ew applicatios of the etropy iequality established i [MV], which, roughly speakig, states that if a set of fuctios has a large etropy i L, it must have a coordiate projectio which cotais a large cube. Defiitio 1.1 We say that a subset σ of Ω is t-shattered by a class of real-valued fuctios F if there exists a level fuctio h o σ such that, give ay subset σ of σ, oe ca fid a fuctio f F with f(x) h(x) t if x σ ad f(x) h(x) + t if x σ \ σ. The shatterig dimesio of A, deoted by vc(f, Ω, t) after Vapik ad Chervoekis, is the maximal cardiality of a subset of Ω which is t-shattered by F. I cases where the uderlyig space is clear we deote the shatterig dimesio by vc(f, t). Research School of Iformatio Scieces ad Egieerig, The Australia Natioal Uiversity, Caberra, ACT 000, Australia, shahar.medelso@au.edu.au Departmet of Mathematical Scieces, Uiversity of Alberta, Edmoto, Alberta T6G G1, Caada, rvershyi@math.ualberta.ca 1

2 Theorem 1. Let A be a class of fuctios bouded by 1, defied o a set Ω. The for every probability measure µ o Ω, ( ) K vc(f, ct), N(F, t, L (µ)) 0 < t < 1, (1.1) t where K ad c are positive absolute costats. Every F R ca be idetified with a class of fuctios o {1,..., } i the atural way: v(i) = v i for v F. If we take µ to be the probability coutig measure o {1,..., } the (1.1) states that for ay 0 < t < 1, N(F, t ( B ) t ) K vc(f, ct). Note that if F happes to be covex ad symmetric with respect to the origi, the vc(f, t) is the maximal cardiality of a subset σ of {1,..., } such that P σ (F ) [ t, t] σ. We apply this result to study covex bodies whose volume ratio is almost maximal. Recall that the volume ratio, itroduced by Szarek ad Tomczak-Jaegerma [S, ST], is defied as vr(d) = ( D / E ) 1/, where E is the ellipsoid of maximal volume cotaied i D. The miimal volume ratio of a symmetric covex body i R is 1 ad is attaied by the Euclidea ball; the maximal is of the order of ad is attaied by the cube B [B]. This pair of extremal bodies is uique up to a liear trasformatio. Ideed, the uiqueess of the miimizer is immediate, while the fact that cube is the uique maximizer was established i [Ba]. The isomorphic versio of this fact describig the bodies whose volume ratio is of order either 1 or is of particular iterest. The questio is whether such bodies iherit ay structure from the Euclidea ball or, respectively, from the cube. If the volume ratio of a body D i R is bouded by a costat, the by the Volume Ratio Theorem [ST], D has a sectio of dimesio proportioal to, which is well isomorphic to the Euclidea ball. O the other had, if vr(d) is of order of, the by [R] ad [V], D has a sectio of dimesio proportioal to, which is well isomorphic to the cube, ad the order of i the dimesio ca ot be improved (the dual of Gluski s polytope is such a example see sectio ). However, we show i sectio that there exists a projectio of D of dimesio proportioal to, which is well isomorphic to the cube.

