varieties of nilpotent elements for simple lie algebras i good primes let g be a simple algebraic group over k or fp where p is good set g lie g given r and a faithful restricted representation g gl v one can define a variety of nilpotent elements nr g x g x r 0 in this paper we determine this variety when is an irreducible representation of minimal dimension or the adjoint representation

concentration of measure inequalities for markov chains and phi mixing processes we prove concentration inequalities for some classes of markov chains and f mixing processes with constants independent of the size of the sample that extend the inequalities for product measures of talagrand the method is based on information inequalities put forward by marton in case of contracting markov chains using a simple duality argument on entropy our results also include the family of logarithmic sobolev inequalities for convex functions applications to bounds on supremum of dependent empirical processes complete this work

displacement convexity of entropy and related inequalities on graphs we introduce the notion of an interpolating path on the set of probability measures on finite graphs using this notion we first prove a displacement convexity property of entropy along such a path and derive prekopa leindler type inequalities a talagrand transport entropy inequality certain hwi type as well as log sobolev type inequalities in discrete settings to illustrate through examples we apply our results to the complete graph and to the hypercube for which our results are optimal by passing to the limit we recover the classical log sobolev inequality for the standard gaussian measure with the optimal constant

principal curve algorithms for partitioning high dimensional data spaces most partitioning algorithms iteratively partition a space into cells that contain underlying linear or nonlinear structures using linear partitioning strategies the compactness of each cell depends on how well the locally linear partitioning strategy approximates the intrinsic structure to partition a compact structure for complex data in a nonlinear context this paper proposes a nonlinear partition strategy this is a principal curve tree pc tree which is implemented iteratively given that a pc passes through the middle of the data distribution it allows for partitioning based on the arc length of the pc to enhance the partitioning of a given space a residual version of the pc tree algorithm is developed denoted here as the principal component analysis tree pcr tree algorithm because of its residual property the pcr tree can yield the intrinsic dimension of high dimensional data comparisons presented in this paper confirm that the proposed pc tree and pcr tree approaches show a better performance than several other competing partitioning algorithms in terms of vector quantization error and nearest neighbor search the comparison also shows that the proposed algorithms outperform competing linear methods in total average coverage which measures the nonlinear compactness of partitioning algorithms

an elementary proof of the polar factorization of vector valued functions we present an elementary proof of an important result of y brenier br1 br2 namely that vector fields in d satisfying a nondegeneracy condition admit the polar factorization u x s x where is a convex function and s is a measure preserving mapping brenier solves a minimization problem using monge kantorovich theory whereas we turn our attention to a dual problem whose euler lagrange equation turns out to be

optimal mass transportation in the heisenberg group in this paper we consider the problem of optimal transportation of absolutely continuous masses in the heisenberg group hn in the case when the cost function is either the square of the carnot caratheodory distance or the square of the koranyi norm in both cases we show existence and uniqueness of an optimal transport map in the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function in the latter case the proof requires some fine properties of bv functions in the heisenberg group

simple lie algebras of small characteristic ii exceptional roots abstract letlbe a finite dimensional simple lie algebra of absolute toral rank 2 over an algebraically closed field of characteristicp 3 andta 2 dimensional torus in the semisimplep envelope ofl suppose thatlis not isomorphic to a melikian algebra it is proved in this paper that for every root l t the subalgebrak generated by i f pki whereki x li x l i 0 acts triangulably onl in particular this implies that in the terminology of r e block and r l wilson 1988 j algebra114 115 259 all roots of l t are nonexceptional

