Empirical Processes and random projections

Transcription

1 Empirical Processes and random projections B. Klartag, S. Mendelson School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA. Institute of Advanced Studies, The Australian National University, Canberra, ACT 000, Australia. Abstract In this note we establish some bounds on the supremum of certain empirical processes indexed by sets of functions with the same L norm. We present several geometric applications of this result, the most important of which is a sharpening of the Johnson-Lindenstrauss embedding Lemma. Our results apply to a large class of random matrices, as we only require that the matrix entries have a subgaussian tail. Introduction The goal of this article is to present bounds on the supremum of some empirical processes and to use these bounds in certain geometric problems. The original motivation for this study was the following problem: let X,..., X be independent random vectors in R n and let Γ : R n R be the random operator Γv = X i, v e i, where {e,..., e } is the standard orthonormal basis in R. Assume for simplicity that E X i, t = for any i and any t S n, where S n denotes the Euclidean sphere in R n. Given a subset T S n, we as whether the random operator Γ : l n l almost preserves the norm of all elements of T. To ensure that Γ is almost norm preserving on T it suffices to analyze the supremum of the random variables Zt = Γt, and to show l that with positive probability, sup Z t is sufficiently small. An important example is when the random vector X i = ξ, i..., ξn, i and ξj in i,j= are independent random variables with expectation 0, variance and have a subgaussian tail, and thus, Γ is a random n subgaussian matrix. A standard concentration argument applied to each Zt individually shows that if T = n and if c log n ε, then with high probability, for every t T, ε Γt l + ε, and thus, Γ almost preserves the norm of all the elements in T. This simple fact is the basis of the celebrated Johnson-Lindenstrauss Lemma [7] which analyzes the ability to almost isometrically embed subsets of l in a Supported by NSF grant DMS-098 and the Bell Companies Fellowship. Supported by an Australian Research Council Discovery grant.

2 Hilbert space of a much smaller dimension; that is, for A l, to find a mapping f : T l such that for any s, t A ε s t l fs ft l + ε s t l. We will call such a mapping an ε-isometry of A. In [7], Johnson and Lindenstrauss proved the following: Theorem. There exists an absolute constant c for which the following holds. If A l, A = n and = c log n ε, there exists an ε-isometry f : l l. The actual proof yields a little more than this statement, from which the connection to the problem we are interested in{ should become clearer. By considering xi x the set of normalized differences T = j x : i j i x j }, if one can find a linear mapping which almost preserves the norms of elements in T, this mapping would be the desired ε-isometry; and indeed, a random n subgaussian matrix is such a mapping. Of course, the complexity parameter for T in the Johnson-Lindenstrauss Lemma is rather unsatisfying - the logarithm of the cardinality of the set T. One might as if there is another, sharper complexity parameter that could be used in this case. Let us consider a more general situation. Let T, d be a metric space, let F T = {f t : t T } L µ be a set of functions on a probability space Ω, µ and denote FT = {f : f F T }. Given X i, independent random variables distributed according to µ, let µ = δ X i be the empirical uniform probability measure supported on X,..., X i.e. µ is a random discrete measure. For any class of functions F L µ, set µ µ F = sup fdµ fdµ = sup fx i Ef f F, and note that µ µ F is a random variable. Our goal is to find a bound for µ µ F T that holds with a positive probability, under the assumption that f dµ = for any f F T. This bound should depend on the geometry of F T and, under some additional mild assumptions, reflect the metric structure of T, d. One possible way of bounding µ µ F which is almost sharp, is based on a symmetrization argument due to Giné and Zinn [5], namely, that for every set F E µ µ F CE X E g g i δ Xi, F where g i are standard independent gaussian random variables and C > 0 is an absolute constant. Let f F P X,..,X F = {fx,..., fx : f F } R

