a p adic simpson correspondence abstract for curves over a p adic field we construct an equivalence between the category of higgs bundles and that of generalised representation of the etale fundamental group the definition of generalised representations uses p adic hodge theory and almost etale coverings and it includes usual representations which form a full subcategory the equivalence depends on the choice of an exponential function for the multiplicative group

convergence rates and decoupling in linear stochastic approximation algorithms almost sure convergence rates for linear algorithms h k 1 h k frac 1 k chi b k a kh k are studied where chi in 0 1 a k k 1 infty are symmetric positive semidefinite random matrices and b k k 1 infty are random vectors it is shown that n gamma h n a 1 b rightarrow 0 a s for the gamma in 0 chi positive definite a and vector b such that frac 1 n chi gamma sum k 1 n a k a to 0 and frac 1 n chi gamma sum k 1 n b k b to 0 a s when chi gamma in left frac12 1 right these assumptions are implied by the marcinkiewicz strong law of large numbers which allows the a k and b k to have heavy tails long range dependence or both finally corroborating experimental outcomes and decreasing gain design considerations are provided

notes on the local p adic simpson correspondence we complete the theory of the local p adic simpson correspondence developed by faltings for rational coefficients it asserts that there is an equivalence between the category of small mathbb q p generalized representations of the geometric fundamental group and that of small mathbb q p higgs bundles for a certain kind of log smooth affine scheme over a complete discrete valuation ring of mixed characteristic 0 p with perfect residue field the difficulty lies in the construction of the latter from the former and we give it via a generalized sen s theory for the log smooth affine scheme which depends on faltings almost purity theorem inspired by a recent work by r liu and x zhu we also give a formulation of local p adic simpson correspondence for mathbb q p generalized representations of the arithmetic fundamental group and a characterization of hodge tate generalized representations in terms of the correspondence

on the impossibility of dimension reduction in l 1 the johnson lindenstrauss lemma shows that any n points in euclidean space i e n with distances measured under the e 2 norm may be mapped down to o log n e2 dimensions such that no pairwise distance is distorted by more than a 1 p e factor determining whether such dimension reduction is possible in e 1 has been an intriguing open question we show strong lower bounds for general dimension reduction in e 1 we give an explicit family of n points in e 1 such that any embedding with constant distortion d requires n 1 d2 dimensions this proves that there is no analog of the johnson lindenstrauss lemma for e 1 in fact embedding with any constant distortion requires n 1 dimensions further embedding the points into e 1 with 1pe distortion requires nh o e log 1 e dimensions our proof establishes this lower bound for shortest path metrics of series parallel graphs we make extensive use of linear programming and duality in devising our bounds we expect that the tools and techniques we develop will be useful for future investigations of embeddings into e 1

rigidity and a riemann hilbert correspondence for p adic local systems we construct a functor from the category of p adic etale local systems on a smooth rigid analytic variety x over a p adic field to the category of vector bundles with an integrable connection on its base change to mathrm b text dr which can be regarded as a first step towards the sought after p adic riemann hilbert correspondence as a consequence we obtain the following rigidity theorem for p adic local systems on a connected rigid analytic variety if the stalk of such a local system at one point regarded as a p adic galois representation is de rham in the sense of fontaine then the stalk at every point is de rham along the way we also establish some basic properties of the p adic simpson correspondence finally we give an application of our results to shimura varieties

on the distributional complexity of disjointness we prove that the distributional communication complexity of the predicate disjointness with respect to a very simple measure on inputs is n

rounds in communication complexity revisited the k round two party communication complexity was studied in the deterministic model by p h papadimitriou and m sipser proc of the 14th stoc 1982 pp 330 337 and p duris z galil and g schnitger proc of the 16th stoc 1984 pp 81 91 and in the probabilistic model by a c yao proc of the 24th focs 1983 pp 420 428 and b halstenberg and r reischuk proc of the 20th stoc 1988 pp 162 172 this paper presents new lower bounds that give 1 randomization is more powerful than determinism in k round protocols and 2 an explicit function which exhibits an exponential gap between its k and k 1 round randomized complexity this paper also studies the three party communication model and exhibits an exponential gap in 3 round protocols that differ in the starting player finally this paper shows new connections of these questions to circuit complexity that motivate further work in this direction

multiple pass streaming algorithms for learning mixtures of distributions in rd we present a multiple pass streaming algorithm for learning the density function of a mixture of k uniform distributions over rectangles in r d for any d 0 our learning model is samples drawn according to the mixture are placed in arbitrary order in a data stream that may only be accessed sequentially by an algorithm with a very limited random access memory space our algorithm makes 2 2 passes for any 0 and requires memory at most o e 2 k 2d 4 4k d where e is the tolerable error of the algorithm this exhibits a strong memory pass tradeoff in terms of e a few more passes significantly lower its memory requirements thus trading one of the two most important resources in streaming computation for the other chang and kannan first considered this problem for d 1 2 our learning algorithm is especially appropriate for situations where massive data sets of samples are available but computation with such large inputs requires very restricted models of computation