3 Two other applicatios we preset are based o the followig corollary of Theorem 1.. Theorem 1.3 [MV] Let F be a class of fuctios bouded by 1, defied o a fiite set I of cardiality. The the gaussia process idexed by F, X f = g if(i) satisfies E = E sup X f K f F 1 ce/ where K ad c are absolute costats. vc(f, t) log(/t) dt, Oe applicatio we preset is a compariso of the average E ε ix i to the miimum over all choices of sigs, mi ±x i. As a cosequece, we compare the type ad the ifratype costats of a Baach space. The, we establish a sharp estimate o the shatterig dimesio of a covex hull of a class of fuctios, based o the shatterig dimesio of the class itself. Namely, we show that for every ε > 0, vc(cov(f ), ε) (C/ε) vc(f, cε), where c ad C are absolute costats. The fial questio we address is whe a radom coordiate projectio a almost isometry. Let (Ω, µ) be a probability space ad f L (µ), ad for simplicity, assume that Ω = {1,..., } ad that µ is the uiform probability measure o Ω. For every ε > 0, our aim is to fid may sets σ {1,..., } of small cardiality such that the atural coordiate projectio P σ f satisfies (1 ε) f L P σ f L σ (1 + ε) f L, (1.) where L k is the L space defied o {1,..., k} with respect to the uiform probability measure. By a stadard cocetratio argumet, if f 1, the with high probability a radom coordiate projectio of dimesio C/ε is a almost isometry i the sese of (1.). We will show that the uiform boudedess of f ca be relaxed; it suffices to assume that f ψ 1, where ψ is the Orlicz orm geerated by the fuctio e t 1. I this case, a radom coordiate projectio of dimesio C/ε will be a almost isometry as 3

4 i (1.) with high probability. Although this result is relatively easy, we decided to preset it because it gives hope that the coditios i stroger cocetratio iequalities (e.g. Talagrad s cocetratio iequality for empirical processes [T 94, L]) ca also be relaxed. As a applicatio, we obtai a coordiate versio of the Johso-Lidestrauss Flatteig Lemma [JL]. Fially, we tur to some otatioal covetios. Throughout, all absolute costats are deoted by c, C, k ad K. Their values may chage from lie to lie or eve withi the same lie. We deote a b if there are absolute costats c ad C such that cb a Cb. ACKNOWLEDGEMENTS: The first author ackowledges partial support by a Australia Research Coucil Discovery Grat. The secod author thaks Nicole Tomczak-Jaegerma for her costat support. A part of this work was doe whe the secod author was visitig the Research School of Iformatio Scieces ad Egieerig at The Australia Natioal Uiversity, which he thaks for its hospitality. He also ackowledges a support from the Pacific Istitute of Mathematical Scieces ad is grateful to the Departmet of Mathematical Scieces of the Uiversity of Alberta for its hospitality. Extremal volume ratios The volume ratio of a covex body D i R is defied as vr(d) = if ( D ) 1/, E where deotes the volume i R, ad the ifimum is over all ellipsoids E cotaied i D. This importat ivariat was itroduced by Szarek ad Tomczak-Jaegerma (see [S], [ST] or [P]). The bodies with extremal volume ratios are the Euclidea ball ad the cube ad these are the oly extreme bodies up to a liear trasformatio (for the uiqueess of the cube, see [Ba]). Oe ca show that for every covex symmetric body i R, 1 = vr(b ) vr(d) vr(b ), (.1) (see [B]), while direct computatio shows that vr(b ) C ad the best value of the costat is C = / πe. 4

5 Ofte, oe ecouters bodies whose volume ratio is almost extremal, i.e. close to oe of sides of (.1). The problem is whether such a body iherits properties of the extremal bodies, the Euclidea ball or the cube. If vr(d) A, the by the Volume Ratio Theorem [ST], D has a sectio of dimesio k = / which is ca -isomorphic to the Euclidea ball B k, ad this result is asymptotically sharp. O the opposite side of the scale, if vr(d) A 1, D has a sectio of D of dimesio k = c(a) which is C(A) log -isomorphic to the cube B k [R, V]. It is ot kow whether the logarithmic term ca be elimiated, but the order of i the dimesio is optimal, as was oticed i [GTT]. Ideed, by a argumet of Figiel ad Johso (see [FJ], cor. 3.), a radom subspace E l (ad thus a dual of Gluski s space) of dimesio at least / satisfies that for ay F E, gl(f ) cdim(f )/, where gl(f ) is the Gordo-Lewis costat of F, ad c is a suitable absolute costat. By [GL], gl(f ) uc(f ), where uc(f ) is the least ucoditioality costat of a basis of F. Sice uc(l k ) = 1 the d(f, l k ) uc(f ) cdim(f ), ad thus, if F is -isomorphic to l k the dim(f ) c. Our ext result shows that D has a projectio of dimesio proportioal to which is ca-isomorphic to the cube B k. Theorem.1 There are absolute costats C ad c for which the followig holds. If D is a covex symmetric body i R for which vr(d) A 1, the there exists a projectio P of rak k c/ log A such that d(p K, B k ) CA. To prove the Theorem, recall the otio of the cubic ratio [B]. For every ball D R oe defies ( B cr(d) = if ) 1, T D where the ifimum is over all liear ivertible operators T o R such that T D B. Lemma. [B] There are absolute costats c ad C such that for every iteger ad every covex symmetric body D R, c vr(d) cr(d) C. 5

