multilinear calderon zygmund theory abstract a systematic treatment of multilinear calderon zygmund operators is presented the theory developed includes strong type and endpoint weak type estimates interpolation a multilinear t 1 theorem and a variety of results regarding multilinear multiplier operators

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

information theoretic lower bounds on the oracle complexity of stochastic convex optimization relative to the large literature on upper bounds on complexity of convex optimization lesser attention has been paid to the fundamental hardn4516420ess of these problems given the extensive use of convex optimization in machine learning and statistics gaining an understanding of these complexity theoretic issues is important in this paper we study the complexity of stochastic convex optimization in an oracle model of computation we introduce a new notion of discrepancy between functions and use it to reduce problems of stochastic convex optimization to statistical parameter estimation which can be lower bounded using information theoretic methods using this approach we improve upon known results and obtain tight minimax complexity estimates for various function classes

on an operator arising in the calderon zygmund method of rotations and the bramble hilbert lemma in this article sharp estimates are obtained for an operator that is a quadratic average with respect to v of one dimensional hardy littlewood maximal functions in the directions v this operator arises naturally in fourier analysis and numerical analysis

finitely additive probability measures on classical propositional formulas definable by godel s t norm and product t norm suppose that e is any 0 1 valued evaluation of the set of propositional letters then e can be uniquely extended to finitely additive probability product and godel s measures on the set of classical propositional formulas those measures satisfy that the measure of any conjunction of distinct propositional letters is equal to the product of or to the minimum of the measures of the propositional letters respectively product measures correspond to the one extreme stochastic or probability independence of elementary events propositional letters while godel s measures correspond to the other extreme logical dependence of elementary events any linear convex combination of a product measure and a godel s measure is also a finitely additive probability measure in that way infinitely many intermediate measures that corresponds to various degrees of dependence of propositional letters can be generated such measures give certain truth functional flavor to probability enabling applications to preferential problems in particular classifications according to predefined criteria some examples are provided to illustrate this possibility we present the proof theoretical and the model theoretical approaches to a probabilistic logic which allows reasoning about the mentioned types of probabilistic functions the logical language enables formalization of classification problems with the corresponding criteria expressible as propositional formulas however more complex criteria for example involving arithmetical functions cannot be represented in that framework we analyze the well known problem proposed by grabisch to illustrate interpretation of such classification problems in fuzzy logic

online estimation of the geometric median in hilbert spaces nonasymptotic confidence balls estimation procedures based on recursive algorithms are interesting and powerful techniques that are able to deal rapidly with very large samples of high dimensional data the collected data may be contaminated by noise so that robust location indicators such as the geometric median may be preferred to the mean in this context an estimator of the geometric median based on a fast and efficient averaged nonlinear stochastic gradient algorithm has been developed by bernoulli19 2013 18 43 this work aims at studying more precisely the nonasymptotic behavior of this nonlinear algorithm by giving nonasymptotic confidence balls in general separable hilbert spaces this new result is based on the derivation of improved l 2 rates of convergence as well as an exponential inequality for the nearly martingale terms of the recursive nonlinear robbins monro algorithm

determination of weights of interacting criteria from a reference set abstract in this paper we present a model allowing to determine the weights related to interacting criteria this is done on the basis of the knowledge of a partial ranking over a reference set of alternatives prototypes a partial ranking over the set of criteria and a partial ranking over the set of interactions between pairs of criteria

quantifiers in the formulation of multiple objective decision functions abstract we discuss the interpretation of linguistically quantified statements using fuzzy sets we indicate that there are two methods of interpreting these statements the algebraic approach and the substitution approach we then use quantified statements to express overall decision functions involving multiple objectives these statements are of the type a solution should satisfy many of the objectives using the algebraic and substitution approach we develop a whole family of forms for combining multiple objectives we show that algebraic approach corresponds to situations in which we are unconcerned about the allocation of satisfaction amongst the objectives but just interested in getting much satisfaction the substitution approach on the other hand leads to forms which are dependent upon the allocation of satisfaction between the objectives

