the image of the thom map for eilenberg maclane spaces fundamental classes in bp cohomology of eilenberg mac lane spaces are defined the image of the thom map from bp cohomology to mod p cohomology is determined for arbitrary eilenberg mac lane spaces this image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of milnor primitives to the fundamental class in mod p cohomology this subalgebra in mod p cohomology is invariant under the action of the steenrod algebra and it is annihilated by all milnor primitives we also show that bp cohomology determines morava k cohomology for eilenberg mac lane spaces

applications of atiyah hirzebruch spectral sequences for motivic cobordism in this paper we study applications of atiyah hirzebruch spectral sequences for motivic cobordism recently found by hopkins and morel for example we compute the mod 2 motivic cohomology of a reduced cech complex of a splitting variety according to orlov vishik and voevodsky the cobordism ring mgl 2 spec mathbb r is computed and compared with the results of the real cobordism theory of hu and kriz moreover we study algebraic cobordism of the classifying spaces bg for algebraic groups over mathbb c for example the chow ring ch bg 2 2 for the exceptional lie group g 2 is determined by using motivic cobordism and motivic cohomology

on the configuration lp for maximum budgeted allocation we study the maximum budgeted allocation problem i e the problem of selling a set of m indivisible goods to n players each with a separate budget such that we maximize the collected revenue since the natural assignment lp is known to have an integrality gap of frac 3 4 34 which matches the best known approximation algorithms our main focus is to improve our understanding of the stronger configuration lp relaxation in this direction we prove that the integrality gap of the configuration lp is strictly better than frac 3 4 34 and provide corresponding polynomial time roundings in the following restrictions of the problem i the restricted budgeted allocation problem in which all the players have the same budget and every item has the same value for any player it can be sold to and ii the graph mba problem in which an item can be assigned to at most 2 players finally we improve the best known upper bound on the integrality gap for the general case from frac 5 6 56 to 2 sqrt 2 2 approx 0 828 22 2 0 828 and also prove hardness of approximation results for both cases

approximation algorithms for scheduling unrelated parallel machines we consider the following scheduling problem there arem parallel machines andn independent jobs each job is to be assigned to one of the machines the processing of jobj on machinei requires timep ij the objective is to find a schedule that minimizes the makespan our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum we also present a polynomial approximation scheme for the case that the number of machines is fixed both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations in particular we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints in contrast to our main result we prove that no polynomial algorithm can achieve a worst case ratio less than 3 2 unlessp np we finally obtain a complexity classification for all special cases with a fixed number of processing times

topological data analysis of financial time series landscapes of crashes abstract we explore the evolution of daily returns of four major us stock market indices during the technology crash of 2000 and the financial crisis of 2007 2009 our methodology is based on topological data analysis tda we use persistence homology to detect and quantify topological patterns that appear in multidimensional time series using a sliding window we extract time dependent point cloud data sets to which we associate a topological space we detect transient loops that appear in this space and we measure their persistence this is encoded in real valued functions referred to as a persistence landscapes we quantify the temporal changes in persistence landscapes via their l p norms we test this procedure on multidimensional time series generated by various non linear and non equilibrium models we find that in the vicinity of financial meltdowns the l p norms exhibit strong growth prior to the primary peak which ascends during a crash remarkably the average spectral density at low frequencies of the time series of l p norms of the persistence landscapes demonstrates a strong rising trend for 250 trading days prior to either dotcom crash on 03 10 2000 or to the lehman bankruptcy on 09 15 2008 our study suggests that tda provides a new type of econometric analysis which complements the standard statistical measures the method can be used to detect early warning signals of imminent market crashes we believe that this approach can be used beyond the analysis of financial time series presented here