6 Proof of Theorem.1. Clearly, we ca assume to be larger tha a suitable absolute costat N, which esures that for every D R, vr(d) 0.8. Sice vr(d) A 1, the by Lemma., cr(d) CA. Hece, there is some T GL such that T D B ad T D 1 c/a. Recall that c 1 B c for some absolute costats c 1, c, ad thus there exists a absolute costat c 3 such that = T D c 3 A 1 ( B ). By a stadard volumetric argumet, the right-had side is bouded by N(T D, c 3 A 1 B ), ad by Theorem 1. there are absolute costats K ad c for which log N(T D, c 3 A 1 B ) K vc(t D, ca 1 ) log(ca). Hece, there is a set σ {1,..., }, such that σ /K log(ca) ad c 1 A 1 B σ P σ (T D) B σ. It oly remais to ote that log(ca) C log A, because A 5/4. Remark. Sice the volume ratio is always greater tha 1, the A 1/. Therefore, the dimesio of the cubic projectio i Theorem.1 is always bouded below by c/ log. I a very similar way, oe ca prove the followig Theorem.3 There are absolute costats C ad c for which the followig holds. Let D be a covex symmetric body i R for which vr(d) A 1. The there exists a projectio P of rak k c such that 3 Type ad Ifratype d(p K, B k ) CA. I this sectio we improve a result of M. Talagrad [T 9] which compares the average over the ± sigs to the miimum over the ± sigs of ±x i. 6

7 Recall that a Baach space X has a (gaussia) type p if there exists some M > 0 such that for all ad all sequeces of vectors (x i ) i, E ( g i x i M x i p) 1 p. (3.1) The best possible costat M i this iequality is deoted by T p (X). We say that X has ifratype p if there exists some M > 0 such that for all ad all sequeces of vectors (x i ) i, mi η i =±1 ( η i x i M x i p) 1 p. (3.) The best possible costat M i this iequality is deoted by I p (X). I [T 9] it was show that if 1 < p < the T p (X) C p I p (X), where C p is a costat which depeds oly o p. It is ot kow whether the square ca be removed. Regardig the case p =, M. Talagrad recetly costructed a symmetric sequece space which has ifratype but ot type [T 03]. Hece oe ca ot obtai dimesio free estimates o T (X) i terms of I (X). Our mai result i this sectio is that is dim(x) = the T (X) CI (X) log 3/. We begi with the followig fact that allows oe to compare Rademacher ad Gaussia averages. Lemma 3.1 There is a absolute costat C for which the followig holds. Let x 1,..., x be vectors i the uit ball of a Baach space ad let 0 < M. If 0 < λ < log 3 (/M ) ad mi η i =±1 i σ for all σ {1,..., } with σ λ, the, E η i x i M σ 1 g i x i CM(/λ) 1/. I the proof of Lemma 3.1 we require the followig observatio from [MS], that if {x 1,..., x } X is ε-shattered by B X, the for ay (a i ) R,. ε a i a i x i (3.3) 7

