Metric Entropy of Convex Hulls

Transcription

1 Metric Entropy of Convex Hulls Fuchang Gao University of Idaho Abstract Let T be a precopact subset of a Hilbert space. The etric entropy of the convex hull of T is estiated in ters of the etric entropy of T, when the latter is of order α = 2. The estiate is best possible. Thus, it answers a question left open in [LL] and [CKP]. 0.1 Introduction Let H be a separable Hilbert space and let T be a precopact subset. Define the covering nuber { } n N(T, ε) := inf n : t 1, t 2,..., t n T, s.t. T B(t k, ε) where B(x, ε) is the open ε-ball centered at x H. The set N ε (T ) := {t 1, t 2,..., t n } is called an ε-net of T. The quantity log N(T, ε) plays an iportant role in the theory of epirical processes (cf.[d]). It is called the etric entropy of T. Let cov(t) denote the convex hull of T. It is natural to ask for good estiates of log N(cov(T ), ε) in ters of log N(T, ε). It is known (cf. [C]) 1

2 that if log N(T, ε) < c ε α for soe α > 0, then log N(cov(T ), ε) c ε 2 (log ε 1 ) 1 2/α, 0 < α < 2, log N(cov(T ), ε) c ε α, α > 2, and those are best possible. As we can see fro the above that the situation is copletely different for α < 2 and α > 2. The case α = 2 was open. In [LL], Li and Linde studied the etric entropy of cov(t ) via certain quantities originated in the theory of ajorizing easures. Aong others, they obtained soe finer estiates of log N(cov(T ), ε), which lead to soe iportant partial results for α = 2. For exaple, the upper bounds for the entropy of cov(t ), T = {t 1, t 2,...}, t i a i, by functions of the a i s only. Their results are optial for the slowly decreasing sequence (a i ). However, in general, the estiate of the etric entropy of cov(t ) for the case α = 2 was left open. In this paper, we give the best possible estiate for the case α = 2. More precisely, we prove the following Theore 1 Let H be a separable Hilbert space and let T be a precopact subset of H. (i) Suppose log N(T, ε) < ε 2, then for soe c > 0, log N(cov(T ), ε) c ε 2 (log ε 1 ) 2 ; (ii) There exists a set T, and a constant c > 0, such that sup ε 2 log N(T, ε) 8, ε>0 2

3 and for all ε < c, 0.2 Proof of (i) log N(cov(T ), ε) cε 2 (log(ε 1 )) 2. Without loss of generality, we assue the diaeter of T is 1. For k 1, let N k be a 2 k -net of T with inial cardinality. Denote D 1 = N 1 {0} and D n = {z N n N n 1 : z 2 n+1 } {0} for n > 1. Then T D 1 + D D n +, where + eans the Minkowsky su. By the assuption of (i), D n consists of no ore than e c22n vectors for soe constant c > 0. Denote C n = cov(d n ) and E n = C 1 + C C n, then we have cov(t ) C 1 + C C n + = E n + C n+1 +. For any 0 < ε < 1/4, suppose 2 n+2 ε < 2 n+3. Because C n+1 + C n+2 + has diaeter at ost 2 n+1, we have log N(cov(T ), ε) log N(E n, 2 n+1 ). To estiate the right side above, we need the following lea, whose proof is standard. Lea 1 There exists a constant c, such that for any λ > 0, log N(E n, λ) cn 2 λ 2. 3

4 Proof: For each k n, suppose D k = {x 1, x 2,..., x dk }, where d k is the cardinality of D k. Thus, d k e c22k. For each z k C k, z k can be expressed as z k = d k Define rando vector Z k, so that a i x i, a i 0, d k a i 1. Pr(Z k = x i ) = a i, 1 i d k, and Pr(Z k = 0) = 1 Let Z k,1, Z k,2,..., Z k,k and Z k,1, Z k,2,..., Z k, k be independent copies of Z k. Then E 1 k Z k,i = z k. k Thus, by convexity and syetrization, we have n n 1 k n 1 k E z k Z k,i = E Z k,i k E k n EE 1 k k n = EE 1 k k n 1 k 2E k d k (Z k,i Z k,i ) a i. n 1 k Z k,i k (Z k,i Z k,i )r k,i (t) Z k,i r k,i (t) where (r k,i (t)), 1 k n, 1 i k, is a Radeacher sequence. Integrating with respect to t over [0, 1], and using Fubini, we obtain ( n n 1 k n ) 1 k 1/2 E z k Z k,i 2E k 2 Z k,i 2 k ( n ) 1/ k+2 = λ, k 4