3 be a random coordinate projection of F. Fixing such a random projection, one needs to bound the expectation of the gaussian process indexed by that projection. To that end, let us remind the reader of the definition of the γ- functionals see [8]: Definition. For a metric space T, d define γ α T, d = inf sup s/α dt, T s, where the infimum is taen with respect to all subsets T s T with cardinality T s s and T 0 =. If the metric is clear we denote the γ functionals by γ α T. By the celebrated majorizing measures Theorem see the discussion before Theorem. for more details, if {X t : t T } is a gaussian process, and d s, t = E X s X t is its covariance structure, then s=0 c γ T, d E sup X t c γ T, d, where c, c { > 0 are absolute constants. Observe that for every X,..., X } the process g ifx i : f F is gaussian and its covariance structure satisfies E g g i f X i g i f X i = f f L µ. Thus, E µ µ F C E X γ P X,..,X F, L µ, and in particular this general bound holds for the set FT. Unfortunately, although this bound is almost sharp and is given solely in terms of γ, it is less than satisfactory for our purposes as it involves averaging γ of random coordinate projections of F with respect to the random L µ structure, and thus could be rather hard to estimate. On the other hand, if one wishes to estimate the expectation of the supremum of a general empirical process E µ µ F with a more accessible geometric parameter of the set F which does not depend on the random coordinate structure of F, the best that one can hope for in general is a bound which is a combination of γ and γ. This too follows from Talagrand s generic chaining approach [8] see Theorem. and Lemma 3. here. The main result we present here is that if a process is indexed by a set of functions with the same L norm, then with non-zero probability the γ part of the bound can be removed. In some sense, the process behaves as if it were a purely subgaussian process, rather than a combination of a subgaussian and sub-exponential. 3

4 To formulate the exact result, we require the notion of the Orlicz norms ψ p. Let X be a random variable and define the ψ p norm of X as X ψp = X p inf C>0 E exp C. A standard argument shows that if X has a bounded p ψ p norm then the tail of X decays faster than exp u p / X p ψ p [9]. In particular, for p =, it means that X has a subgaussian tail. Theorem.3 Let Ω, µ be a probability space and let X, X, X,..., X be independent random variables distributed according to µ. Set T to be a collection of functions, such that for every f T, Ef X = f L = and f ψ β. Define the random variable Z f = f X i f L. Then for any e c γ T, ψ < δ <, with probability larger than δ, sup Zf cδ, β γ T, ψ f T where c > 0 is an absolute constant and cδ, β depends solely on δ, β. The three applications we present have a geometric flavor. The first, which motivated our study, is concerned with random projections and follows almost immediately from Theorem.3. Theorem.4 For every β > 0 there exists a constant cβ for which the following holds. Let T S n be a set and let Γ : R n R be a random matrix whose entries are independent, identically distributed random variables, that satisfy EΓ i,j = 0, EΓ i,j = and Γ i,j ψ β. Then, with probability larger than /, for any x T and ε c γt, ε Γx l < + ε. In the case of a Gaussian projection, our result was proved by Gordon Corollary.3 in [6]. However, that proof uses the special structure of Gaussian variables and does not seem to generalize to arbitrary subgaussian random variables. Note that for a set T S N of cardinality n, γ T, c log n. In particular, Theorem.4 improves the Johnson-Lindenstrauss embedding result for any random operator with i.i.d ψ entries which are centered and have variance. From here on we shall refer to such an operator as a ψ operator. The novelty in Theorem.4 is that the logn term can be improved to γt in the general case, for any ψ operator. 4

5 The second application that follows from the theorem is an estimate on the so called Gelfand numbers of a convex, centrally symmetric body, that is, on the ability to find sections of the body of a small Euclidean diameter a question which is also nown as the low M -estimate. Indeed, with a similar line of reasoning to the one used in [, 6, 9] where such an estimate on the Gelfand numbers was obtained using a random orthogonal projection or a gaussian operator, one can establish the next corollary. Corollary.5 There exists an absolute constant c for which the following holds. Let V R n be a convex, centrally symmetric body, set n and let Γ be a random, n ψ matrix. Then, with probability at least /, diam V erγ cγ V /. Let us point out that this corollary does not use the full strength of our result, as its proof only requires that for ρ γ V /, 0 ΓT, where T = V/ρ S n. Thus, Corollary.5 is implied by a one-sided isomorphic condition, that 0 < inf Γt, rather than the two-sided quantitative estimate we actually have in Theorem.4. The third application is related to a Dvorezy-type theorem. As T R n, the parameter γ T has a natural geometric interpretation: it is approximately proportional to the mean width of T. Indeed, if T S n R n then γ T is equivalent to nwt, where wt is the mean-width of T R n see, for example, [0], chapter 9. Note that exactly the same estimate as in Theorem.4, which is c γt ε, appears in a result related to Dvorezy s Theorem due to Milman e.g. [9], Section.3.5. There, random -dimensional projections of a convex body are analyzed for much larger than the critical value - that is, is larger than the dimension in which a random projection is almost Euclidean. It turns out that even though a typical projection is far from being Euclidean, a regular behavior may be observed for the diameter of the projected body. In our setting, this result could be described as follows: if T S n is a centrally-symmetric set and Γ is a random orthogonal projection, then for C γ T ε with high probability ε max Γt l + ε. 3 This consequence of a Dvorezy-type theorem also follows from our arguments, even for an arbitrary ψ -matrix Γ. Let us emphasize that for γt the convex hull of ΓT is approximately a Euclidean ball. Thus, for << γt it is impossible to obtain an estimate in the spirit of 3, even for ε =, as the Euclidean ball cannot be well embedded in a lower-dimensional space. We end this introduction with a notational convention. Throughout, all absolute constants are positive and are denoted by c or C. Their values may change from line to line or even within the same line. Cϕ and C ϕ denote a constant which depends only on the parameter ϕ which is usually a real number, and a ϕ b means that c ϕ b a C ϕ b. If the constants are absolute we use the notation a b. 5

