a quadratic convergence yielding iterative method for the implementation of lavrentiev regularization method for ill posed equations george and elmahdy 2012 considered an iterative method which converges quadratically to the unique solution is a nonlinear monotone operator defined on a real hilbert space x the convergence analysis of the method was based on a majorizing sequence in this paper we are concerned with the problem of expanding the applicability of the method considered by george and elmahdy 2012 by weakening the restrictive conditions imposed on the radius of the convergence ball and also by weakening the popular lipschitz type hypotheses considered in earlier studies such as george and elmahdy 2012 mahale and nair 2009 mathe and perverzev 2003 nair and ravishankar 2008 semenova 2010 and tautanhahn 2002 we show that the adaptive scheme considered by perverzev and schock 2005 for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate in the concluding section the method is applied to numerical solution of the inverse gravimetry problem

small ball probability estimates ψ2 behavior and the hyperplane conjecture we introduce a method which leads to upper bounds for the isotropic constant we prove that a positive answer to the hyperplane conjecture is equivalent to some very strong small probability estimates for the euclidean norm on isotropic convex bodies as a consequence of our method we obtain an alternative proof of the result of j bourgain that every 2 body has bounded isotropic constant with a slightly better estimate if k is a symmetric convex body in rn such that q 2 for every sn 1 and every q 2 then lk c log where c 0 is an absolute constant

saturating constructions for normed spaces ii we prove several results of the following type given finite dimensional normed space v there exists another space x with log dim x o log dim v and such that every subspace or quotient of x whose dimension is not too small contains a further subspace isometric to v this sheds new light on the structure of such large subspaces or quotients resp large sections or projections of convex bodies and allows us to solve several problems stated in the 1980s by v milman

hastings s additivity counterexample via dvoretzky s theorem the goal of this note is to show that hastings s counterexample to the additivity of minimal output von neumann entropy can be readily deduced from a sharp version of dvoretzky s theorem

ill posed problems with unbounded operators abstract let a be a linear closed densely defined unbounded operator in a hilbert space assume that a is not boundedly invertible if eq 1 a u f is solvable and f f then the following results are provided problem f u a u f 2 u 2 has a unique global minimizer u for any f u a a a i 1 f there is a function lim 0 0 such that lim 0 u y 0 where y is the unique minimal norm solution to 1 a priori and a posteriori choices of are given dynamical systems method dsm is justified for eq 1

entropy and asymptotic geometry of non symmetric convex bodies abstract we extend to the general not necessarily centrally symmetric setting a number of basic results of local theory which were known before for centrally symmetric bodies and were using very essentially the symmetry in their proofs some of these extensions look surprising the main additional tool is a study of volume behavior around the centroid of the body

gaussian processes and almost spherical sections of convex bodies on presente une preuve simple avec des estimations pointues du theoreme de dvoretzky sur l existence des sections presque spheriques ayant une grande dimension dans des corps convexes arbitraires dans r n

almost euclidean sections in symmetric spaces and concentration of order statistics abstract we prove that for any symmetric n dimensional normed space e and any e 0 e contains a 1 e euclidean subspace of dimension at least c ln n ln 1 e where c is an absolute constant the proof is based on a concentration property of order statistics of random vectors with i i d coordinates

isoperimetry of waists and local versus global asymptotic convex geometries existence of nicely bounded sections of two symmetric convex bodies k and l implies that the intersection of random rotations of k and l is nicely bounded for l subspace this main result immediately yields the unexpected phenomenon if k has one nicely bounded section then most sections of k are nicely bounded this existence implies randomness consequence was proved independently in giannopoulos milman and tsolomitis the main result represents a new connection between the local asymptotic convex geometry study of sections of convex bodies and the global asymptotic convex geometry study of convex bodies as a whole the method relies on the new isoperimetry of waists on the sphere due to gromov