8 Proof of Lemma 3.1. Clearly we ca assume that the give Baach space is X = (R, ) ad that (x i ) i are the uit coordiate vectors i R. Set B = B X ad by the hypothesis of the lemma ad (3.3), vc(b, Mv 1/ ) v if 0 v λ. Hece, for ay M(λ) 1/ t 1, Set E = E vc(b, t) (M/t). (3.4) g i x i X = E sup b B g i b(i). By Theorem 1.3, there are absolute costats C ad c such that E K 1 vc(b, t) log(/t) dt. ce/ If ce/ M(λ) 1/, the lemma trivially follows. Otherwise, if the coverse iequality holds, the by (3.4) ad sice λ < 1, E K 1 (M/t) log(/t) dt K M log 3 (/M ), ce/ ad by the assumptio o λ, as claimed. E K M log 3 (/M ) K M/ λ, Usig Lemma 3.1, oe ca compare the type ad ifratype of a Baach space X. Let T () (X) ad I () (X) deote the best possible costats M i (3.1) ad (3.) respectively (with p = ). So, T () (X) ad I () (X) measure the type/ifratype computed o vectors. Clearly, I (X) T (X) ad I () (X) T () (X). Theorem 3. Let X be a -dimesioal Baach space. The, for every umber 0 < λ < log 3 (/I (X) ), T (X) C I (λ) (X)/ λ. 8

9 I particular, ( T (X) I (X) C log 3 ) I (X) I (X) C log 3. Proof. By [TJ] ad [BKT] Theorem 3.1, the Gaussia type ca be computed o vectors of orm oe. Precisely, this meas that T (X) is the smallest possible costat M for which the iequality E g i x i M 1/ holds for all vectors x 1,..., x of orm oe. Now, the assertio follows from Lemma The shatterig dimesio of covex hulls I this sectio we preset a sharp estimate which compares the shatterig dimesios of a class ad of its covex hull. To that ed, we coect the shatterig dimesio to the growth rate of the expectatio of the supremum of the gaussia process {X a, a P σ F } as a fuctio of σ. Defiitio 4.1 Let F be a class of fuctios bouded by 1 ad set l (F ) = sup E g sup g i f(x i ), (x 1,...,x ) Ω f F where g 1,..., g are idepedet, stadard gaussia radom variables. Hece, l (F ) is the largest gaussia average associated with a coordiate projectio of F o poits. Sice F B ( L (Ω) ) the for every σ = (x 1,..., x ), { (f(x1 P σ F = ),..., f(x ) ) } : f F B, ad the largest projectio oe might ecouter is whe P σ F = B, i which case l(p σ F ). We defie a scale-sesitive parameter which measures for every ε > 0 the largest cardiality of a projectio which has a large l-orm: t(f, ε) = sup{ : l (F ) ε}. 9

10 Theorem 4. There are absolute costats K, c ad c such that for ay F B ( L (Ω) ) ad every ε > 0, vc(f, c ε) t(f, ε) (K/ε ) vc(f, cε). I the proof, we will use the followig wordig: the fuctio f associated to a set σ i the Defiitio 1.1 will be called the fuctio that shatters σ. Proof of Theorem 4.. Assume that {x 1,..., x } is ε-shattered by F. For every J {x 1,..., x }, let f J be the fuctio shatterig J, ad for each (ε 1,..., ε ) { 1, 1} set I = {x i ε i = 1}. By the triagle iequality ad lettig f = f I, f = f I c i the secod iequality below, Hece, ε i f(x i ) 1 sup ( ε i f(xi ) f (x i ) ) f,f F 1 ( ε i fi (x i ) f I c(x i ) ) ε. sup f F sup f F ε i f(x i ) ε, ad i particular this holds for the average. The first boud is evidet because of the kow coectios betwee gaussia ad Rademacher averages [TJ1], amely, that there is a absolute costat C such that for ay class F ad ay set σ = (x 1,..., x ), l(p σ F ) = E g g i e i (PσF ) C E ε = C E ε sup f F ε i f(x i ). ε i e i (PσF ) The reverse iequality follows from Theorem 1.3 i a similar way to the proof of Elto s Theorem i [MV]. If l(p σ F ) ε, the E sup f F X f ε, where X f = g if(x i ). By Theorem 1.3, ε E sup X f K f F 1 cε vc(pσ F, t) log(/t) dt. 10