constraint local principal curve existing principal curve algorithms have some drawbacks such as time consuming and narrow application scope in practice since these algorithms are mainly based on global optimization in this paper we present the concept of constraint local principal curve clpc which uses local optimization methods and restricts the principal curve with two fixed endpoints to reduce the computational complexity in addition we propose three clpc algorithms by local optimization and adaptive radius to expand the range of applications and increase the solution quality the first algorithm i e clpcg is based on greedy thinking the second algorithm i e clpcs uses one dimensional search and the last algorithm clpcc combines the greedy thinking and one dimensional search then we define six performance indices to evaluate the performance of the clpc algorithms finally we present some numerical experiments with three simulation data sets and two gps measured data sets in both highway and railway the results indicate that all of the three clpc algorithms can obtain high accuracy data from multiple low accuracy data efficiently the clpc algorithms can improve the accuracy and computational speed compared with the existing k segment principal curve kpc algorithm in addition clpcc outperforms clpcg and clpcs according to the comprehensive experiments while clpcg runs much faster than other ones we present the concept of constraint local principal curve clpc to reduce the computational complexity we propose three clpc algorithms that combine local optimization and adaptive radius to expand the range of applications and increase the solution quality we present some numerical experiments with three simulation data sets and two measured gps data sets in highway and railway the clpc algorithms can improve the accuracy and computational speed compared with the existing kpc algorithms the features of the each clpc algorithm are analyzed according to the comprehensive experiments

simple lie algebras of small characteristic vi completion of the classification let l be a finite dimensional simple lie algebra over an algebraically closed field of characteristic p 3 it is proved in this paper that if the p envelope of adl in derl contains a torus of maximal dimension whose centralizer in adl acts nontriangulably on l then p 5 and l is isomorphic to one of the melikian algebras m m n in conjunction with a premet h strade simple lie algebras of small characteristic v the non melikian case j algebra 314 2007 664 692 theorem 1 2 this implies that up to isomorphism any finite dimensional simple lie algebra over an algebraically closed field of characteristic p 3 is either classical or a filtered lie algebra of cartan type or a melikian algebra of characteristic 5 this result finally settles the classification problem for finite dimensional simple lie algebras over algebraically closed fields of characteristic 2 3

some applications of mass transport to gaussian type inequalities as discovered by brenier mapping through a convex gradient gives the optimal transport in n in the present article this map is used in the setting of gaussian like measures to derive an inequality linking entropy with mass displacement by a straightforward argument as a consequence logarithmic sobolev and transport inequalities are recovered finally a result of caffarelli on the brenier map is used to obtain gaussian correlation inequalities

learning and design of principal curves principal curves have been defined as self consistent smooth curves which pass through the middle of a d dimensional probability distribution or data cloud they give a summary of the data and also serve as an efficient feature extraction tool we take a new approach by defining principal curves as continuous curves of a given length which minimize the expected squared distance between the curve and points of the space randomly chosen according to a given distribution the new definition makes it possible to theoretically analyze principal curve learning from training data and it also leads to a new practical construction our theoretical learning scheme chooses a curve from a class of polygonal lines with k segments and with a given total length to minimize the average squared distance over n training points drawn independently convergence properties of this learning scheme are analyzed and a practical version of this theoretical algorithm is implemented in each iteration of the algorithm a new vertex is added to the polygonal line and the positions of the vertices are updated so that they minimize a penalized squared distance criterion simulation results demonstrate that the new algorithm compares favorably with previous methods both in terms of performance and computational complexity and is more robust to varying data models

the hesselink stratification of nullcones and base change let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 we give a case free proof of lusztig s conjectures lusztig in transform groups 10 449 487 2005 on so called unipotent pieces this presents a uniform picture of the unipotent elements of g which can be viewed as an extension of the dynkin kostant theory but is valid without restriction on p we also obtain analogous results for the adjoint action of g on its lie algebra mathfrak g and the coadjoint action of g on mathfrak g

bounding bar d distance by informational divergence a method to prove measure concentration there is a simple inequality by pinsker between variational distance and informational divergence of probability measures defined on arbitrary probability spaces we shall consider probability measures on sequences taken from countable alphabets and derive from pinsker s inequality bounds on the bar d distance by informational divergence such bounds can be used to prove the concentration of measure phenomenon for some nonproduct distributions

monotonicity properties of optimal transportation and the fkg and related inequalities optimal transportation between densities f x g y can be interpreted as a joint probability distribution with marginally f x and g y we prove monotonicity and concavity properties of optimal transportation y x under suitable assumptions on f and g as an application we obtain the fortuin kasteleyn ginibre correlation inequalities as well as some generalizations of the brascamp lieb momentum inequalities