applications of a kushner and clark lemma to general classes of stochastic algorithms two general classes of stochastic algorithms are considered including algorithms considered by ljung as well as algorithms of the form theta n 1 theta n gamma n 1 v n 1 theta n z where z is a stationary ergodic process it is shown how one can apply a lemma of kushner and clark to obtain properties of these algorithms this is done by using in particular martingale arguments in the generalized ljung case in these various situations the convergence is obtained by the method of the associated ordinary differential equation under the classical boundedness assumptions in the case of linear algorithms the boundedness assumptions are dropped

noise conditions for prespecified convergence rates of stochastic approximation algorithms we develop deterministic necessary and sufficient conditions on individual noise sequences of a stochastic approximation algorithm for the error of the iterates to converge at a given rate specifically suppose spl rho sub n is a given positive sequence converging monotonically to zero consider a stochastic approximation algorithm x sub n 1 x sub n a sub n a sub n x sub n b sub n a sub n e sub n where x sub n is the iterate sequence a sub n is the step size sequence e sub n is the noise sequence and x is the desired zero of the function f x ax b then under appropriate assumptions we show that x sub n x o spl rho sub n if and only if the sequence e sub n satisfies one of five equivalent conditions these conditions are based on well known formulas for noise sequences kushner and clark s 1978 condition chen s see proc ifac world congr p 375 80 1996 condition kulkarni and horn s see ieee trails automat contr vol 41 p 419 24 1996 condition a decomposition condition and a weighted averaging condition our necessary and sufficient condition on e sub n to achieve a convergence rate of spl rho sub n is basically that the sequence e sub n spl rho sub n satisfies any one of the above five well known conditions we provide examples to illustrate our result in particular we easily recover the familiar result that if a sub n a n and e sub n is a martingale difference process with bounded variance then x sub n x o n sup 1 2 log n sup spl beta for any spl beta 1 2

sampling methods for the nystrom method the nystrom method is an efficient technique to generate low rank matrix approximations and is used in several large scale learning applications a key aspect of this method is the procedure according to which columns are sampled from the original matrix in this work we explore the efficacy of a variety of fixed and adaptive sampling schemes we also propose a family of ensemble based sampling algorithms for the nystrom method we report results of extensive experiments that provide a detailed comparison of various fixed and adaptive sampling techniques and demonstrate the performance improvement associated with the ensemble nystrom method when used in conjunction with either fixed or adaptive sampling schemes corroborating these empirical findings we present a theoretical analysis of the nystrom method providing novel error bounds guaranteeing a better convergence rate of the ensemble nystrom method in comparison to the standard nystrom method

the fourth moment method higher moment analysis has typically been used to upper bound certain functions in this paper we introduce a new combinatorial method to lower bound the expectation of the absolute value of a random variable x by the expectation of a quartic in x in the special case where we are looking at the absolute value of a weighted sum of 1 1 unbiased random variables we achieve tight bounds using only a fourth moment for the total discrepancy of a set system because the fourth moment depends only on 4 wise independence our bounds will hold over polynomially sized distributions and so these bounds will be directly applicable in removing randomness to obtain nc algorithms we obtain the first nc algorithms for the problems of total discrepancy maximum acyclic subgraph tournament ranking the gale berlekamp switching game and edge discrepancy we show that for most of these applications it is truly necessary to consider a fourth moment by exhibiting a 3 wise independent distribution which does not achieve the required bounds our method is strong enough to give a new combinatorial bound on tournament ranking

convergence of the robbins monro method for linear problems in a banach space on etudie la convergence de la methode de robbins monro pour les problemes lineaires dans un espace de banach

on finding common neighborhoods in massive graphs we consider the problem of finding pairs of vertices that share large common neighborhoods in massive graphs we prove lower bounds on the resources needed to solve this problem on resource bounded models of computation in streaming models in which algorithms can access the input only a constant number of times and only sequentially we show that even with randomization any algorithm that determines if there exists any pair of vertices with a large common neighborhood must essentially store and process the input graph off line in sampling models in which algorithms can only query an oracle for the common neighborhoods of specified vertex pairs we show that any algorithm must sample almost every pair of vertices for their respective common neighborhoods

analysis of stochastic gradient algorithms for linear regression problems parameter estimation problems that can be formulated as linear regressions are quite common in many applications recursive on line sequential estimation of such parameters can be performed using the recursive least squares rls algorithm or a stochastic gradient version of this algorithm in this paper the convergence properties of the gradient algorithm are analyzed under the assumption that the gain tends to zero the technique is the same as the so called ordinary differential equation approach but the treatment here is self contained and includes a proof of the boundedness of the estimates a main result is that the convergence conditions for the gradient algorithm are the same as those for the recursive least squares algorithm