6 Generic chaining Our analysis is based on Talagrand s generic chaining approach [7, 8], which is a way of controlling the expectation of the supremum of centered processes indexed by a metric space compare also with [] for another application of a similar flavor. The generic chaining was introduced by Talagrand [8] as a way of simplifying the majorizing measures approach used in the analysis of some stochastic processes in particular of gaussian processes, and in numerous other applications see [8, 6, 3] and references therein. We will not discuss this beautiful approach in detail, but rather mention the results we require. The majorizing measures Theorem in its modern version states, as shown in, that for any T l, γ T, is equivalent to the expectation of the supremum of the gaussian process {X t : t T }, where X t = g it i. The upper bound in its original form is due to Fernique building on earlier ideas of Preston [4,, 3], while the lower was established by Talagrand in [5]. Let us reformulate the corresponding result for subgaussian processes as well as for processes whose tails are a mixture of a subgaussian part and a sub-exponential part. Theorem. [8] Let d and d be metrics on T space and let {Z t : t T } be a centered stochastic process.. If there is a constant α such that for every s, t T and every u > 0, P r { Z s Z t > u} α exp, u d s,t then for any t 0 T, E sup Z t Z t0 Cαγ T, d.. If there is a constant α such that for every s, t T and every u > 0, u P r { Z s Z t > u} α exp min d s,t, u d s,t, then for any t 0 T, E sup Z t Z t0 Cα γ T, d + γ T, d. The main result of this article, formulated in Theorem.3, follows as a particular case of the next proposition and the rest of this section is devoted to its proof. Proposition. Let T, d be a metric space and let {Z x } x T be a stochastic process. Let > 0, ϕ : [0, R and set W x = ϕ Z x and ε = γt,d Assume that for some η > 0 and e cη < δ < 4 For any x, y T and t < δ 0 = 4 η log δ, the following holds: P r Z x Z y > tdx, y < exp ηδ0 t.. 6

7 For any x, y T and t >, P r W x W y > tdx, y < exp ηt. 3 For any x T, with probability larger than δ, Z x < ε. 4 ϕ is increasing, differentiable at zero and ϕ 0 > 0. Then, with probability larger than δ, sup Z x < Cε x T where C = Cϕ, δ, η > 0 depends solely on its arguments. Note the difference between requirements and in Proposition.. In we are interested in the domain t < C, while in the domain of interest is t >. The proof is composed of two steps. The first reduces the problem of estimating sup Z t to the easier problem of estimating sup Z t, where T T is an ε-cover of T that is, a set T such that for any t T there is some s T for which dt, s ε. Lemma.3 Let {W t } be a stochastic process that satisfies requirement in Proposition.. Let also ε,, η be as in Proposition.. Then there exist constants c and c which depend solely on η from Proposition. and a set T T with T 4, such that with probability larger than e c, sup W t < sup W t + c ε. Proof. Let {T s : s 0} be a collection of subsets of T such that T s = s, T 0 = and for which γ T, d is almost attained. For any x T, let π s x be a nearest point to x in T s with respect to the metric d. We may assume that γ T, d <, and thus the sequence π s x converges to x for any x T. Clearly, by the definition of γ and the triangle inequality, s/ dπ s x, π s+ x Cγ T, d. 4 i=0 Let s 0 be the minimal integer such that s0 >, put T = T s0 and note that T = s 0 < = 4. It remains to prove that with high probability, for any x T W x W πs0 x = Wπs+x W πsx < Cε, 5 s=s 0 as this implies that sup W t sup W t + Cε. Fix s > s 0 and x T s+, y log Ts Ts+ T s. Since > then by the increment assumption on W, the probability that log W x W y < c Ts T s+ η dx, y 6 7