kantorovich duality for general transport costs and applications we introduce a general notion of transport cost that encompasses many costs used in the literature including the classical one and weak transport costs introduced by talagrand and marton in the 90 s and prove a kantorovich type duality theorem as a by product we obtain various applications in different directions we give a short proof of a result by strassen on the existence of a martingale with given marginals we characterize the associated transport entropy inequalities together with the log sobolev inequality restricted to convex concave functions some explicit examples of discrete measures satisfying weak transport entropy inequalities are also given

new concentration inequalities in product spaces we introduce three new ways to measure the distance from a point to a subset of a product space and we prove corresponding concentration inequalities each of them allows to control the fluctuation of a new class of random variables

a modular analogue of morozov s theorem on maximal subalgebras of simple lie algebras abstract let g be a simple algebraic group over an algebraically closed field of characteristic p 0 and suppose that p is a very good prime for g in this paper we prove that any maximal lie subalgebra m of g lie g with rad m 0 has the form m lie p for some maximal parabolic subgroup p of g this means that morozov s theorem on maximal subalgebras is valid under mild assumptions on g we show that such assumptions are necessary by providing a counterexample to morozov s theorem for groups of type e 8 over fields of characteristic 5 our proof relies on the main results and methods of the classification theory of finite dimensional simple lie algebras over fields of prime characteristic

simple lie algebras of small characteristic iii the toral rank 2 case abstract it is proved that every finite dimensional simple lie algebra of absolute toral rank 2 over an algebraically closed field of characteristic p 3 is of classical or cartan type or a melikian algebra

information inequalities and concentration of measure we derive inequalities of the form p q h p r h q r which hold for every choice of probability measures p q r where h p r denotes the relative entropy of p with respect to r and p q stands for a coupling type distance between p and q using the chain rule for relative entropies and then specializing to q with a given support we recover some of talagrand s concentration of measure inequalities for product spaces

a second order equation for schrodinger bridges with applications to the hot gas experiment and entropic transportation cost the schrodinger problem is obtained by replacing the mean square distance with the relative entropy in the monge kantorovich problem it was first addressed by schrodinger as the problem of describing the most likely evolution of a large number of brownian particles conditioned to reach an unexpected configuration its optimal value the entropic transportation cost and its optimal solution the schrodinger bridge stand as the natural probabilistic counterparts to the transportation cost and displacement interpolation moreover they provide a natural way of lifting from the point to the measure setting the concept of brownian bridge in this article we prove that the schrodinger bridge solves a second order equation in the riemannian structure of optimal transport roughly speaking the equation says that its acceleration is the gradient of the fisher information using this result we obtain a fine quantitative description of the dynamics and a new functional inequality for the entropic transportation cost that generalize talagrand s transportation inequality finally we study the convexity of the fisher information along schrodigner bridges under the hypothesis that the associated reciprocal characteristic is convex the techniques developed in this article are also well suited to study the feynman kac penalisations of brownian motion

measure concentration for a class of random processes let x x i i be a stationary random process with a countable alphabet and distribution q let q x k 0 denote the conditional distribution of x x 1 x 2 x n given the k length past write d 1 x 1 0 if 1 x 1 and d 1 x 1 1 otherwise we say that the process x admits a joining with finite distance u if for any two past sequences k 0 k 1 0 and x k 0 x k 1 x 0 there is a joining of q k 0 and q x k 0 say dist 0 x 0 k 0 x k 0 such that the main result of this paper is the following inequality for processes that admit a joining with finite distance

existence and uniqueness of optimal maps on alexandrov spaces the purpose of this paper is to show that in a finite dimensional metric space with alexandrov s curvature bounded below monge s transport problem for the quadratic cost admits a unique solution

extremal properties of principal curves in the plane principal curves were introduced to formalize the notion of a curve passing through the middle of a dataset vaguely speaking a curve is said to pass through the middle of a dataset if every point on the curve is the average of the observations projecting onto it this idea can be made precise by defining principal curves for probability densities in this paper we study principal curves in the plane like linear principal components principal curves are critical points of the expected squared distance from the data however the largest and smallest principal components are extrema of the distance whereas all principal curves are saddle points this explains why cross validation does not appear to be a viable method for choosing the complexity of principal curve estimates