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

on dvoretzky s theorem for subspaces of lp abstract we prove that for any 2 p and for every n dimensional subspace x of l p represented on r n whose unit ball b x is in lewis position one has the following two level gaussian concentration inequality p z e z e e z c exp c min p e 2 n e n 2 p 0 e 1 where z is the standard n dimensional gaussian vector p 0 is a constant depending only on p and c c 0 are absolute constants as a consequence we show optimal lower bound on the dimension of random almost spherical sections for these spaces in particular for any 2 p and every n dimensional subspace x of l p the euclidean space l 2 k can be 1 e embedded into x with k c p min e 2 n e n 2 p where c p 0 is a constant depending only on p this improves upon the previously known estimate due to figiel lindenstrauss and milman

dynamical systems method for solving nonlinear equations with monotone operators a version of the dynamical systems method dsm for solving ill posed nonlinear equations with monotone operators in a hilbert space is studied in this paper an a posteriori stopping rule based on a discrepancy type principle is proposed and justified mathematically the results of two numerical experiments are presented they show that the proposed version of dsm is numerically efficient the numerical experiments consist of solving nonlinear integral equations

averages of norms and quasi norms we compute the number of summands in q averages of norms needed to approximate an euclidean norm it turns out that these numbers depend on the norm involved essentially only through the maximal ratio of the norm and the euclidean norm particular attention is given to the case q in which the average is replaced with the maxima this is closely connected with the behavior of certain families of projective caps on the sphere

two observations regarding embedding subsets of euclidean spaces in normed spaces abstract this paper contains two results concerning linear embeddings of subsets of euclidean space in low dimensional normed spaces the first is an improvement of the known dependence on in dvoretzky s theorem from order of 2 to order of except for log factors the second is a joint generalization of milman s version of dvoretzky s theorem and a recent generalization by klartag and mendelson of the johnson lindenstrauss lemma

an achievable rate region for the multiple access channel with feedback an achievable rate region r 1 leq i x 1 y x 2 u r 2 leq i x 2 y x 1 u r 1 r 2 leq i x 1 x 2 y where p u x l x 2 y p u p x l u p x 2 u p y x l x 2 is established for the multiple access channel with feedback time sharing of these achievable rates yields the rate region of this paper this region generally exceeds the achievable rate region without feedback and exceeds the rate point found by gaarder and wolf for the binary erasure multiple access channel with feedback the presence of feedback allows the independent transmitters to understand each other s intended transmissions before the receiver has sufficient information to achieve the desired decoding this allows the transmitters to cooperate in the transmission of information that resolves the residual uncertainty of the receiver at the same time independent information from the transmitters is superimposed on the cooperative correction information the proof involves list codes and block markov encoding

explicit euclidean embeddings in permutation invariant normed spaces abstract let x be a real normed space of dimension n n with a basis e i 1 n such that the norm is invariant under coordinate permutations assume for simplicity that the basis constant is at most 2 consider any n n and 0 e 1 4 such that n c log e 1 1 log n we provide an explicit construction of a matrix that generates a 1 e embedding of l 2 n into x

when random proportional subspaces are also random quotients we discuss when a generic subspace of some fixed proportional dimension of a finite dimensional normed space can be isomorphic to a generic quotient of some proportional dimension of another space we show in theorem 4 1 that if this happens for some natural random structures then for any proportion arbitrarily close to 1 the first space has a lot of euclidean subspaces and the second space has a lot of euclidean quotients

high dimensional random sections of isotropic convex bodies we study two properties of random high dimensional sections of convex bodies in the first part of the paper we estimate the central section function k f n k1 k for random f gn k and k rn a centrally symmetric isotropic convex body this partially answers a question raised by v d milman and a pajor see v d milman a pajor isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n dimensional space in lecture notes in math vol 1376 springer 1989 p 88 in the second part we show that every symmetric convex body has random high dimensional sections f gn k with outer volume ratio bounded by ovr k f cnn klog 1 nn k

small ball probability estimates for log concave measures we establish a small ball probability inequality for isotropic log concave probability measures there exist absolute constants c1 c2 0 such that if x is an isotropic log concave random vector in r with 2 constant bounded by b and if a is a non zero n n matrix then for every e 0 c1 and y r p ax y 2 6 e a hs 6 e c2 b a hs a op 2 where c1 c2 0 are absolute constants