analysis of online composite mirror descent algorithm we study the convergence of the online composite mirror descent algorithm which involves a mirror map to reflect the geometry of the data and a convex objective function consisting of a loss and a

on multilinear square function and its applications to multilinear littlewood paley operators with non convolution type kernels abstract let m 2 n 1 and x r n define the multilinear square function t by t f x 0 r n m k t x y 1 y m j 1 m f j y j d y 1 d y m 2 d t t 1 2 where the kernel k satisfies a class of integral smooth conditions which is much weaker than the standard calderon zygmund kernel conditions in this paper we first established the l p 1 w 1 l p m w m l p estimate of t when each p i 1 and weak type l p 1 w 1 l p m w m l p estimate of t when there is a p i 1 where i 1 m i p p i and each w i is a nonnegative function on r n as applications of the above results we obtained the boundedness of multilinear littlewood paley operators with non convolution type kernels including multilinear g function marcinkiewicz integral and g function

learning theory of randomized kaczmarz algorithm a relaxed randomized kaczmarz algorithm is investigated in a least squares regression setting by a learning theory approach when the sampling values are accurate and the regression function conditional means is linear such an algorithm has been well studied in the community of non uniform sampling in this paper we are mainly interested in the different case of either noisy random measurements or a nonlinear regression function in this case we show that relaxation is needed a necessary and sufficient condition on the sequence of relaxation parameters or step sizes for the convergence of the algorithm in expectation is presented moreover polynomial rates of convergence both in expectation and in probability are provided explicitly as a result the almost sure convergence of the algorithm is proved by applying the borel cantelli lemma

a uniform estimate in this article we study bilinear operators whose input consists of two functions on the real line and whose output is again a function on the real line we shall be concerned with a priori estimates for these operators for simplicity we initially require the two input functions to be schwartz functions and allow the output function to be a tempered distribution by abstract generalities such a bilinear operation b can be formally written as

polynomial growth estimates for multilinear singular integral operators on demontre un theoreme pour des mesures de carleson et on l applique a l operateur de kato en dimension 1 on demontre un critere de limitation general on donne une application aux commutateurs de coolderon on etudie les d commutateurs

shannon sampling and function reconstruction from point values i first met rene at the well known 1956 meeting on topology in mexico city he then came to the university of chicago where i was starting my job as instructor for the fall of 1956 he suzanne clara and i became good friends and saw much of each other for many decades especially at ihes in paris thom s encouragement and support were important for me especially in my first years after my ph d i studied his work in cobordism singularities of maps and transversality gaining many insights i also enjoyed listening to his provocations for example his disparaging remarks on complex analysis 19th century mathematics and bourbaki there was also a stormy side in our relationship neither of us could hide the pain that our public conflicts over catastrophe theory caused rene thom was a great mathematician leaving his impact on a wide part of mathematics i will always treasure my memories of him steve smale

online learning as stochastic approximation of regularization paths optimality and almost sure convergence in this paper an online learning algorithm is proposed as sequential stochastic approximation of a regularization path converging to the regression function in reproducing kernel hilbert spaces rkhss we show that it is possible to produce the best known strong rkhs norm convergence rate of batch learning through a careful choice of the gain or step size sequences depending on regularity assumptions on the regression function the corresponding weak mean square distance convergence rate is optimal in the sense that it reaches the minimax and individual lower rates in the literature in both cases we deduce almost sure convergence using bernstein type inequalities for martingales in hilbert spaces to achieve this we develop a bias variance decomposition similar to the batch learning setting the bias consists in the approximation and drift errors along the regularization path which display the same rates of convergence and the variance arises from the sample error analysed as a reverse martingale difference sequence the rates above are obtained by an optimal trade off between the bias and the variance

piecewise linear regularized solution paths we consider the generic regularized optimization problem argmin l y x j efron hastie johnstone and tibshirani ann statist 32 2004 407 499 have shown that for the lasso that is if l is squared error loss and j 1 is the l 1 norm of the optimal coefficient path is piecewise linear that is is piecewise constant we derive a general characterization of the properties of loss l penalty j pairs which give piecewise linear coefficient paths such pairs allow for efficient generation of the full regularized coefficient paths we investigate the nature of efficient path following algorithms which arise we use our results to suggest robust versions of the lasso for regression and classification and to develop new efficient algorithms for existing problems in the literature including mammen and van de geer s locally adaptive regression splines