on the almost sure rate of convergence of linear stochastic approximation algorithms the almost sure rate of convergence of linear stochastic approximation algorithms is analyzed as the main result it is demonstrated that their almost sure rate of convergence is equivalent to the almost sure rate of convergence of the averages of their input data sequences as opposed to most of the existing results on the rate of convergence of stochastic approximation which cover only algorithms with the noise decomposable as the sum of a martingale difference vanishing and telescoping sequence the main results of this paper hold under assumptions not requiring the input data sequences to admit any particular decomposition although no decomposition of the input data sequences is required the results on the almost sure rate of convergence of linear stochastic approximation algorithms obtained in this correspondence are as tight as the rate of convergence in the law of iterated logarithm moreover the main result of this correspondence yields the law of iterated logarithm for linear stochastic approximation if the law of iterated logarithm holds for the input data sequences the obtained general results are illustrated with two nontrivial examples where the input data sequences are strongly mixing strictly stationary random processes or functions of a uniformly ergodic markov chain these results are also applied to the analysis of least mean square lms algorithms

near linear time construction of sparse neighborhood covers this paper introduces a near linear time sequential algorithm for constructing a sparse neighborhood cover this implies analogous improvements from quadratic to near linear time for any problem whose solution relies on network decompositions including small edge cuts in planar graphs approximate shortest paths and weight and distance preserving graph spanners in particular an o log n approximation to the k shortest paths problem on an n vertex e edge graph is obtained that runs in soh n e k time

convergence of an adaptive linear estimation algorithm in this work we prove the almost sure convergence of an adaptive linear estimator governed by a stochastic gradient algorithm with decreasing step size in the presence of correlated observations two complementary contributions are added to the famous 1977 ljung theorem first we drop the condition of nondivergence of the algorithm assumed by ljung while that condition can be ensured by adding a barrier the convergence of the suitably bounded algorithm itself is not established even on the basis of ljung theorem here the barrier problem is overcome by proving that it is not necessary for the convergence our second contribution is to generalize the model describing the correlated observations no state space model is used and no linear relationship between the observations and the signal to be estimated needs to be assumed instead we use a decreasing covariance model that agrees with a very wide class of practical applications

sparsification a technique for speeding up dynamic graph algorithms we provide data strutures that maintain a graph as edges are inserted and deleted and keep track of the following properties with the following times minimum spanning forests graph connectivity graph 2 edge connectivity and bipartiteness in time o n 1 2 per change 3 edge connectivity in time o n 2 3 per change 4 edge connectivity in time o n n per change k edge connectivity for constant k in time o n log n per change 2 vertex connectivity and 3 vertex connectivity in the o n per change and 4 vertex connectivity in time o n n per change further results speed up the insertion times to match the bounds of known partially dynamic algorithms all our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph which we call sparsification

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

graph distances in the data stream model we explore problems related to computing graph distances in the data stream model the goal is to design algorithms that can process the edges of a graph in an arbitrary order given only a limited amount of working memory we are motivated by both the practical challenge of processing massive graphs such as the web graph and the desire for a better theoretical understanding of the data stream model in particular we are interested in the trade offs between model parameters such as per data item processing time total space and the number of passes that may be taken over the stream these trade offs are more apparent when considering graph problems than they were in previous streaming work that solved problems of a statistical nature our results include the following 1 spanner construction there exists a single pass tilde o tn 1 1 t space tilde o t 2n 1 t time per edge algorithm that constructs a 2t 1 spanner for t omega log n log log n the algorithm satisfies the semistreaming space restriction of o n operatorname polylog n and has per edge processing time o operatorname polylog n this resolves an open question from j feigenbaum et al theoret comput sci 348 2005 pp 207 216 2 breadth first search bfs trees for any even constant k we show that any algorithm that computes the first k layers of a bfs tree from a prescribed node with probability at least 2 3 requires either greater than k 2 passes or tilde omega n 1 1 k space since constructing bfs trees is an important subroutine in many traditional graph algorithms this demonstrates the need for new algorithmic techniques when processing graphs in the data stream model 3 graph distance lower bounds any t approximation of the distance between two nodes requires omega n 1 1 t space we also prove lower bounds for determining the length of the shortest cycle and other graph properties 4 techniques for decreasing per edge processing we discuss two general techniques for speeding up the per edge computation time of streaming algorithms while increasing the space by only a small factor