pointwise estimates for marginals of convex bodies we prove a pointwise version of the multi dimensional central limit theorem for convex bodies namely let be an isotropic log concave probability measure on rn for a typical subspace e rn of dimension nc consider the probability density of the projection of onto e we show that the ratio between this probability density and the standard gaussian density in e is very close to 1 in large parts of e here c 0 is a universal constant this complements a recent result by the second named author where the total variation metric between the densities was considered

m matrices and bounds for reliable transmission of information in communication systems and economic markets abstract a multivariate normal density n 0 whose covariance matrix is positive with an m matrix inverse is shown to give an upper bound to the region in which communication systems and economic markets can reliably transmit information the upper bound is known to be tight when the density is bivariate

dynamical systems method dsm for selfadjoint operators abstract let a be a selfadjoint linear operator in a hilbert space h the dsm dynamical systems method for solving equation a v f consists of solving the cauchy problem u t u u 0 u 0 where is a suitable operator and proving that i u t t 0 ii u and iii a u f it is proved that if equation a v f is solvable and u solves the problem u i a i a u i f u 0 u 0 where a 0 is a parameter and u 0 is arbitrary then lim a 0 lim t u t a y where y is the unique minimal norm solution of the equation a v f stable solution of the equation a v f is constructed when the data are noisy i e f is given in place of f f f the case when a a t 0 0 a t d t a t 0 as t is considered it is proved that in this case lim t u t y and if f is given in place of f then lim t u t y where t is properly chosen

lowm estimates on coordinate subspaces letkbe a symmetric convex body inrn it is well known that for every 0 1 there exists a subspacefofrnwith dim f 1 n such thatpf k c mkdn f where pfdenotes the orthogonal projection ontof consider a fixed coordinate system inrn we study the question whether an analogue of can be obtained when one is restricted to choosefamong the coordinate subspacesr 1 n with 1 n we prove several coordinate versions of in terms of the cotype 2 constant of the volume ratio and other parameters ofk the basic source of our estimates is an exact coordinate analogue of in the ellipsoidal case applications to the computation of the number of lattice points inside a convex body are considered throughout the paper

m matrices as covariance matrices of multinormal distributions abstract the class of multivariate normal densities n 0 whose inverse covariance matrix 1 is an m matrix is equivalent to this normal density being multivariate totally positive of order 2 mtp 2 equivalent characterizations are given in terms of certain partial correlation coefficients being positive it is further shown that related partial and multiple regression coefficients and canonical correlation are positive when is an m matrix the corresponding normal random vector components are negatively associated this concept and some extensions are discussed

lp john ellipsoids it is shown that the classical john ellipsoid the petty ellipsoid and a recently discovered dual of the legendre ellipsoid are all special cases p infty p 1 and p 2 of a family of l p ellipsoids which can be associated with a fixed convex body this insight allows for a unified view of alternative approaches to and extensions of some basic results in convex geometry

the plank problem for symmetric bodies given a symmetric convex bodyc andn hyperplanes in an euclidean space there is a translate of a multiple ofc at least frac 1 n 1 times as large insidec whose interior does not meet any of the hyperplanes the result generalizes bang s solution of the plank problem of tarski and has applications to diophantine approximation

lp moments of random vectors via majorizing measures abstract for a random vector x in r n we obtain bounds on the size of a sample for which the empirical pth moments of linear functionals are close to the exact ones uniformly on a convex body k r n we prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis in particular we find a short lewis type decomposition for any finite dimensional subspace of l p we also prove that for an isotropic log concave random vector we only need n p 2 log n sample points so that the empirical pth moments of the linear functionals are almost isometrically the same as the exact ones we obtain a concentration estimate for the empirical moments the main ingredient of the proof is the construction of an appropriate majorizing measure to bound a certain gaussian process

a central limit theorem for convex sets we show that there exists a sequence varepsilon n searrow0 for which the following holds let k n be a compact convex set with a non empty interior let x be a random vector that is distributed uniformly in k then there exist a unit vector in n t0 and 0 such that sup a subset mathbb r left textit prob langle x theta rangle in a frac 1 sqrt 2 pi sigma int ae frac t t 0 2 2 sigma 2 dt right leq varepsilon n qquad ast where the supremum runs over all measurable sets a and where denotes the usual scalar product in n furthermore under the additional assumptions that the expectation of x is zero and that the covariance matrix of x is the identity matrix we may assert that most unit vectors satisfy with t0 0 and 1 corresponding principles also hold for multi dimensional marginal distributions of convex sets