torsion in cohomology of compact lie groups and chow rings of reductive algebraic groups the purpose of the present note is to give a general procedure for calculating the singular cohomology h k ifp with coefficients in an arbitrary finite field ifv of a connected compact lie group k simultaneously we calculate the chow ring a g ifp a g p of the corresponding complex algebraic group g and the degrees of the basic generalized invariants of the weyl group w in arbitrary characteristic p a mysterious connection between the degrees of generators of a g ifp and the kernel of the principal nilpotent element in characteristic p is pointed out the computation of real cohomology of compact lie groups started by e cartan in 1929 was completed for classical groups by r brauer and by l pontryagin in 1935 h hopf and h samelson showed in 1941 that h k f is an exterior algebra on generators of degrees 2 d 1 1 2dr l where l r a n k k but only in 1949 c t yen managed to compute the d for all exceptional groups case by case on the other hand c chevalley and j leray showed that the d i are nothing else but the degrees of the basic invariants over of the weyl group w a simple procedure for calculating the dl was given in the late fifties by a j coleman and by b kostant nowadays this is common knowledge the problem of computing the torsion in h k z proved to be more recalcitrant this was done by several authors c e miller settled so in 1953 23 a borel settled spin ad spin and the quotients of su and sp in 1954 6 p f baum and w browder settled the remaining classical group the semispin group in 1965 4 g 2 and f 4 were settled in 1954 6 and the p 3 torsion for e 6 and p 5tors ion for ev e s in 1961 7 by a borel s araki computed the 2 torsion for e 6 e 7 1 and the 3 torsion for e 7 e s 2 and announced the 2 torsion for e 8 in a joint paper with y shikata in 1961 3 the details of the proof for the last result have appeared only recently 18 there was an error in 1 in the case of ad e 7 which has been fixed in 13

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

impossibility results for truthful combinatorial auctions with submodular valuations a long standing open question in algorithmic mechanism design is whether there exist computationally efficient truthful mechanisms for combinatorial auctions with performance guarantees close to those possible without considerations of truthfulness in this article we answer this question negatively the requirement of truthfulness can impact dramatically the ability of a mechanism to achieve a good approximation ratio for combinatorial auctions more precisely we show that every universally truthful randomized mechanism for combinatorial auctions with submodular valuations that approximates optimal social welfare within a factor of m1 2 e must use exponentially many value queries where m is the number of items furthermore we show that there exists a class of succinctly represented submodular valuation functions for which the existence of a universally truthful polynomial time mechanism that provides an m1 2 e approximation would imply np e rp in contrast ignoring truthfulness there exist constant factor approximation algorithms for this problem and ignoring computational efficiency the vcg mechanism is truthful and provides optimal social welfare these are the first hardness results for truthful polynomial time mechanisms for any type of combinatorial auctions even for deterministic mechanisms our approach is based on a novel direct hardness technique that completely skips the notoriously hard step of characterizing truthful mechanisms the characterization step was the main obstacle for proving impossibility results in algorithmic mechanism design so far

on the local behavior of spaces of natural images in this study we concentrate on qualitative topological analysis of the local behavior of the space of natural images to this end we use a space of 3 by 3 high contrast patches we develop a theoretical model for the high density 2 dimensional submanifold of showing that it has the topology of the klein bottle using our topological software package plex we experimentally verify our theoretical conclusions we use polynomial representation to give coordinatization to various subspaces of we find the best fitting embedding of the klein bottle into the ambient space of our results are currently being used in developing a compression algorithm based on a klein bottle dictionary