11 c Set v(t) = 0 t log 1.1 (/t) where c 0 > 0 is chose so that 1 0 v(t) dt = 1. Hece, there is some cε t 1 such that K vc(p σ F, t) ε v (t)/ log(/t), implyig that vc(f, cε) vc(p σ F, cε) vc(p σ F, t) c ε t log 3. (/t) c ε. The previous result ca be used to estimate the shatterig dimesio of a covex hull of a class. Corollary 4.3 There are absolute costats K ad c such that for ay F B ( L (Ω) ) ad every ε > 0, vc(cov(f), ε) (K/ε) vc(f, cε). Proof. Sice the l-orm of a set ad of its covex hull are the same, the for ay ε > 0, t(f, ε) = t(cov(f ), ε). By Theorem 4., vc(cov(f ), ε) t(cov(f ), ε) = t(f, ε) (K/ε) vc(f, cε). Next, we show that this estimate is sharp, i the sese that the expoet of 1/ε ca ot be improved. To that ed, we require some properties of the shatterig dimesio of classes of liear fuctioals metioed before, which was ivestigated i [MS]. If X is a ormed space the B X ca be viewed as a subset of L (B X ) i the atural way. It is ot difficult to characterize the shatterig dimesio i this case. Lemma 4.4 A set {x 1,..., x } B X is ε-shattered by B X if ad oly if (x i ) are liearly idepedet ad ε-domiate the l 1 uit-vector basis; i.e., ε a i a i x i a i for every a 1,..., a R. I particular, if X is -dimesioal, ad if the Baach-Mazur distace satisfies that d(x, l 1 ) α, the vc(b X, B X, 1/α) =. 11

12 Corollary 4.5 There exists a absolute costat k for which the followig holds. For every 0 < ε < 1/ there is a class F B ( L (Ω) ) such that vc(cov(f ), ε) k vc(f, ε) ε log(1/ε). Proof. For every iteger, let Ω = B ad set F = {e 1,..., e }, that is, the stadard uit vectors i R, whe cosidered as liear fuctioals o B. Sice F =, it follows that for every ε > 0, vc(f, Ω, ε) log. O the other had, cov(f ) = B1 whe cosidered as fuctioals o B. By Lemma 4.4 applied to X = l, ad sice d(l, l 1 ) K [TJ1], it is evidet that there is a subset o cardiality i B which is k/ -shattered by B1. Thus, for ε = k/, vc(cov(f ), Ω, ε ) k vc(f, Ω, ε ) ε, log(1/ε ) from which the proof easily follows. 5 Almost isometric coordiate projectios Give real-valued fuctio f o a probability space, its ψ p -orm (p 1) is defied as the Orlicz orm correspodig to the fuctio exp(t p ) 1. Precisely, f ψp is the ifimum of all umbers λ satisfyig E exp( f p /λ p ) e. It is possible to compare the ψ p with other ψ q orms ad the L p orms. Ideed, oe ca show that if 1 p q <, f ψp C p,q f ψq, ad f Lp C p f ψ1 (see, for example, [VW]). A fuctio f is bouded i the ψ orm if ad oly if f has a subgaussia tail. Namely, if f ψ 1 the by Chebychev s iequality P{ f > t} e t +1 for all t > 0. Coversely, if for some A 1 oe has P{ f > t} Ae t for all t > 1, the itegratig by parts it follows that E exp(f/) 1 + A/3 A, ad by Jese s iequality oe ca coclude that f ψ A (we did ot attempt here to give the right depedece o A). Aother simple but useful fact which follows from Jese s iequality is that f ψ C E exp(f ), where C is a absolute costat. We will focus o fuctios defied o a fiite domai, which we idetify with {1,..., }, equipped with a uiform measure, where each atom carries a weight of 1/. We deote the ψ orm of a fuctio f o this 1

