Geometry of log-concave Ensembles of random matrices

Transcription

1 Geometry of log-concave Ensembles of random matrices Nicole Tomczak-Jaegermann Joint work with Radosław Adamczak, Rafał Latała, Alexander Litvak, Alain Pajor Cortona, June 2011 Nicole Tomczak-Jaegermann () Cortona, / 21

2 Basic definitions and Notation Let X = (X(1),..., X(N)) be a random vector in R N with full dimensional support. We say that the distribution of X is log-concave, if X has density of the form e h(x) with h: R N (, ] convex; isotropic, if EX i = 0 and EX i X j = δ i,j. For x, y R N we put x = x 2 = ( N i=1 x2 i x, y is the inner product ) 1/2 For I {1,..., N}, by P I we denote the coordinate projection onto {y R N : supp(y) I}. Nicole Tomczak-Jaegermann () Cortona, / 21

3 Examples: log-concave random vectors 1. Let K R n be a convex body ( = compact convex, with non-empty interior) (symmetric means K = K). X a random vector uniformly distributed in K. Then the corresponding probability measure on R N K A µ K (A) = K is log-concave (by Brunn-Minkowski). Moreover, for every convex body K there exists an affine map T such that µ T K is isotropic. 2. The Gaussian vector G = (g 1,..., g n ), where g i s have N(0, 1) distribution, is isotropic and log-concave. 3. Similarly the vector X = (ξ 1,..., ξ n ), where ξ i s are independent with the exponential distribution (i.e., with density f(t) = 1 2 exp( 2 t ), for t R) is isotropic and log-concave. Nicole Tomczak-Jaegermann () Cortona, / 21

4 Matrices Let n, N be integers; X R N a random vector, X 1,..., X n independent, distributed as X. A is an n N matrix which has X i s as rows A = X 1 X 2 X n = For J {1,..., n}, I {1,..., N}, let A(J, I) be the sub-matrix with rows from J and columns from I. For k n, m N, let A k,m = sup A(J, I), J,I over all J k, I m. This is maximum of norms of sub-matrices of k columns and m rows (the operator norms from l N 2 to ln 2 ). Question: Upper bounds for A k,m with high probability Nicole Tomczak-Jaegermann () Cortona, / 21

5 Example: Norms of sub-matrices, uniform version of P. Earlier case: m = N, i.e., A k,n. We select k rows and take all columns. Solved by ALPT (JAMS 2010); connected with length of good approximations of the covariance matrices Case k = 1: For any J with J = 1, and I with I m, sub-matrix of one row has a form ( ) A(J, I) = P I X, so A 1,m = sup I m P I X. A bound for A 1,m will imply a uniform bound for P I X, for t 1. sup P I X bound for A 1,m, ( ) I =m with high probability. Compare with Paouris theorem for a fixed I with I m: for s 1, ( P P I X Cs ) ( N 1 exp s ) N. However the complexity of the family of subsets is ( N m) exp(c N). To prove ( ) we need more advanced technique. Nicole Tomczak-Jaegermann () Cortona, / 21

6 Order Statistics For an N dimensional random vector X by X 1 X 2... X N we denote the nonincreasing rearrangement of X(1),..., X(N) (in particular X 1 = max{ X(1),..., X(N) } and X N = min{ X(1),..., X(N) }). Random variables X k, 1 k N, are called order statistics of X. Problem: Find an upper bound for P(X k t). If coordinates of X i are independent symmetric exponential r.v. with variance 1 then Med(X k ) log(en/k) for k N/2. Nicole Tomczak-Jaegermann () Cortona, / 21

7 Order Statistics for isotropic log-concave vectors Theorem ( ) en Let X be N-dim. log-concave isotropic vector. Then for all t C log k, ( ) ( P X k t exp 1 ) kt. C Actually we need a stronger theorem that uses the following important weak parameter Nicole Tomczak-Jaegermann () Cortona, / 21

8 Weak parameter For a vector X in R N we define We also let σ 1 X Examples σ X (p) := sup (E t, X p ) 1/p p 2. t S N 1 to be the left-inverse function. σ 1 X (s) := sup { t: σ X (t) s } For isotropic log-concave vectors X, σ X (p) p/ 2. For subgaussian vectors X, σ X (p) C p. We say that an isotropic vector X is ψ α if σ X (p) Cp 1/α (uniform distributions on suitable normalized B N p balls are ψ α with α = min(p, 2)) Nicole Tomczak-Jaegermann () Cortona, / 21

9 Order Statistics with weak parameter Theorem ( ) en For any N-dim. log-concave isotropic vector X, l 1 and t C log l, ( ) ( P X l t exp ( 1 σ 1 X l)) C t This theorem implies uniform estimates for norms of projections. Nicole Tomczak-Jaegermann () Cortona, / 21

10 Uniform bound for norms of projections Theorem Let X be an isotropic log-concave vector in R N, and m N. Then for any t 1, ( P sup P I X Ct ( en ) ) m log I =m m is less than or equal to ( (t ( ) en m log ) ) ( exp σ 1 m m ( en ) ) X exp t log. log(em) log(em) m The bound is sharp, except for log in the probability estimate. We conjecture this factor is not needed. Nicole Tomczak-Jaegermann () Cortona, / 21

11 Uniform bound for norms of projections, idea Idea of the proof. We want to estimate ( We have P sup P I X = I =m sup P I X Ct m log I =m ( en m ) ). ( m ) 1/2 ( s 1 ) 1/2, X k i X 2 i 2 k=1 where s = log 2 m. We then use the estimates for order statistics. i=0 Nicole Tomczak-Jaegermann () Cortona, / 21