uniform bounds for the bilinear hilbert transforms i it is shown that the bilinear hilbert transforms h f g x p v r f x t g x t dt t map l p1 r l p2 r l p r uniformly in the real parameters when 2 p 1 p 2 oo and 1 p p 1 p 2 p 1 p 2 2 combining this result with the main result in 9 we deduce that the operators h 1 map l 2 r x l r l 2 r uniformly in the real parameter 0 1 this completes a program initiated by a calderon

square roots of elliptic second order divergence operators on strongly lipschitz domains l p theory we study l p estimates for square roots of second order elliptic non necessarily selfadjoint operators in divergence form l mathrm div a nabla on lipschitz domains subject to dirichlet or to neumann boundary conditions pursuing our work 4 where we considered operators on mathbb r n we obtain among other things l 1 2 f p le c nabla f p for all 1 p infty if l is real symmetric and the domain bounded which is new for 1 p 2 we also obtain similar results for perturbations of constant coefficients operators our methods rely on a singular integral representation calderon zygmund theory and quadratic estimates a feature of this study is the use of a commutator between the resolvent of the laplacian dirichlet and neumann and partial derivatives which carries the geometry of the boundary

on aggregation functions based on linguistically quantified propositions and finitely additive set functions we study aggregation functions defined as convex combinations of the linguistically quantified propositions at least k criteria are satisfied our approach is similar to the towa function in spirit but instead of using a maxitive measure we propose to rely on a real valued finitely additive set function this assumption leads to a new framework however it is combinatorial by nature and in general it produces functions with high computational costs therefore we analyze some particular settings and introduce new aggregation functions which can alleviate the combinatorial burden thanks to several combinatorial identities these methods have interesting features and in particular some of them make it possible to set different types of relationships between criteria by allowing the use of different t norms the interest of our proposals is illustrated with a famous example which cannot be modeled by classical aggregation functions such as the choquet integral

online pairwise learning algorithms pairwise learning usually refers to a learning task that involves a loss function depending on pairs of examples among which the most notable ones are bipartite ranking metric learning and auc maximization in this letter we study an online algorithm for pairwise learning with a least square loss function in an unconstrained setting of a reproducing kernel hilbert space rkhs that we refer to as the online pairwise learning algorithm opera in contrast to existing works kar sriperumbudur jain karnick 2013 wang khardon pechyony jones 2012 which require that the iterates are restricted to a bounded domain or the loss function is strongly convex opera is associated with a non strongly convex objective function and learns the target function in an unconstrained rkhs specifically we establish a general theorem that guarantees the almost sure convergence for the last iterate of opera without any assumptions on the underlying distribution explicit convergence rates are derived under the condition of polynomially decaying step sizes we also establish an interesting property for a family of widely used kernels in the setting of pairwise learning and illustrate the convergence results using such kernels our methodology mainly depends on the characterization of rkhss using its associated integral operators and probability inequalities for random variables with values in a hilbert space

a note onbmoand bloch space of generalized α β harmonic functions abstract in this paper we study functions in the unit ball in c n annihilated by the differential operators c given by formula we obtain two equivalent carleson measure characterizations of those of such functions having boundary value in bmo we also give analogous carleson bergman measure characterizations of harmonic bloch functions

online learning with kernels kernel based algorithms such as support vector machines have achieved considerable success in various problems in batch setting where all of the training data is available in advance support vector machines combine the so called kernel trick with the large margin idea there has been little use of these methods in an online setting suitable for real time applications in this paper we consider online learning in a reproducing kernel hilbert space by considering classical stochastic gradient descent within a feature space and the use of some straightforward tricks we develop simple and computationally efficient algorithms for a wide range of problems such as classification regression and novelty detection in addition to allowing the exploitation of the kernel trick in an online setting we examine the value of large margins for classification in the online setting with a drifting target we derive worst case loss bounds and moreover we show the convergence of the hypothesis to the minimizer of the regularized risk functional we present some experimental results that support the theory as well as illustrating the power of the new algorithms for online novelty detection