on the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and gap in this paper we consider the following maximum budgeted allocation mba problem given a set of m indivisible items and n agents with each agent i willing to pay b ij on item j and with a maximum budget of b i the goal is to allocate items to agents to maximize revenue the problem naturally arises as auctioneer revenue maximization in budget constrained auctions and as the winner determination problem in combinatorial auctions when utilities of agents are budgeted additive our main results are as follows i we give a 3 4 approximation algorithm for mba improving upon the previous best of simeq0 632 n andelman and y mansour proceedings of the 9th scandinavian workshop on algorithm theory swat 2004 pp 26 38 j vondrak proceedings of the 40th annual acm symposium on the theory of computing stoc 2008 pp 67 74 also implied by the result of u feige and j vondrak proceedings of the 47th ieee symposium on foundations of computer science focs 2006 pp 667 676 our techniques are based on a natural lp relaxation of mba and our factor is optimal in the sense that it matches the integrality gap of the lp ii we prove it is np hard to approximate mba to any factor better than 15 16 previously only np hardness was known t sandholm and s suri games econom behav 55 2006 pp 321 330 b lehmann d lehmann and n nisan proceedings of the 3rd acm conference on electronic commerce ec 2001 pp 18 28 our result also implies np hardness of approximating maximum submodular welfare with demand oracle to a factor better than 15 16 improving upon the best known hardness of 275 276 u feige and j vondrak proceedings of the 47th ieee symposium on foundations of computer science focs 2006 pp 667 676 iii our hardness techniques can be modified to prove that it is np hard to approximate the generalized assignment problem gap to any factor better than 10 11 this improves upon the 422 423 hardness of c chekuri and s khanna proceedings of the 11th annual acm siam symposium on discrete algorithms soda 2000 pp 213 222 m chlebik and j chlebikova proceedings of the 8th scandinavian workshop on algorithm theory swat 2002 pp 170 179 we use iterative rounding on a natural lp relaxation of the mba problem to obtain the 3 4 approximation we also give a 3 4 epsilon factor algorithm based on the primal dual schema which runs in tilde o nm time for any constant epsilon 0

splitting multidimensional necklaces the well known splitting necklace theorem of alon n alon splitting necklaces adv math 63 1987 247 253 says that each necklace with kai beads of color i 1 n can be fairly divided between k thieves by at most n k 1 cuts alon deduced this result from the fact that such a division is possible also in the case of a continuous necklace 0 1 where beads of given color are interpreted as measurable sets ai 0 1 or more generally as continuous measures i we demonstrate that alon s result is a special case of a multidimensional consensus division theorem about n continuous probability measures 1 n on a d cube 0 1 d the dissection is performed by m1 md n k 1 hyperplanes parallel to the sides of 0 1 d dividing the cube into m1md elementary cuboids parallelepipeds where the integers mi are prescribed in advance

torsion cohomology classes and algebraic cycles on complex projective manifolds abstract atiyah and hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold which are not poincare dual to an algebraic cycle we notice that the order of these classes must be small compared to the dimension of the manifold however building upon a construction of kollar one can provide such examples with arbitrary high prime order the dimension being fixed this method also provides examples of torsion algebraic cycles which are non trivial in the griffiths group and lie in a arbitrary high level of the h saito filtration on chow groups

drinfeld modules and torsion in the chow groups of certain threefolds let e b be an elliptic surface defined over the algebraic closure of a finite field of characteristic greater than 5 let w be a resolution of singularities of e b e we show that the l adic abel jacobi map from the l power torsion in the second chow group of w to h 3 w zl 2 ql zl is an isomorphism for almost all primes l a main tool in the proof is the assertion that certain cm cycles in fibres of w b are torsion which is proven using results from the theory of drinfeld modular curves

topological analysis of variance and the maxillary complex it is common to reduce the dimensionality of data before applying classical multivariate analysis techniques in statistics persistent homology a recent development in computational topology has been shown to be useful for analyzing high dimensional nonlinear data in this article we connect computational topology with the traditional analysis of variance and demonstrate the value of combining these approaches on a three dimensional orthodontic landmark dataset derived from the maxillary complex indeed combining appropriate techniques of both persistent homology and analysis of variance results in a better understanding of the data s nonlinear features over and above what could have been achieved by classical means supplementary material for this article is available online

the chow ring of the classifying space bso 2n c we compute the chow ring of the classifying space bso 2n c in the sense of totaro using the fibration gl 2n so 2n bso 2n bgl 2n and a computation of the chow ring of gl 2n so 2n in a previous paper we find this chow ring is generated by chern classes and a characteristic class defined by edidin and graham which maps to 2n 1 times the euler class under the usual class map from the chow ring to ordinary cohomology moreover we show this class represents 1 2n 1 n 1 times the nth chern class of the representation of so 2n whose highest weight vector is twice that of the half spin representation