13 probability space by f ψ. Sice f is defied o {1,..., }, we sometimes idetify f with the sequece of scalars (f(i)). We shall use the followig stadard probabilistic model for radom coordiate projectios. Give 0 < δ 1/, let δ 1,..., δ be selectors, i.e. idepedet {0, 1}-valued radom variables with mea δ. The σ = {i 1 i, δ i = 1} is a radom subset of the iterval {1,..., } with average cardiality δ. By Berstei s iequality [VW], for every 0 < ε < 1, { P (1 ε) f L 1 δ } δ i f(i) (1 + ε) f L ( 1 exp cε δ ), f ad by aother applicatio of Berstei s iequality, { 1 } P δ i δ ε exp( cε δ), (5.1) δ implyig that if f 1, the with probability at least 4 exp( cε σ ), (1 ε) f L P σ f L σ (1 + ε) f L. I this sectio we relax the assumptio that f is bouded i the uiform orm, ad assume that f is bouded i the ψ orm. Roughly speakig, we show that for every 1 p <, the set of vectors i S(L p ) which will be almost isometrically projected oto L σ p are those with a small ψ p orm. Propositio 5.1 Let (δ i ) be idepedet {0, 1}-valued radom variables with mea δ > 0. Set a = (a i ) R ad put M = a ψ 1. The, for every positive umber t < M/, { } P (δ i δ)a i > tδ where c is a absolute costat. ( exp ct δ ) M, The proof starts with the followig stadard lemma. Lemma 5. Let Z be a radom variable ad assume that for some b, λ > 0, E exp(λz) e b λ. The P{Z > b λ} e b λ. 13

14 Proof. For t > 0, P{Z > t} = P{exp(λ(Z t)) > 1} E exp(λ(z t)) = e λt E exp(λz) e b λ λt. Settig t = b λ completes the proof. Proof of Propositio 5.1. By homogeeity, we ca assume that M = 1, ad we shall evaluate E exp ( t (δ ) i δ)a i. To that ed, let δ i be a idepedet copy of δ i ad set δ i = δ i δ i. By Jese s iequality, E exp ( t ) ( (δ i δ)a i E exp t (δ i δ i)a ) i = E exp(t δ i a i ) = E. Set δ = δ(1 δ) ad ote that δ i is 0 with probability 1 δ, ad 1 ad 1, each with probability δ. Therefore, E exp(t δ i a i ) = (1 δ) + δe ta i + δe ta i = 1 + δ(cosh(ta i ) 1). Sice cosh x x e x for all real x, the E (1 + δt a i e t ai ) exp( δt a i e t ai ) = exp ( δt 1 ) a i e t a i. The ormalized sum is estimated by Cauchy-Schwartz ad usig the fact that t 1: 1 ( 1 a i e t ai a i 4) 1 ( 1 ) 1 e t a i ( 1 ) 1 a L e a i 4 a L 4 C, because c a L 4 a ψ 1 1. Hece, E exp(c δt ) exp(c δt ). We put this i a form coveiet for applyig Lemma 5.: ( 1 E exp tδ δ ) ( C (δ i δ)a i exp δ (tδ)) 14