8 is larger than e c η log Ts Ts+ > T s T s+ for the appropriate choice of the constants c η, c η > 0 which depend only on η. Hence, with probability larger than T, condition 6 holds for all x T s s+ and y T s. It follows that with probability larger than exp c η, for all x T and any s s 0, Wπs+x W πsx < Cη s/ dπ s+ x, π s x, since log T s = s. Applying 4, with probability at least exp c η, for any x T, W x W πs0 x W πs+x W πsx < C η s/ dπ s+ x, π s x s=s 0 s=s 0 < C η γ T and thus 5 holds, as γt < ε. The next step is to bound the maximum of a stochastic process, when indexed by a small set. Lemma.4 Let T, d be a metric space, assume that T 4 and let {Z t } be a stochastic process that satisfies requirements and 3 in Proposition.. If η, ε, δ, are as in Proposition. then with probability larger than 3 δ, sup Z t Cη, δε where Cη, δ depends solely on δ and η. Proof. Let {T s : s N} be a collection of subsets of T such that T s s, T 0 = and for which γ T is almost attained. Again, for any x E, let π s x be a nearest point to x in T s, and because T < 4, we may assume that T = T s for s > s 0 = log +. Thus, it is evident that Z x Z π0x = Zπsx Z πs x s= log + = s= Zπsx Z πs x. Fix some u δ0 = η log δ and s s 0. Selecting t = u s+/ / δ 0 in condition in Proposition., it follows that for a fixed x T, P r Z πsx Z πs x > u s+/ dπ sx, π s x e η δ u s+ 0. Hence, with probability larger than s= s s e η δ 0 u s+, for every x 8

9 T s Z x Z π0x 0+ = Zπsx Z πs x u s/ dπ sx, π s x s= cu γ T. By condition 3 in Proposition., with probability larger than δ we have Z π0x < ε. Hence, with probability larger than s= e η s+ δ u s 0 δ, for every x T, Z x < Z π0x + cu γ T s < Z π0x + cuε < cu + ε. It remains to estimate the probability, and to that end, put u = η log δ and observe that s+ e η δ u s+ 0 δ δ δ s > 3 δ, as claimed. s= Proof of Proposition.. By condition 4, and hence for any 0 < ε <, s= C = Cϕ ϕ ϕt + t = sup < 0<t< t ϕε + ε < ϕ Cε. 7 Let T T be the set from Lemma.3. Applying Lemma.3 and since ϕ is increasing, sup Z t = ϕ sup W t ϕ sup W t + c ε = ϕ ϕ sup Z t + c ε with probability larger than e c > δ. Since T < 4 then by Lemma.4, with probability larger than 3 δ, sup Z t < Cη, δε. and hence, by 7, with probability larger than δ [ ] sup Z t < ϕ [ϕ Cη, δε + c ε] < ϕ ϕ Cϕ max{cη, δ, c }ε = cη, δ, ϕε. 9

10 3 Empirical Processes Next we show how Theorem.3 follows from Proposition.. A central ingredient in the proof is the well nown Bernstein s inequality. Theorem 3. [9, ] Let X,..., X be independent random variables with zero mean such that for every i and every m, E X i m m!m m v i /. Then, for any v v i and any u > 0, P r { } X i > u i=i exp u v + um It is easy to see that if E exp X /b, i.e., if X ψ b then m= b m m!, and the assumptions of Theorem 3. are satisfied for M = X ψ and v = 4 X ψ. Hence, { } { } u u P r X i > u exp c min X, 8 ψ X ψ i=i. E X m Lemma 3. Using the notation of Theorem.3, for any f, g T and u > 0, P r Z f Zg > u f g ψ exp c min{u, u } and P r Z f > u exp c β min{u, u } where c is a universal constant and c β depends solely on β from Theorem.3. Proof. Clearly, Z f Z g = f gx i f + gx i. Let Y i = f gx i f + gx i = f X i g X i. Then for any u > 0, P r Y i > 4u f g ψ f + g ψ P r f gx i > u f g ψ + P r { f + gxi > u f + g ψ } e u, which implies that Y i ψ c f g ψ f + g ψ c f g L, as f L = g L =. In particular, since Y,..., Y are independent and EY i = 0, then by 8 P r { Z f Zg u > u exp c } u min f g,. L f g L Z The estimate for P r f > u follows the same path, where we define Y i = f X i, use the fact that fx ψ β and apply 8. 0