fast monte carlo algorithms for matrices i approximating matrix multiplication motivated by applications in which the data may be formulated as a matrix we consider algorithms for several common linear algebra problems these algorithms make more efficient use of computational resources such as the computation time random access memory ram and the number of passes over the data than do previously known algorithms for these problems in this paper we devise two algorithms for the matrix multiplication problem suppose a and b which are m times n and n times p respectively are the two input matrices in our main algorithm we perform c independent trials where in each trial we randomly sample an element of 1 2 ldots n with an appropriate probability distribution cal p on 1 2 ldots n we form an m times c matrix c consisting of the sampled columns of a each scaled appropriately and we form a c times n matrix r using the corresponding rows of b again scaled appropriately the choice of cal p and the column and row scaling are crucial features of the algorithm when these are chosen judiciously we show that cr is a good approximation to ab more precisely we show that left ab cr right f o left a right f left b right f sqrt c where cdot f denotes the frobenius norm i e a 2 f sum i j a ij 2 this algorithm can be implemented without storing the matrices a and b in ram provided it can make two passes over the matrices stored in external memory and use o c m n p additional ram to construct c and r we then present a second matrix multiplication algorithm which is similar in spirit to our main algorithm in addition we present a model the pass efficient model in which the efficiency of these and other approximate matrix algorithms may be studied and which we argue is well suited to many applications involving massive data sets in this model the scarce computational resources are the number of passes over the data and the additional space and time required by the algorithm the input matrices may be presented in any order of the entries and not just row or column order as is the case in many applications where e g the data has been written in by multiple agents in addition the input matrices may be presented in a sparse representation where only the nonzero entries are written

purity for hodge tate representations we prove that a kind of purity holds for hodge tate representations of the fundamental group of the generic fiber of a semi stable scheme over a complete discrete valuation ring of mixed characteristic with perfect residue field as an application we see that the relative p adic etale cohomology with proper support of a scheme separated of finite type over the generic fiber is hodge tate if it is locally constant

stable distributions pseudorandom generators embeddings and data stream computation in this article we show several results obtained by combining the use of stable distributions with pseudorandom generators for bounded space in particular we show that for any p 0 2 one can maintain using only o log n e2 words of storage a sketch c q of a point q lnp under dynamic updates of its coordinates the sketch has the property that given c q and c s one can estimate vq svp up to a factor of 1 p e with large probability this solves the main open problem of feigenbaum et al 1999 we show that the aforementioned sketching approach directly translates into an approximate algorithm that for a fixed linear mapping a and given x n and y m estimates vax yvp in o n p m time for any p 0 2 this generalizes an earlier algorithm of wasserman and blum 1997 which worked for the case p e 2 we obtain another sketch function c which probabilistically embeds ln1 into a normed space lm1 the embedding guarantees that if we set m e log 1 o 1 e then for any pair of points q s ln1 the distance between q and s does not increase by more than 1 p e with constant probability and it does not decrease by more than 1 e with probability 1 this is the only known dimensionality reduction theorem for the l1 norm in fact stronger theorems of this type i e that guarantee very low probability of expansion as well as of contraction cannot exist brinkman and charikar 2003 we give an explicit embedding of ln2 into lno log n 1 with distortion 1 p 1 n 1

high dimensional variable selection with sparse random projections measurement sparsity and statistical efficiency we consider the problem of high dimensional variable selection given n noisy observations of a k sparse vector rp estimate the subset of non zero entries of a significant body of work has studied behavior of l1 relaxations when applied to random measurement matrices that are dense e g gaussian bernoulli in this paper we analyze sparsified measurement ensembles and consider the trade off between measurement sparsity as measured by the fraction of non zero entries and the statistical efficiency as measured by the minimal number of observations n required for correct variable selection with probability converging to one our main result is to prove that it is possible to let the fraction on non zero entries 0 at some rate yielding measurement matrices with a vanishing fraction of non zeros per row while retaining the same statistical efficiency as dense ensembles a variety of simulation results confirm the sharpness of our theoretical predictions

maintaining stream statistics over sliding windows we consider the problem of maintaining aggregates and statistics over data streams with respect to the last n data elements seen so far we refer to this model as the sliding window model we consider the following basic problem given a stream of bits maintain a count of the number of 1 s in the last n elements seen from the stream we show that using o frac 1 epsilon log 2 n bits of memory we can estimate the number of 1 s to within a factor of 1 epsilon we also give a matching lower bound of omega frac 1 epsilon log 2 n memory bits for any deterministic or randomized algorithms we extend our scheme to maintain the sum of the last n positive integers and provide matching upper and lower bounds for this more general problem as well we also show how to efficiently compute the lp norms p in 1 2 of vectors in the sliding window model using our techniques using our algorithm one can adapt many other techniques to work for the sliding window model with a multiplicative overhead of o frac 1 epsilon log n in memory and a 1 epsilon factor loss in accuracy these include maintaining approximate histograms hash tables and statistics or aggregates such as sum and averages