classification of balanced toral elements of exceptional lie algebras abstract let g be a simple lie algebra of exceptional type over an algebraically closed field k and let g be a simple linear algebraic group with lie algebra g for char k p 0 we present a complete listing of the g conjugacy classes of balanced toral elements of g as a result we also obtain the classification of conjugacy classes of balanced inner torsion automorphisms of g of order p when char k 0

sub principal homomorphisms in positive characteristic let g be a reductive group over an algebraically closed field of characteristic p and let u g be a unipotent element of order p suppose that p is a good prime for g we show in this paper that there is a homomorphism sl 2 k g whose image contains u this result was first obtained by d testerman j algebra 1995 using case considerations for each type of simple group and using in some cases computer calculations with explicit representatives for the unipotent orbits the proof we give is free of case considerations except in its dependence on the bala carter theorem our construction of generalizes the construction of a principal homomorphism made by j p serre in invent math 1996 in particular is obtained by reduction modulo from a homomorphism of group schemes over a valuation ring in a number field this permits us to show moreover that the weight spaces of a maximal torus of sl 2 k on lie g are the same as in characteristic 0 the existence of a with this property was previously obtained again using case considerations by lawther and testerman memoirs ams 1999 and has been applied in some recent work of g seitz invent math 2000

simple modules with character height at most one for the restricted witt algebras abstract the simple modules with character height at most one for the restricted witt algebras are considered their classification construction and dimension formulas are reduced to the same for the general linear algebra results of chang and shen are recovered in the process

structure of optimal martingale transport plans in general dimensions given two probability measures mu and nu in convex order on r d we study the profile of one step martingale plans pi on r d x r d that optimize the expected value of the modulus of their increment among all martingales having mu and nu as marginals while there is a great deal of results for the real line i e when d 1 much less is known in the richer and more delicate higher dimensional case that we tackle in this paper we show that many structural results can be obtained provided the initial measure mu is absolutely continuous with respect to the lebesgue measure one such a property is that mu almost every x in r d is transported by the optimal martingale plan into a probability measure pi x concentrated on the extreme points of the closed convex hull of its support this will be established for the distance cost c x y vertical bar x y vertical bar in the two dimensional case and also for any d 3 as long as the marginals are in subharmonic order in some cases pi x is supported on the vertices of a k x dimensional polytope such as when the target measure is discrete duality plays a crucial role in our approach even though in contrast to standard optimal transports the dual extremal problem may not be attained in general we show however that martingale supporting borel subsets of r d x r d can be decomposed into a collection of mutually disjoint components by means of a convex paving of the source space in such a way that when the martingale is optimal for a general cost function each of the components then supports a restricted optimal martingale transport whose dual problem is attained this decomposition is used to obtain structural results in cases where global duality is not attained on the other hand it shows that certain optimal martingale supporting borel sets can be viewed as higher dimensional versions of nikodym type sets the paper focuses on the distance cost but much of the results hold for general lipschitz cost functions

locally defined principal curves and surfaces principal curves are defined as self consistent smooth curves passing through the middle of the data and they have been used in many applications of machine learning as a generalization dimensionality reduction and a feature extraction tool we redefine principal curves and surfaces in terms of the gradient and the hessian of the probability density estimate this provides a geometric understanding of the principal curves and surfaces as well as a unifying view for clustering principal curve fitting and manifold learning by regarding those as principal manifolds of different intrinsic dimensionalities the theory does not impose any particular density estimation method can be used with any density estimator that gives continuous first and second derivatives therefore we first present our principal curve surface definition without assuming any particular density estimation method afterwards we develop practical algorithms for the commonly used kernel density estimation kde and gaussian mixture models gmm results of these algorithms are presented in notional data sets as well as real applications with comparisons to other approaches in the principal curve literature all in all we present a novel theoretical understanding of principal curves and surfaces practical algorithms as general purpose machine learning tools and applications of these algorithms to several practical problems