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

multilinear operators with non smooth kernels and commutators of singular integrals we obtain endpoint estimates for multilinear singular integral operators whose kernels satisfy regularity conditions significantly weaker than those of the standard calder6n zygmund kernels as a consequence we deduce endpoint l 1 x l 1 to weak l 1 m estimates for the mth order commutator of calderon our results reproduce known estimates for m 1 2 but are new for m 3 we also explore connections between the 2nd order higher dimensional commutator and the bilinear hilbert transform and deduce some new off diagonal estimates for the former

optimum bounds for the distributions of martingales in banach spaces a general device is proposed which provides for extension of exponential inequalities for sums of independent real valued random variables to those for martingales in the 2 smooth banach spaces this is used to obtain optimum bounds of the rosenthal burkholder and chung types on moments of the martingales in 2 smooth banach spaces in turn it leads to best order bounds on moments of sums of independent random vectors in any separable banach spaces although the emphasis is put on infinite dimensional martingales most of the results seem to be new even for one dimensional martingales moreover the bounds on moments of the rosenthal burkholder type seem to be to a certain extent new even for sums of independent real valued random variables analogous inequalities for one dimensional supermartingales are given

incremental proximal methods for large scale convex optimization we consider the minimization of a sum sum i 1 mf i x consisting of a large number of convex component functions f i for this problem incremental methods consisting of gradient or subgradient iterations applied to single components have proved very effective we propose new incremental methods consisting of proximal iterations applied to single components as well as combinations of gradient subgradient and proximal iterations we provide a convergence and rate of convergence analysis of a variety of such methods including some that involve randomization in the selection of components we also discuss applications in a few contexts including signal processing and inference machine learning

on complexity issues of online learning algorithms in this paper some new probabilistic upper bounds are presented for the online learning algorithm proposed in and more generally for linear stochastic approximations in hilbert spaces with these upper bounds not only does one recover almost sure convergence but also relaxes the square summable condition on the step size appeared in the early work furthermore two probabilistic upper bounds are given for an averaging process both of which achieve the same rate with respect to sample size as in batch learning algorithms and one of which is tight in both sample size and regularization parameter

the multilinear marcinkiewicz interpolation theorem revisited the behavior of the constant abstract we provide a self contained proof of the multilinear extension of the marcinkiewicz real method interpolation theorem with initial assumptions a set of restricted weak type estimates considering possible degenerate situations that may arise the advantage of this proof is that it yields a logarithmically convex bound for the norm of the operator on the intermediate spaces in terms of the initial restricted weak type bounds it also provides an explicit estimate in terms of the exponents of the initial estimates the constant blows up like a negative power of the distance from the intermediate point to the boundary of the convex hull of the initial points

existence and boundedness of multilinear littlewood paley operators on campanato spaces abstract in this paper we established the boundedness of multilinear littlewood paley operators on campanato spaces including the multilinear g function multilinear lusin s area integral and multilinear littlewood paley g function we proved that if the above operators are finite for one point then they are finite almost everywhere moreover norm inequalities do hold for the above operators on the corresponding campanato spaces

multilinear operator valued calderon zygmund theory abstract we develop a general theory of multilinear singular integrals with operator valued kernels acting on tuples of umd banach spaces this in particular involves investigating multilinear variants of the r boundedness condition naturally arising in operator valued theory we proceed by establishing a suitable representation of multilinear operator valued singular integrals in terms of operator valued dyadic shifts and paraproducts and studying the boundedness of these model operators via dyadic probabilistic banach space valued analysis in the bilinear case we obtain a t 1 type theorem without any additional assumptions on the banach spaces other than the necessary umd higher degrees of multilinearity are tackled via a new formulation of the rademacher maximal function rmf condition in addition to the natural umd lattice cases our rmf condition covers suitable tuples of non commutative l p spaces we employ our operator valued theory to obtain new multilinear multi parameter operator valued theorems in the natural setting of umd spaces with property