KRIVINE SCHEMES ARE OPTIMAL

Transcription

1 KRIVINE SCHEMES ARE OPTIMAL ASSAF NAOR AND ODED REGEV Abstract. It is shown that for every N there exists a Borel probability measure µ on { 1, 1} R { 1, 1} R such that for every m, n N and x 1,..., x m, y 1,..., y n S m+n 1 there exist x 1,..., x m, y 1,..., y n S m+n 1 such that if G : R m+n R is a random (m + n) matrix whose entries are i.i.d. standard Gaussian random variables then for all (i, j) {1,..., m} {1,..., n} we have E G [ { 1,1} R { 1,1} R f(gx i)g(gy j)dµ(f, g) ] = x i, y j (1 + C/)K G, where K G is the real Grothendiec constant and C (0, ) is a universal constant. This establishes that Krivine s rounding method yields an arbitrarily good approximation of K G. 1. introduction Grothendiec s inequality [4, 7] asserts that there exists a constant K (0, ) such that for every n, m N, every m n matrix A = (a ij ) with real entries, and every choice of unit vectors x 1,..., x m, y 1,..., y n S m+n 1, there are signs ε 1,..., ε m, δ 1,..., δ n { 1, 1} satisfying a ij x i, y j K a ij ε i δ j. (1) The infimum over those K (0, ) for which (1) holds true is called the (real) Grothendiec constant and is denoted K G. The exact value of K G remains a long-standing mystery ever since Grothendiec s original 1953 wor [4]. The best nown bounds on K G are due to [9, ]. We call a Borel probability measure µ on { 1, 1} R { 1, 1} R a -dimensional Krivine scheme of quality K (0, ) if for every m, n N and every choice of unit vectors x 1,..., x m, y 1,..., y n S m+n 1 there exist new unit vectors x 1,..., x m, y 1,..., y n S m+n 1 such that if G : R m+n R is a random (m + n) matrix whose entries are i.i.d. standard Gaussian random variables then for all (i, j) {1,..., m} {1,..., n} we have ] E G f(gx [ { 1,1} i)g(gy j)dµ(f, g) = 1 R { 1,1} K x i, y j. () R For () to mae sense we need to require that the function of G appearing under the expectation in () is Lebesgue measurable. The required measurability will be immediate for the Krivine schemes that we discuss below, but in general this assumption should be taen to be part of the definition of a Krivine scheme. A.N. was supported by NSF grant CCF , BSF grant 01001, the Pacard Foundation and the Simons Foundation. O. R. was supported by a European Research Council (ERC) Starting Grant. 1

2 The existence of a -dimensional Krivine scheme of quality K implies that K G K. Indeed, since f(gx i), g(gy j) { 1, 1} point-wise, [ ( m ) ] n a ij x i, y j = KE G a ij f(gx i)g(gy j) dµ(f, g) { 1,1} R { 1,1} R K max ε 1,...,ε m,δ 1,...,δ n { 1,1} m n a ij ε i δ j. (3) A different way to describe the above reasoning is via the following procedure that, given unit vectors x 1,..., x m, y 1,..., y n S m+n 1, produces signs ε 1,..., ε m, δ 1,..., δ n { 1, 1} so as to satisfy the desired inequality (1). Identify f { 1, 1} R with the bipartition of R given by f 1 ( 1), f 1 (1) R. Thus µ could be thought of as a carefully chosen notion of a random pair ({A, R A}, {B, R B}) of bipartitions of R. The new unit vectors x 1,..., x m, y 1,..., y n S m+n 1 should be thought of as the result of preprocessing the initial vectors x 1,..., x m, y 1,..., y n S m+n 1, in anticipation of the random projection step to come. In this step one chooses a random (m+n) Gaussian matrix G, and, independently of G, a pair of bipartitions ({A, R A}, {B, R B}) whose law is µ. One then defines random signs ε 1,..., ε m, δ 1,..., δ n { 1, 1} by { { def 1 if Gx ε i = i A, def 1 if Gy 1 if Gx i R and δ A, j = j B, 1 if Gy j R B. Inequality (3) means that for this choice of random signs the desired Grothendiec inequality (1) holds in expectation, with respect to the randomness of the matrix G and the pair of bipartitions ({A, R A}, {B, R B}). The preprocessing step of a Krivine scheme has been somewhat suppressed in the above discussion because it is essentially determined by the measure µ itself. Indeed, by rotation invariance the left hand side of () depends only on x i, y j ; say, it equals Ψ( x i, y j ) for some function Ψ : [ 1, 1] [ 1, 1]. If Ψ were invertible on [ 1/K, 1/K] then () would become x i, y j = Ψ 1 ( x i, y j /K). This means that the mutual angles between the new unit vectors (if they indeed exist) are determined by () and the initial configuration of vectors. Only these angles matter for the purpose of applying a Krivine scheme to bound K G. We note that in the nown examples of Krivine schemes (including those described below) the preprocessing step that associates new unit vectors x 1,..., x m, y 1,..., y n S m+n 1 to the initial unit vectors x 1,..., x m, y 1,..., y n S m+n 1 is oblivious to the initial configuration in the following sense. There exist (nonlinear) mappings S, T : S S, where S is the unit sphere of l, such that if we identify the span of T (x 1 ),..., T (x m ), S(y 1 ),..., S(y n ) with a subspace of R m+n then () holds true with x i = T (x i ) and y j = S(y j ). We call such Krivine schemes oblivious Krivine schemes. It is possible to show that any Krivine scheme satisfying an additional continuity assumption is oblivious. For all purposes that we can imagine, one can tae the definition of a Krivine scheme to include the assumption that it is oblivious. Theorem 1.1 (Optimality of Krivine schemes). For every N there exists a -dimensional oblivious Krivine scheme of quality (1 + O(1/))K G. One might initially believe that restricting attention to Krivine schemes in order to prove Grothendiec s inequality is too restrictive: the role of random Gaussian matrices is not