local linear regression on manifolds and its geometric interpretation high dimensional data analysis has been an active area and the main focus areas have been variable selection and dimension reduction in practice it occurs often that the variables are located on an unknown lower dimensional nonlinear manifold under this manifold assumption one purpose of this article is regression and gradient estimation on the manifold and another is developing a new tool for manifold learning as regards the first aim we suggest directly reducing the dimensionality to the intrinsic dimension d of the manifold and performing the popular local linear regression llr on a tangent plane estimate an immediate consequence is a dramatic reduction in the computational time when the ambient space dimension p d we provide rigorous theoretical justification of the convergence of the proposed regression and gradient estimators by carefully analyzing the curvature boundary and nonuniform sampling effects we propose a bandwidth selector that can handle heteroscedastic errors with re

statistical topological data analysis using persistence landscapes we define a new topological summary for data that we call the persistence landscape since this summary lies in a vector space it is easy to combine with tools from statistics and machine learning in contrast to the standard topological summaries viewed as a random variable with values in a banach space this summary obeys a strong law of large numbers and a central limit theorem we show how a number of standard statistical tests can be used for statistical inference using this summary we also prove that this summary is stable and that it can be used to provide lower bounds for the bottleneck and wasserstein distances

adwords and generalized online matching how does a search engine company decide what ads to display with each query so as to maximize its revenueq this turns out to be a generalization of the online bipartite matching problem we introduce the notion of a trade off revealing lp and use it to derive an optimal algorithm achieving a competitive ratio of 1 1 e for this problem

clique topology reveals intrinsic geometric structure in neural correlations detecting meaningful structure in neural activity and connectivity data is challenging in the presence of hidden nonlinearities where traditional eigenvalue based methods may be misleading we introduce a novel approach to matrix analysis called clique topology that extracts features of the data invariant under nonlinear monotone transformations these features can be used to detect both random and geometric structure and depend only on the relative ordering of matrix entries we then analyzed the activity of pyramidal neurons in rat hippocampus recorded while the animal was exploring a 2d environment and confirmed that our method is able to detect geometric organization using only the intrinsic pattern of neural correlations remarkably we found similar results during nonspatial behaviors such as wheel running and rapid eye movement rem sleep this suggests that the geometric structure of correlations is shaped by the underlying hippocampal circuits and is not merely a consequence of position coding we propose that clique topology is a powerful new tool for matrix analysis in biological settings where the relationship of observed quantities to more meaningful variables is often nonlinear and unknown

truthful mechanism design via correlated tree rounding a powerful algorithmic technique for truthful mechanism design is the maximal in distributional range midr paradigm unfortunately many such algorithms use heavy algorithmic machinery e g the ellipsoid method and approximate solution of convex programs in this paper we present a correlated rounding technique for designing mechanisms that are truthful in expectation it is elementary and can be implemented quickly the main property we rely on is that the domain offers fractional optimum solutions with a tree structure in auctions based on the generalized assignment problem each bidder has a publicly known knapsack constraint that captures the subsets of items that are of value to him he has a private valuation for each item and strives to maximize the value of assigned items minus payment for this domain we design a truthful 2 approximate midr mechanism for social welfare maximization it avoids using the ellipsoid method or convex programming in contrast to some previous work our mechanism achieves exact truthfulness in restricted related scheduling with selfish machines each job comes with a public weight and it must be assigned to a machine from a public job specific subset each machine has a private speed and strives to maximize payments minus workload of jobs assigned to it here we design a mechanism for makespan minimization this is a single parameter domain but the approximation status of the optimization problem is similar to unrelated machine scheduling the best known algorithm obtains a non truthful 2 approximation for unrelated machines and there is 1 5 hardness our mechanism matches this bound with a truthful 2 approximation