15 ad apply the lemma for λ = tδ ad b = C δ. It follows that for every t > 0, { 1 P (δ i δ)a i > ct} exp( cδt ), δ which completes the proof. Corollary 5.3 Applyig Propositio 5.1 to a i it is evidet that where M = a ψ 1 { } ( P (δ i δ)a i > tδ exp ct δ ) M ad 0 < t < M/. A easy applicatio of this corollary is the fact that the ψ -orm of poits o the sphere determies the cardiality of a almost isometric projectio. Corollary 5.4 There is a absolute costat C for which the followig holds. For every iteger, ay f S(L ) ad every ε > 0, a radom set σ {1,..., } of average cardiality (CM/ε) satisfies with probability at least 1/ that 1 ε P σ f L σ 1 + ε, where M = f ψ. Proof. The proof follows immediately from Corollary 5.3, by takig a i = f (i) ad δ = (CM/ε), ad applyig (5.1). Note that a similar result ca be easily derived for ay 1 p <, simply by the fact that (a i ) ψ p = (a p i ) ψ 1. Corollary 5.3 ca be used to preset a ew isight to the well kow Johso-Lidestrauss Flatteig Lemma [JL], which states that every set {x 1,..., x } l ca be 1 + ε isometrically embedded i lm, where m (C/ε) log. Oe ca formulate the Johso-Lidestrauss Lemma as follows: Theorem 5.5 There is a absolute costat C for which the followig holds. For every f 1,..., f S(L ) ad every ε > 0 there is a orthogoal operator O ad a set σ {1,..., } of cardiality at most (C/ε) log, such that for all 1 i, 1 ε P σ Of i L σ 1 + ε. 15

16 As Corollary 5.4 shows, a almost isometric coordiate projectio of f is possible, as log as f ψ is small; hece, the ψ orm defies a good regio o the sphere for which a radom coordiate projectio will be a almost isometry. I a similar way, this ca also be performed with may fuctios simultaeously: Corollary 5.6 There is a absolute costat C for which the followig holds. For every f 1,..., f S(L ) ad every ε > 0 a radom set σ {1,..., } of cardiality (CM/ε) log satisfies that with probability at least 1/, 1 ε P σ f i L σ 1 + ε, 1 i. where M = max i f i ψ. Proof. As i Corollary 5.4, but takig δ = (CM/ε) log, we obtai the for every 1 i The P r{1 ε P σ f i L σ 1 + ε} 1 1. P r{ 1 i, 1 ε P σ f i L σ 1 + ε} 1/, which completes the proof. The coectio to the Johso-Lidestrauss Lemma is easy: with high probability, a radom orthogoal operator O will map ay set of vectors o the sphere to the good regio, i.e. to the regio where the ψ orm is bouded by a absolute costat. Lemma 5.7 There is a absolute costat C such that for every iteger ad ay x S(L ), P r O { Ox ψ C } < 1, where the probability measure is the Haar measure o the orthogoal group. As a cosequece, for every f 1,..., f S(L ), max Of i ψ i C with probability greater tha 1/, ad thus Theorem 5.5 is implied by Corollary

17 Proof. Clearly, it suffices to show that there is a absolute costat C such that P r { x S 1 : x ψ C } 1. Cosider the fuctio g : S 1 R defied by g(x) = x ψ. To estimate its Lipschitz costat, observe that for every x S 1, x ψ / log. Ideed, for 0 x 1, x / x + 1; hece, 1 exp( x i log ) = 1 x i / 1 (x i + 1). To boud the expectatio of g (with respect to the Haar measure o the sphere), recall the media of the fuctio f(x) = x 1 satisfies that M f c, ad that f lip 1. Hece, by cocetratio of measure o the sphere [MiS], for ay s > c ad every 1 i, P r { x S 1 : x i s } π / e s, ad thus, E exp(cx i ) for a appropriate absolute costat c. Recall that there is a absolute costat K such that for every fuctio f, f ψ KE exp(f ); therefore, for x = (x 1,..., x ), cx ψ K exp(cx i ). Takig the expectatio with respect to x o the sphere, E x ψ K E exp(cx i ) K for a absolute costat K. By the cocetratio of measure o the sphere applied to the fuctio g, P r { x S 1 : x ψ C + t } π log e ct, ad the claim follows by selectig t = C /. 17

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