11 Proof of Theorem.3. We will show that the conditions -4 of Proposition. are satisfied, for the choice of ϕt = + t and df, g = f g ψ. Fix η c, the constant from Lemma 3.. Assume that t < δ 0 = 4 η log δ. By Lemma 3., P r Z f Z g > t f g ψ < exp η min{t, t } < exp η t. By the triangle inequality, W f W g = / / f X i g X i Applying 8 for t >, P r W f W g > t f g ψ P r = P r δ 0 f g X i f g X i > t f g ψ f g X i > t f g ψ < exp ct exp ηt since we assumed that η is smaller than the constant in 8. 3 For any x T, by Lemma 3., 4 ϕ 0 = > 0. P r Z x > ε < exp ηε < δ. / Remar: We may formulate a variant of Theorem.3, as follows Theorem 3.3 Let Ω, µ be a probability space and let X, X, X,..., X be independent random variables distributed according to µ. Set T to be a collection of functions such that for every f T, EfX = 0 and fx ψ β. Define a metric on T by df, g = max{ f gx ψ, f gx ψ } and consider the random variable Zf = fx i. Then for any e c γ T,d < δ <, with probability larger than δ, sup Zf cβ, δ γ T, d 9 f T where cβ, δ depends solely on β and δ and c is an absolute constant.

12 The proof of Theorem 3.3 is almost identical to that of Theorem.3, with the only difference being the fact that requirement in Proposition. is satisfied. In this case, and for t >, W f W g = + fx i f gx i P r W f W g > t f g ψ by 8. P r + gx i f gx i > t f g ψ < exp ct 4 Applications to random Embeddings Theorem 4. For every β > 0 there exists a constant cβ for which the following holds. Let T S n be a set, and let Γ : R n R be a random operator whose rows are independent vectors Γ,.., Γ R n. Assume that for any x R n and i, E Γ i, x = x and Γ i, x ψ β Γ i, x L. Then, with probability larger than /, for any x T and any ε cβ γt, ε Γx l < + ε. Proof. For x T define Z x = Γx l n = Γ i, x. Then EZx = 0 and Γ i, x y ψ β x y l n. The assumptions of Theorem.3 are satisfied, and hence for δ =, with probability at least /, as claimed. sup x T Γx l < ε,

13 An important example is when the elements of the matrix Γ are independent random variables Γ i,j, such that EΓ i,j = 0, V arγ i,j = and Γ i,j ψ β Γ i,j L, since in that case, Theorem.4 holds. Indeed, one just needs to verify that n Γ i,jx i ψ cβ x l n. To that end, recall that for any random variable X, X ψ < c u E expux < exp 3c u X ψ < 0c. Hence, the fact that n Γ i,jx i ψ cβ x l n follows from Ee u n Γi,jxi = n Ee uxiγi,j n e cβu x c i = e βu x l n. As was mentioned in the introduction, in the special case where Γ i,j are independent, standard gaussian variables, shorter proofs of Theorem.4 exist. The first proof in that case is due to Gordon [6] and is based on a comparison Theorem for Gaussian variables. Another simple proof which uses Talagrand s generic chaining approach may be described as follows. Let Γ be an n gaussian matrix. Since the gaussian measure is rotation invariant, it follows that for any t S n, E Γt l is independent of t, and we denote it by A. It is standard to verify that A is equivalent to an absolute constant. Consider the process Z t = A Γt l indexed by T Sn and note that this process is centered. In [4], G. Schechtman proved that there is some absolute constant c such that for any s, t S n, { } Γt l P r Γs u l u exp c s t. 0 l n Hence, the process Z t is subgaussian with respect to the Euclidean metric, implying that for any t 0 T E sup Z t E sup Z t Z t0 + E A Γt 0 l γ T C A + C A ε, where the last inequality follows from Theorem., combined with a standard estimate on E Γt l E Γt l and the fact that Cγ T /ε. A second, analogous case in which the preceding argument wors, is that of projections onto random subspaces. Consider G n,, the Grassmanian of -dimensional subspaces of R n and recall that there exists a unique rotation invariant probability measure on G n,. Assume that P is an orthogonal projection onto a random -dimensional subspace in R n and let Γ = np. Note that for a suitable A, which is equivalent to an absolute constant, the process Z t = A Γt l, indexed by T Sn, is centered. Next, we shall prove 3