3 mandated by the problem at hand, and moreover it might be beneficial to choose the signs ε 1,..., ε m, δ 1,..., δ n { 1, 1} via a procedure that depends on the given matrix (a ij ). Nevertheless, Theorem 1.1 shows that oblivious Krivine rounding schemes capture K G exactly. To date, the best nown upper estimates on K G were obtained via oblivious Krivine schemes. Krivine introduced [6] this approach to Grothendiec s inequality in 1977 in order to obtain a new upper bound on K G. His argument used a one-dimensional Krivine scheme that is based on a deterministic partition of R: in Krivine s original wor the measure µ is supported on the pair (sign( ), sign( )) { 1, 1} R { 1, 1} R. Krivine conjectured [6] that the bound thus obtained is actually the exact value of the Grothendiec constant. Clearly, if true, Krivine s conjecture would have implied that oblivious Krivine schemes yield an exact evaluation of K G, and would have thus made Theorem 1.1 redundant. However, in 011 Krivine s conjecture was refuted [], yielding the best nown upper bound on K G via a two-dimensional Krivine rounding scheme. It was this development that motivated us to investigate the possible validity of Theorem 1.1. While at present there is no compelling conjecture as to the exact value of K G (though in [] a candidate for this value is proposed based on a certain geometric conjecture that at this point seems quite speculative), Theorem 1.1 justifies the focus on oblivious Krivine schemes. In essence, Theorem 1.1 says that in order to understand the Grothendiec constant it suffices to focus on ways to partition Euclidean space so as to ensure the validity of () with K as small as possible.. Proof of Theorem 1.1 Fix N. By a well-nown duality argument (see [8, Thm. 3.4] and [1, Prop. 1.1]) there exists a Borel probability measure ν on { 1, 1} S 1 { 1, 1} S 1 such that x, y S 1, f(x)g(y)ν (f, g) = 1 x, y. (4) { 1,1} S 1 { 1,1} K S 1 G By identifying f { 1, 1} S 1 with its radial extension x f(x/ x ) (with the convention that the radial extension equals 1 at x = 0), the measure ν induces a probability measure µ on { 1, 1} R { 1, 1} R that satisfies x, y R {0}, f(x)g(y)µ (f, g) = 1 x y,. (5) { 1,1} R { 1,1} K R G x y Our goal will be to show that µ is the desired Krivine scheme (note that, due to (5), the measurability of the function of G appearing under the expectation in () is immediate). To this end it would be beneficial to rewrite the ey requirement () using (5). Let G 1, G R be i.i.d. standard Gaussian random vectors in R. For t [ 1, 1] define [ f (t) def G 1 = E G1,G, G 1 tg t G tg1 + 1 t G ]. (6) Using this notation, combined with (5) and rotation invariance, the desired identity () becomes 1 ( ) (i, j) {1,..., m} {1,..., n}, f x K i, y j 1 = G K x i, y j. (7) 3