12 Estimates for order statistics Our approach to estimates for order statistics is based on the suitable estimate of moments of the process N X (t), where for t 0, N X (t) := N i=1 1 {Xi t} That is, N X (t) is equal to the number of coordinates of X larger than or equal to t. Nicole Tomczak-Jaegermann () Cortona, / 21

13 Estimate for N X Theorem For any isotropic log-concave vector X and p 2 we have E(t 2 N X (t)) p (Cp) 2p E(t 2 N X (t)) p (Cσ X (p)) 2p ( Nt 2 ) for t C log. p 2 ( Nt 2 ) for t C log σ 2 X (p). Nicole Tomczak-Jaegermann () Cortona, / 21

14 Estimate for N X Theorem For any isotropic log-concave vector X and p 2 we have E(t 2 N X (t)) p (Cp) 2p E(t 2 N X (t)) p (Cσ X (p)) 2p ( Nt 2 ) for t C log. p 2 ( Nt 2 ) for t C log σ 2 X (p). To get estimate for order statistics we observe that X k t implies that N X (t) k/2 or N X (t) k/2 and vector X is also isotropic and log-concave, and σ X = σ X. Estimates for N X and Chebyshev s inequality give ( 2 ) p(enx P(X k t) (t) p + EN X (t) p) ( CσX (p) ) 2p k t k ( ) provided that t C log(nt 2 /σ 2 1 X (p)). Set p = σ 1 X ec t k and notice that the restriction on t follows by the assumption that t C log(en/k). Nicole Tomczak-Jaegermann () Cortona, / 21

15 Convolutions of measures Let X 1,..., X n be independent isotropic log-concave vectors in R N, and let x = (x i ) R n. Consider the vector Y = Probability for bounds of the process n x i X i R N. i=1 sup P I Y I =m depends on the vector x. Specifically, it is convenient to assume the normalization x 1 and x b 1. Nicole Tomczak-Jaegermann () Cortona, / 21

16 Uniform bound for projections of convolutions Theorem Let Y = n i=1 x ix i, where X 1,..., X n are independent isotropic N-dimensional log-concave vectors. Assume that x 1 and x b 1. i) If b 1 m, then for any t 1, P ( sup P I Y Ct ( en m log I =m m ii) If b 1 m then for any t 1, P ) ) exp ( t ( ) en ) m log m b. log(e 2 b 2 m) ( sup P I Y Ct ( en ) ) m log I =m m ( { ( en ) exp min t 2 m log 2, t ( en )}) m log. m b m Nicole Tomczak-Jaegermann () Cortona, / 21

17 Uniform bound for norms of submatrices Let A be an n N random matrix with independent log-concave isotropic rows X 1,..., X n R N. For k n, m N we let A k,m = the maximal operator norm of a k m submatrix of A. For simplicity assume n N. Theorem For any integers n N, k n, m N and any t 1, we have ) ( P (A k,m Ctλ mk exp tλ mk ), log(3m) where λ mk = log log(3m) ( en ) m log + ( en ) k log. m k Nicole Tomczak-Jaegermann () Cortona, / 21

18 Uniform bound for norms of submatrices, cont. The threshold value of λ km is optimal, up to the factor log log(3m). Under the additional assumption of unconditionality of the rows we can remove this factor and get the sharp estimate. For example, for expectations: Theorem Let A be an n N matrix whose rows are independent isotropic log-concave unconditional random vectors. Then ( m 3N EA km C log m + k log 3n ). k Nicole Tomczak-Jaegermann () Cortona, / 21

19 Reconstruction Let n, N be integers; X R N a random vector, X 1,..., X n independent, distributed as X. A is an n N matrix which has X i s as rows X 1 A = X 2 = X n We can treat A : R N R n given by x ( X i, x ) R n. Let m N. A vector x R N is m-sparse if supp x m. Problem from Compressed Sensing theory: Reconstruct any m-sparse vector x R N from the data Ax R n, with a fast algorithm. Given Ax, find x, knowing that it is sparse. Note that of course A is not-invertible. Nicole Tomczak-Jaegermann () Cortona, / 21

20 RIP and Geometry pf Polytopes Define δ m = δ m (A) as the infimum of δ > 0 such that Ax 2 E Ax 2 δ n x 2 holds for all m-sparse vectors x R N. δ m is the Restricted Isometry Property (RIP) parameter of order m. Candes and Tao (2006): if δ 2m is sufficiently small then ( ) whenever y = Ax has a m-sparse solution x, then x is the unique solution of the l 1 -minimization program: min t l1 with the min over all t such that At = y. Geometry of Polytopes: By Donoho (2005), ( ) is equivalent to the condition that the centrally symmetric polytope A(B N 1 ) is m-centrally neighborly (i.e., any set of less than m vertices containing no opposite pairs, is a vertex set of a face). Nicole Tomczak-Jaegermann () Cortona, / 21

21 Estimate for δ m Lemma Let X 1,..., X n be independent isotropic random vectors in R N. Let 0 < θ < 1 and B 1. Then with probability at least ( ) N 1 exp ( 3θ 2 n/8b 2) m one has δ m θ + 1 n ( ) A 2 k,m + EA 2 k,m, where k n is the largest integer satisfying k (A k,m /B) 2 ; Nicole Tomczak-Jaegermann () Cortona, / 21

22 RIP theorem for matrices with independent rows: Theorem Let n N and 0 < θ < 1. Let A be an n N matrix, whose rows are independent isotropic log-concave random vectors X i, i n. There exists an absolute constant c > 0, such that if m N satisfies ( m log log(3m) log 3N ) 2 ( ) 2 θ c n m log(3/θ) then δ m θ with overwhelming probability. Optimal up to a log log factor. For unconditional distributions can be removed Nicole Tomczak-Jaegermann () Cortona, / 21