some remarks concerning mod nk theory the spectral sequence predicted by a beilinson relating motivic cohomology to algebraic k theory has been established for smooth quasi projective varieties over a field cf fs l1 among other properties verified this spectral sequence has the expected multiplicative behavior involving cup product in motivic cohomology and product in algebraic k theory and a good multiplicative mod n reduction relating mod n motivic cohomology to mod n algebraic k theory the purpose of this short paper is to draw two simple conclusions concerning mod n algebraick theory of smooth quasi projective varieties over an algebraically closed field by comparing this spectral sequence to its localization which we show converges to mod n etale k theory in theorem 2 1 we verify that the map from mod n algebraic k theory to mod n etale k theory of a smooth quasi projective variety x over an algebraically closed field in which n is invertible is surjective in degrees greater than or equal to twice the dimension of x this refines the surjectivity result of dfst and extends an argument by a suslin in s1 then in example 3 4 we show that certain projective smooth 3 folds x constructed by s bloch and h esnault have non zero elements in k0 x z n which are killed by multiplication by a sufficiently high power of the bott element k2 x z n although the existence of such examples is perhaps not surprising there have been attempts to prove that multiplication by is always injective the authors gratefully acknowledge useful conversations with h esnault and a suslin

transfers of chern classes in bp cohomology and chow rings the bp module structure of bp bg for extraspecial 2 groups is studied using transfer and chern classes these give rise to p torsion elements in the kernel of the cycle map from the chow ring to ordinary cohomology first obtained by totaro

prime discriminant simplicial complex the structure representation of data distribution plays an important role in understanding the underlying mechanism of generating data in this paper we propose the prime discriminant simplicial complex pdsc by utilizing persistent homology to capture such structures assuming that each class is represented with a prime simplicial complex we classify unlabeled samples based on the nearest projection distances from the samples to the simplicial complexes we also extend the extrapolation ability of these complexes with a projection constraint term experiments in simulated and practical datasets indicate that compared with several published algorithms the proposed pdsc approaches achieve promising performance without losing structure representation

spectra of linear relations _ series and cohomology of eilenberg mac lane spaces on brown peterson cohomology groups of a space we introduce a natural inherent topology bp topology which is always complete hausdorff for any space we then construct a spectra map which calculates infinite bp linear sums convergent with respect to the bp topology and a spectrum which describes infinite sum bp linear relations in bp cohomology the mod p cohomology of this spectrum is a cyclic module over the steenrod algebra with relations generated by products of exactly two milnor primitives we show a close relationship between bp linear relations in bp cohomology and the action of the milnor primitives on mod p cohomology we prove main relations in the bp cohomology of eilenberg mac lane spaces these are infinite sum bp linear relations convergent with respect to the bp topology using bp fundamental classes we define v series which are v analogues of the p series finally we show that the above main relations come from the vn series

shapley homology topological analysis of sample influence for neural networks data samples collected for training machine learning models are typically assumed to be independent and identically distributed i i d recent research has demonstrated that this assumption can be problematic as it simplifies the manifold of structured data this has motivated different research areas such as data poisoning model improvement and explanation of machine learning models in this work we study the influence of a sample on determining the intrinsic topological features of its underlying manifold we propose the shapley homology framework which provides a quantitative metric for the influence of a sample of the homology of a simplicial complex our proposed framework consists of two main parts homology analysis where we compute the betti number of the target topological space and shapley value calculation where we decompose the topological features of a complex built from data points to individual points by interpreting the influence as a probability measure we further define an entropy that reflects the complexity of the data manifold furthermore we provide a preliminary discussion of the connection of the shapley homology to the vapnik chervonenkis dimension empirical studies show that when the zero dimensional shapley homology is used on neighboring graphs samples with higher influence scores have a greater impact on the accuracy of neural networks that determine graph connectivity and on several regular grammars whose higher entropy values imply greater difficulty in being learned

information topological characterization of periodically correlated processes by dilation operators giving process information through spectral considerations has been tackled for decades we propose in this paper a new way of dealing with such an objective by giving hidden information topology of the spectral measure of non stationary and periodically correlated processes we used first the kolmogorov decomposition which is a natural extension of the naimark operator theory to obtain a sequence of rotation matrices called the dilation matrices these matrices carry all the spectral information of the process and belong to so n or su n with respect to respectively the real or complex nature of the periodically correlated processes studied in order to give a topological interpretation of the positioning of these matrices on the space of rotation matrices we have applied a persistent homology technique and next exposed fundamental attributes we showed that different types of periodically correlated processes are endowed with a point cloud structure that can be easily discriminated by topological and information features