14 an analog of 0 for such random operators. As in the gaussian case, Theorem. implies that all of our results also hold for random operators which are orthogonal projections onto random subspaces. The proof of the following lemma is similar to Schechtman s proof in [4]. Lemma 4. There exists an absolute constant c for which the following holds. Fix s, t S n and let P be a random projection of ran. Then, { P } P r s l P t l > u n s t l n exp cu. Proof. Let d = s t l n and set Ω d = {x, y : x, y S n, x y l n = d}. There exists a unique rotation invariant probability measure µ d on Ω d see, for example, the first pages of [0]. Rather than fixing s, t and randomly selecting P, one may equivalently fix an orthogonal projection P of ran and prove that } µ d { P s l P t l > u n s t l n exp cu. Note that if one conditions on x = s+t then y = s t is distributed uniformly on the sphere d Sn. Thus it is enough to show that for any fixed x, { P r y S n : P x + d l P y P x d l } P y > du exp cu. n Since the function fy = P x + d P y l P x d P y l is Lipschitz on S n with a constant d and since its expectation is zero, then by the standard concentration inequality on the sphere see, e.g. [0], which completes the proof. P r y S n : fy > ud e cun, Note that this argument relays heavily on the structure of the gaussian random variables particularly, the rotation invariance, which is the reason why one can have a purely subgaussian behavior for the process Z t. For a general ψ operator, the best that one could hope for off-hand is a mixture of subgaussian and sub-exponential tails, and in order to obtain an upper bound solely in terms of γ one requires the more general argument. References [] S. Artstein, The change in the diameter of a convex body under a random sign-projection, preprint. 4

15 [] G. Bennett, Probability inequalities for the sum of independent random variables, J. Am. Stat. Assoc. 57, 33-45, 96. [3] R.M. Dudley, Uniform Central Limit Theorems, Cambridge Studies in Advanced Mathematics 63, Cambridge University Press, 999. [4] X. Fernique, Régularité des trajectoires des fonctiones aléatoires gaussiennes, Ecole d Eté de Probabilités de St-Flour 974, Lecture notes in Mathematics 480, Springer-Verlag -96, 975. [5] E. Giné and J. Zinn: Some limit theorems for empirical processes, Ann. Probab. 4, , 984. [6] Y. Gordon, On Milman s inequality and random subspaces which escape through a mesh in R n. Geom. Aspects of Funct. Anal., Israel seminar, Lecture Notes in Math., 37, Springer-Verlag, 84-06, 988. [7] W.B. Johnson, J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, Contemp. math. 6, 89-06, 984. [8] V.D. Milman, Geometrical inequalities and mixed volumes in the local theory of Banach spaces, Astérisque 3, , 985. [9] V.D. Milman, Topics in Asymptotic Geometric Analysis, Geom. Funct. Anal., Special Volume - GAFA000, Birhäuser, 79-85, 000. [0] V.D. Milman, G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Lecture Notes in Mathematics 00, Springer, 986. [] A. Pajor, N. Tomcza-Jaegermann, Subspaces of small codimension of finite-dimensional Banach spaces, Proc. AMS, 97 4, , 986. [] C. Preston, Banach spaces arising from some integral inequalities, Indiana Math. J. 0, , 97. [3] C. Preston, Continuity properties of some Gaussian processes, Ann. Math. Statist. 43, 85-9, 97. [4] G. Schechtman, A remar concerning the dependence on ɛ in Dvoretzy s theorem. Geom. Aspects of Funct. Anal , Lecture Notes in Math., 376, Springer, Berlin, 74 77, 989. [5] M. Talagrand, Regularity of Gaussian processes, Acta Math. 59, 99-49, 987. [6] M. Talagrand, Majorizing Measures, the generic chaining, Ann. Probab. 4, , 996. [7] M. Talagrand, Majorizing Measures without measures, Ann. Probab. 9, 4-47, 00. 5

16 [8] M. Talagrand, The generic chaining, forthcoming. [9] A. W. Van der Vaart, J. A. Wellner, Wea convergence and Empirical Processes, Springer-Verlag,