5 taking k = 4n2 2k+2 λ 2. This in particular iplies that for soe realization, But, there is no ore than n n 1 k z k Z k,i λ. k n (d k ) k e cn2 λ 2 possible realizations of n k Z k,i/ k. The lea follows. Applying Lea 1 with λ = 2 n+1, and keeping in ind that 2 n+2 ε < 2 n+3, we obtain log N(cov(T ), ε) log N(E n, 2 n+1 ) c n 2 2 2n+2 = c ε 2 (log ε 1 ) 2. Reark 1 Both Li and Linde pointed out to e that the result (i) can be derived fro a result in [CKP]. We include the proof because the current proof sees ore transparent, and holds for any Banach space of type 2. Also, it is ore convenient to the readers. 0.3 Proof of (ii) Let (e k ) be a standard basis of H. For each integer k 1, we define D k = { 2 k e i : e 22k 2 i e 22k} {0}, 5

6 and T = D 1 + D D k +. For any 0 < ε < 1, suppose 2 n ε < 2 n+1. Define S n = D 1 + D D n. Because S n is an 2 n -net of T, and S n has cardinality no ore than e 22k e 22n+1, k n we have log N(T, ε) < 2 2n+1. Thus ε 2 log N(T, ε) < 2 2n+2 2 2n+1 = 8. To obtain a lower bound for log N(cov(T ), ε), we need the following lea. Lea 2 There exists c > 0, such that for e 22k 3 < δ < c 2 k, log N(cov(D k ), δ) > c δ 2. Proof: Denote I k = {i : e 22k 2 i < e 22k }, and let I k be the cardinality of I k. Consider the set A = a i εe i : a i is non-negative integer, a i 2 k /ε. i I k i I k Let be the largest integer, such that 2 k /ε. Then A has cardinality no less than I k /! > I k /2. For each t A, and 2 l <, consider B(t, l) = {s A : t s 1 lε}. B(t, l) contains no ore than 2 l I k l I k 2l eleents. Thus A contains a subset U of cardinality ore than ( I k /2 ) ( I k 2l ), whose utual l 1-6

7 distance between any two eleents is at least lε. Thus, the utual l 2 - distance is at least lε. Let l /6. Because A cov(d k ), we have log N(cov(D k ), lε) log N(co(D k ), /6 ε) log ( I k /2 / I k /3) 6 log I k, which iplies that log N(cov(D k ), δ) c δ 2 for soe c > 0 and e 22k 3 < δ < c 2 k. Lea 3 For n 12, let = [n/6], and E n = cov(d ) + cov(d +1 ) + cov(d +2 ) + + cov(d n ). Then for soe constant c > 0, log N(E n, n 2 2n 1 ) cn 2 4n. Proof: By Lea 2, for each k n, there exists a set S k cov(d k ) of cardinality L = e c 24n whose utual distance between any two eleents is at least 2 2n. Consider the set F n = S + S S n. For t, s F n, suppose t = t + t t n, and s = s + s s n 7

8 with t k S k and s k S k. Define the Haing distance h(t, s) = cardinality of {k : t k s k, k n}. For each t F n, the ball B h (t, n/3) := {s F n : h(t, s) n/3} contains no ore than (nl) n/4 < L n/3 eleents. Thus F n contains a subset of cardinality L n L n/3 L n/2, whose utual Haing distance between any two eleents is at least n/4. Thus the utual l 2 -distance is at least n 2 2n 1. This iplies that log N(F n, n 2 2n 1 ) n 2 log L = cn 2 24n. Now we finish the proof of (ii). For any 0 < ε < 2 24, there exists n 12, such that n n 3 < ε n 2 2n 1. Because F n cov(t ), we have log N(cov(T ), ε) log N(F n, n 2 2n 1 ) cn 2 24n c ε 2 (log ε 1 ) 2. Acknowledgent The author thanks J. Creutzig, W. Li and W. Linde for there interest and valuable coents. 8

9 0.4 Reference [BLL] B. Bühler, W. Li and W. Linde, Location of Majorizing Measures, Asyptotic Methods in Probability and Statistics with Applications, Birkhauser, Boston.(to appear) [BP] K. Ball and A. Pajor, The entropy of convex bodies with few extree points, London Math. Soc. Lecture Note Ser. 158(1990), [C] B. Carl, Metric entropy of convex hulls in Hilbert spaces, Bull. London Math. Soc. 29(1997), [CKP] B. Carl, I. Kyrezi and A. Pajor, Metric entropy of convex hulls in Banach spaces, preprint, [D] R. M. Dudley, The sizes of copact subsets of Hilbert space and continuity of Gaussian processes, J. Funct. Anal. 1(1967), [LL] W. Li and W. Linde, Metric Entropy of Convex Hulls in Hilbert Spaces, Studia Matheatica (to appear). Fuchang Gao Departent of Matheatics University of Idaho Moscow, Idaho