cohomologie non ramifiee et conjecture de hodge entiere en nous appuyant sur la conjecture de bloch kato en k theorie de milnor nous etablissons un lien general entre le defaut de la conjecture de hodge entiere pour la cohomologie de degre 4 et le troisieme groupe de cohomologie non ramifiee a coefficients q z ceci permet de montrer que sur un solide en anglais threefold uniregle le troisieme groupe de cohomologie non ramifiee a coefficients q z s annule ce que la k theorie algebrique ne permet d obtenir que dans certains cas ceci permet a l inverse de deduire d exemples ayant leur source en k theorie que la conjecture de hodge entiere pour la cohomologie de degre 4 peut etre en defaut pour les varietes rationnellement connexes pour certaines familles a un parametre de surfaces on etablit un lien entre la conjecture de hodge entiere et l indice de la fibre generique

high dimensional statistical measure for region of interest tracking this paper deals with region of interest roi tracking in video sequences the goal is to determine in successive frames the region which best matches in terms of a similarity measure a roi defined in a reference frame some tracking methods define similarity measures which efficiently combine several visual features into a probability density function pdf representation thus building a discriminative model of the roi this approach implies dealing with pdfs with domains of definition of high dimension to overcome this obstacle a standard solution is to assume independence between the different features in order to bring out low dimension marginal laws and or to make some parametric assumptions on the pdfs at the cost of generality we discard these assumptions by proposing to compute the kullback leibler divergence between high dimensional pdfs using the k th nearest neighbor framework in consequence the divergence is expressed directly from the samples i e without explicit estimation of the underlying pdfs as an application we defined 5 7 and 13 dimensional feature vectors containing color information including pixel based gradient based and patch based and spatial layout the proposed procedure performs tracking allowing for translation and scaling of the roi experiments show its efficiency on a movie excerpt and standard test sequences selected for the specific conditions they exhibit partial occlusions variations of luminance noise and complex motion

examples for the mod p motivic cohomology of classifying spaces let bg be the classifying space of a compact lie group g some examples of computations of the motivic cohomology h eg z p are given by comparing with h bg z p ch eg and bp bg

equivariant cobordism for torus actions abstract we study the equivariant cobordism groups for the action of a split torus t on varieties over a field k of characteristic zero we show that for t acting on a variety x there is an isomorphism t x b t l x as applications we show that for a connected linear algebraic group g acting on a k variety x the forgetful map g x x is surjective with rational coefficients as a consequence we describe the rational algebraic cobordism ring of algebraic groups and flag varieties we prove a structure theorem for the equivariant cobordism of smooth projective varieties with torus action using this we prove various localization theorems and a form of bott residue formula for such varieties as an application we show that the equivariant cobordism of a smooth variety x with torus action is generated by the invariant cobordism cycles in x as b t module

torsion algebraic cycles and etale cobordism we prove that the classical integral cycle class map from algebraic cycles to etale cohomology factors through a quotient of l adic etale cobordism over an algebraically closed field of positive characteristic this shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of atiyah hirzebruch and totaro also work in positive characteristic

an adaptable k nearest neighbors algorithm for mmse image interpolation we propose an image interpolation algorithm that is nonparametric and learning based primarily using an adaptive k nearest neighbor algorithm with global considerations through markov random fields the empirical nature of the proposed algorithm ensures image results that are data driven and hence reflect ldquoreal worldrdquo images well given enough training data the proposed algorithm operates on a local window using a dynamic k nearest neighbor algorithm where k differs from pixel to pixel small for test points with highly relevant neighbors and large otherwise based on the neighbors that the adaptable k provides and their corresponding relevance measures a weighted minimum mean squared error solution determines implicitly defined filters specific to low resolution image content without yielding to the limitations of insufficient training additionally global optimization via single pass markov approximations similar to cited nearest neighbor algorithms provides additional weighting for filter generation the approach is justified in using a sufficient quantity of training per test point and takes advantage of image properties for in depth analysis we compare to existing methods and draw parallels between intuitive concepts including classification and ideas introduced by other nearest neighbor algorithms by explaining manifolds in low and high dimensions