logarithmic reduction of the level of randomness in some probabilistic geometric constructions many of the surprising phenomena occurring in high dimensions are proved by use of probabilistic arguments which show the existence of organized and regular structures but do not hint as to where exactly do these structures lie it is an intriguing question whether some of them could be realized explicitly in this paper we show that the amount of randomness used can be reduced significantly in many of these questions from asymptotic convex geometry and most of the random steps can be substituted by completely explicit algorithmic steps the main tool we use is random walks on expander graphs

iterative solution of linear equations with unbounded operators a convergent iterative process is constructed for solving any solvable linear equation in a hilbert space including equations with unbounded closed densely defined linear operators the method is proved to be stable towards small perturbation of the data some abstract results are established and used in an analysis of variational regularization method for equations with unbounded linear operators the dynamical systems method dsm is justified for unbounded closed densely defined linear operators the stopping time is chosen by a discrepancy principle equations with selfadjoint operators are considered separately numerical examples illustrating the efficiency of the proposed method are given

a discrepancy principle for equations with monotone continuous operators a discrepancy principle for solving nonlinear equations with monotone operators given noisy data is formulated the existence and uniqueness of the corresponding regularization parameter a are proved convergence of the solution obtained by the discrepancy principle is justified the results are obtained under natural assumptions on the nonlinear operator

dynamical systems method dsm for solving equations with monotone operators without smoothness assumptions on f u abstract a version of the dynamical systems method dsm for solving ill posed nonlinear equations f u f with monotone operators f in a hilbert space is studied in this paper under less restrictive assumptions on the nonlinear operators f than the assumptions used earlier a new method of proof of the basic results is used an a posteriori stopping rule based on a discrepancy type principle is proposed and justified mathematically under weaker assumptions on the nonlinear operator f than the assumptions used earlier

diameters of sections and coverings of convex bodies abstract we study the diameters of sections of convex bodies in r n determined by a random n n matrix either as kernels of or as images of entries of are independent random variables satisfying some boundedness conditions and typical examples are matrices with gaussian or bernoulli random variables we show that if a symmetric convex body k in r n has one well bounded k codimensional section then for any m ck random sections of k of codimension m are also well bounded where c 1 is an absolute constant it is noteworthy that in the gaussian case when determines randomness in sense of the haar measure on the grassmann manifold we can take c 1

global convergence for ill posed equations with monotone operators the dynamical systems method consider an operator equation f u 0 in a real hilbert space let us call this equation ill posed if the operator f u is not boundedly invertible and well posed otherwise if f is monotone c2loc h operator then we construct a cauchy problem which has the following properties 1 it has a global solution for an arbitrary initial data 2 this solution tends to a limit as time tends to infinity and 3 the limit is the minimum norm solution to the equation f u 0 an example of applications to linear ill posed operator equation is given

asymptotic formulas for the diameter of sections of symmetric convex bodies sharpening work of the first two authors for every proportion 0 1 we provide exact quantitative relations between global parameters of n dimensional symmetric convex bodies and the diameter of their random n dimensional sections using recent results of gromov and vershynin we obtain an asymptotic formula for the diameter of random proportional sections

a new discrepancy principle the aim of this note is to prove a new discrepancy principle the advantage of the new discrepancy principle compared with the known one consists of solving a minimization problem approximately rather than exactly and in the proof of a stability result

centroid bodies and the logarithmic laplace transform a unified approach abstract we unify and slightly improve several bounds on the isotropic constant of high dimensional convex bodies in particular a linear dependence on the body s 2 constant is obtained along the way we present some new bounds on the volume of l p centroid bodies and yet another equivalent formulation of bourgain s hyperplane conjecture our method is a combination of the l p centroid body technique of paouris and the logarithmic laplace transform technique of the first named author