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

optimal transport maps for monge kantorovich problem on loop groups let g be a compact lie group and consider the loop group leg l c 0 1 g l 0 l 1 e let be the heat kernel measure at the time 1 for any density function f on leg such that ent f we shall prove that there exists a unique optimal transportation map t leg leg which pushes forward to f

on a problem of optimal transport under marginal martingale constraints the basic problem of optimal transportation consists in minimizing the expected costs e c x 1 x 2 by varying the joint distribution x 1 x 2 where the marginal distributions of the random variables x 1 and x 2 are fixed inspired by recent applications in mathematical finance and connections with the peacock problem we study this problem under the additional condition that x i i 1 2 is a martingale that is e x 2 x 1 x 1 we establish a variational principle for this problem which enables us to determine optimal martingale transport plans for specific cost functions in particular we identify a martingale coupling that resembles the classic monotone quantile coupling in several respects in analogy with the celebrated theorem of brenier the following behavior can be observed if the initial distribution is continuous then this monotone martingale is supported by the graphs of two functions t 1 t 2 r r

monge kantorovitch measure transportation and monge ampere equation on wiener space let w h be an abstract wiener space assume two i i 1 2 probabilities on w w we give some conditions for the wasserstein distance between 1 and 2 with respect to the cameron martin space to be finite where the infimum is taken on the set of probability measures on w w whose first and second marginals are 1 and 2 in this case we prove the existence of a unique cyclically monotone map t iw with w h such that t maps 1 to 2 moreover if 2 then t is stochastically invertible i e there exists s w w such that s t iw 1 a s and t s iw 2 a s if in addition 1 then there exists a 1 convex function in the gaussian sobolev space such that these results imply that the quasi invariant transformations of the wiener space with finite wasserstein distance from can be written as the composition of a transport map t and a rotation i e a measure preserving map we give also 1 convex sub solutions and ito type solutions of the monge ampere equation on w

a unified model for probabilistic principal surfaces principal curves and surfaces are nonlinear generalizations of principal components and subspaces respectively they can provide insightful summary of high dimensional data not typically attainable by classical linear methods solutions to several problems such as proof of existence and convergence faced by the original principal curve formulation have been proposed in the past few years nevertheless these solutions are not generally extensible to principal surfaces the mere computation of which presents a formidable obstacle consequently relatively few studies of principal surfaces are available we previously 2000 proposed the probabilistic principal surface pps to address a number of issues associated with current principal surface algorithms pps uses a manifold oriented covariance noise model based on the generative topographical mapping gtm which can be viewed as a parametric formulation of kohonen s self organizing map building on the pps we introduce a unified covariance model that implements pps 0 1 by varying the clamping parameter spl alpha then we comprehensively evaluate the empirical performance of pps gtm and the manifold aligned gtm on three popular benchmark data sets it is shown in two different comparisons that the pps outperforms the gtm under identical parameter settings convergence of the pps is found to be identical to that of the gtm and the computational overhead incurred by the pps decreases to 40 percent or less for more complex manifolds these results show that the generalized pps provides a flexible and effective way of obtaining principal surfaces

nilpotent orbits in good characteristic and the kempf rousseau theory let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 image and suppose that p is a good prime for the root system of g in this paper we give a fairly short conceptual proof of pommerening s theorem pommerening j algebra 49 1977 525 536 j algebra 65 1980 373 398 which states that any nilpotent element in image is richardson in a distinguished parabolic subalgebra of the lie algebra of a levi subgroup of g as a by product we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent g orbits in image for earlier proofs of this theorem see kawanaka invent math 84 1986 575 616 premet trans amer math soc 347 1995 2961 2988 spaltenstein j fac sci univ tokyo sect ia math 31 1984 283 286 we extend recent results of sommers internal math res notices 11 1998 539 562 to reductive lie algebras of good characteristic thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in image and unipotent elements in g earlier computations of these groups in positive characteristics relied mostly on work of mizuno j fac sci univ tokyo sect ia math 24 1977 525 563 tokyo j math 3 1980 391 459 our approach is based on the theory of optimal parabolic subgroups for g unstable vectors also known as the kempf rousseau theory which provides a good substitute for the image theory prominent in the characteristic zero case

on a mixture of brenier and strassen theorems we give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in 33 optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling we also establish some connections with caffarelli s contraction theorem 14

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

mixing properties of a class of bernoulli processes we prove that stationary very weak bernoulli processes with rate o 1 n vwb o 1 n are strictly very weak bernoulli with rate o 1 n furthermore we discuss the relation between vwb o 1 n and the classical mixing properties for countable state processes in particular we show that vwb o 1 n implies mixing

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159