4 By direct computation (see [5, page 03] for = and [3, Lemma.1] for general ) for every t [ 1, 1] we have f (t) = n=0 a n()t n+1, where a n () def = 1 ( ) ( n Γ (( + 1)/) Γ (n + 1/) 4 n = ( Γ +1 )) π n Γ (/) Γ (n /) Γ ( n (j 1) ) j( + j). (8) By Stirling s formula, for every integer n we have Hence, Also, a n () n= (+1)/ / n(n + /) (+1)/ (1 + /(n)) (+1)/. (9) n / (n + /) (+1)/ a n () (+1)/ / a 0 () = 1 1 ( ) 1 + O 1 dx (x + /) (+1)/ (+1)/ 3 / (/) ( 1)/ 1. (10) and a 1 () = a 0() ( + ) 1. (11) Write D = {z C : z < 1}. By (10) for every z D we have f (z) = n=0 a n()z n+1, and f is continuous on D and analytic on D. Moreover, by (10) and (11) we now that there exists a universal constant C (1, ) such that f (z) a 0 ()z < C/ for every z D. We assume from now on that is large enough so that a 0 () C/ > 0. Then a 0 ()z f (z) < a 0 ()z ζ for every ζ (a 0 () C/)D and z D. By Rouché s theorem it follows that a 0 ()z ζ and f (z) ζ have the same number of zeros in D. Hence : (a 0 () C/)D D is well defined and analytic on (a 0 () C/)D. For w (a 0 () C/)D write f 1 (w) = n=0 b n()w n+1 for some {b n ()} n=0 R. Thus b 0 () = 1/a 0 () and b 1 () = a 1 ()/a 0 () 4 = 1/(( + )a 0 () 3 ). Observe that for large f 1 enough we have 1 4C/ (a 0 () C/)/a 0 () n=0 b n() (a 0 () C/) n+1 <, so that by continuity there exists c (0, 1 C/] satisfying n=0 b n() c n+1 = 1 4C/. Lemma.1. We have c 1 O(1/). Assuming the validity of Lemma.1 for the moment, we conclude the proof of Theorem 1.1. Proof of Theorem 1.1. This is a slight variant of Krivine s original argument [6]. Define two mappings S, T : S ( n=0 (l ) (n+1)) R def = H = l by ( ( ) ( )) S(x) def = b n () 1/ c (n+1)/ 4C x (n+1), n=0, 0, and T (x) def = ( ( ) sign(b n ()) b n () 1/ c (n+1)/ x (n+1), n=0 ( 0, )) 4C. For every x S we have S(x) H = T (x) H = ( n=0 b n() c n+1 +4C/ = 1. Moreover, f ( S(x), T (y) H ) = f n=0 b n()c n+1 x, y n+1) = f (f 1 (c x, y )) = c x, y. Thus, recalling the formulation of the desired identity appearing in (7), µ is an oblivious - dimensional Krivine scheme of quality K G /c = (1 + O(1/))K G, as desired. It remains to prove Lemma.1. 4

5 Proof of Lemma.1. For every w (a 0 () C/)D write z = f 1 (w) D. Then using (11), f 1 (w) b 0()w b 1 ()w 3 = f 1 (w) w a 0 () + w 3 ( + )a 0 () 3 = f (z) a 0 ()z a 1 ()z 3 a 0 () (a 0()z f (z))(a 0 () z + a 0 ()zf (z) + f (z) ) ( + )a 0 () 3. (1) By (10) and (11) we have f (z) a 0 ()z a 1 ()z 3 1/ and f (z) a 0 ()z 1/. Moreover, due to (11) we have a 0 () 1 and therefore a 0 () z + a 0 ()zf (z) + f (z) 1. Consequently, it follows from (1) that f 1 (w) b 0()w b 1 ()w 3 1. Hence, by Cauchy s integral formula for every integer n and every r (0, a 0 () C/), b n () = 1 f 1 (w) b 0()w b 1 ()w 3 dw πi w n+ 1 r. n+1 r D Thus there is a universal constant A (0, ) such that b n () A (a 0 () C/) (n+1) for n. If c a 0 () C/ then we are done. Assume therefore that c < a 0 () C/, in which case, recalling that b 0 () = 1/a 0 () and b 1 () = 1/(( + )a 0 () 3 ), we have 1 = n=0 b n () c n+1 c a 0 () + c 3 ( + )a 0 () + A = c a 0 () + c 3 ( + )a 0 () + A ( n= c a 0 () C/ ) n+1 (c /(a 0 () C/)) 5 1 (c /(a 0 () C/)). (13) By (11), a straightforward computation shows that (13) implies that c 1 O(1/). Acnowledgements. We are grateful to Thomas Vidic for helpful suggestions. References [1] R. Blei. The Grothendiec inequality revisited. Preprint available at [] M. Braverman, K. Maarychev, Y. Maarychev, and A. Naor. The Grothendiec constant is strictly smaller than Krivine s bound. In IEEE 5nd Annual Symposium on Foundations of Computer Science, pages , 011. [3] J. Briët, F. M. de Oliveira Filho, and F. Vallentin. Grothendiec inequalities for semidefinite programs with ran constraint. Preprint available at [4] A. Grothendiec. Résumé de la théorie métrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paulo, 8:1 79, [5] U. Haagerup. A new upper bound for the complex Grothendiec constant. Israel J. Math., 60():199 4, [6] J.-L. Krivine. Sur la constante de Grothendiec. C. R. Acad. Sci. Paris Sér. A-B, 84(8):A445 A446, [7] J. Lindenstrauss and A. Pe lczyńsi. Absolutely summing operators in L p -spaces and their applications. Studia Math., 9:75 36, [8] G. Pisier. Grothendiec s theorem, past and present. Bull. Am. Math. Soc., New Ser., 49():37 33, 01. [9] J. A. Reeds. A new lower bound on the real Grothendiec constant. Unpublished manuscript, available at