the nonlinear statistics of high contrast patches in natural images recently there has been a great deal of interest in modeling the non gaussian structures of natural images however despite the many advances in the direction of sparse coding and multi resolution analysis the full probability distribution of pixels values in a neighborhood has not yet been described in this study we explore the space of data points representing the values of 3 3 high contrast patches from optical and 3d range images we find that the distribution of data is extremely sparse with the majority of the data points concentrated in clusters and non linear low dimensional manifolds furthermore a detailed study of probability densities allows us to systematically distinguish between images of different modalities optical versus range which otherwise display similar marginal distributions our work indicates the importance of studying the full probability distribution of natural images not just marginals and the need to understand the intrinsic dimensionality and nature of the data we believe that object like structures in the world and the sensor properties of the probing device generate observations that are concentrated along predictable shapes in state space our study of natural image statistics accounts for local geometries such as edges in natural scenes but does not impose such strong assumptions on the data as independent components or sparse coding by linear change of bases

barcodes the persistent topology of data this article surveys recent work of carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high dimensional data the primary mathematical tool considered is a homology theory for point cloud data sets persistent homology and a novel representation of this algebraic characterization barcodes we sketch an application of these techniques to the classification of natural images 1 the shape of data when a topologist is asked how do you visualize a four dimensional object the appropriate response is a socratic rejoinder how do you visualize a threedimensional object we do not see in three spatial dimensions directly but rather via sequences of planar projections integrated in a manner that is sensed if not comprehended we spend a significant portion of our first year of life learning how to infer three dimensional spatial data from paired planar projections years of practice have tuned a remarkable ability to extract global structure from representations in a strictly lower dimension the inference of global structure occurs on much finer scales as well with regards to converting discrete data into continuous images dot matrix printers scrolling led tickers televisions and computer displays all communicate images via arrays of discrete points which are integrated into coherent global objects this also is a skill we have practiced from childhood no adult does a dot to dot puzzle with anything approaching anticipation 1 1 topological data analysis problems of data analysis share many features with these two fundamental integration tasks 1 how does one infer high dimensional structure from low dimensional representations and 2 how does one assemble discrete points into global structure the principal themes of this survey of the work of carlsson de silva edelsbrunner harer zomorodian and others are the following 1 it is beneficial to replace a set of data points with a family of simplicial complexes indexed by a proximity parameter this converts the data set into global topological objects 2 it is beneficial to view these topological complexes through the lens of algebraic topology specifically via a novel theory of persistent homology adapted to parameterized families 3 it is beneficial to encode the persistent homology of a data set in the form of a parameterized version of a betti number a barcode the author gratefully acknowledges the support of darpa hr0011 07 1 0002 the work reviewed in this article is funded by the darpa program tda topological data analysis

k theoretic chern class formulas for vexillary degeneracy loci abstract using raising operators and geometric arguments we establish formulas for the k theory classes of degeneracy loci in classical types we also find new determinantal and pfaffian expressions for classical cases considered by giambelli the loci where a generic matrix drops rank and where a generic symmetric or skew symmetric matrix drops rank in an appendix we construct a k theoretic euler class for even rank vector bundles with quadratic form refining the chow theoretic class introduced by edidin and graham we also establish a relation between top chern classes of maximal isotropic subbundles which is used in proving the type d degeneracy locus formulas

on the cohomology of elementary abelian groups the structure of the bp cohomology of elementary abelian p groups is studied obtaining a presentation expressed in terms of bp cohomology and mod p singular cohomology using the milnor derivations the arguments are based on a result on multi koszul complexes which is related to margolis s criterion for freeness of a graded module over an exterior algebra

persistence based clustering in riemannian manifolds we present a clustering scheme that combines a mode seeking phase with a cluster merging phase in the corresponding density map while mode detection is done by a standard graph based hill climbing scheme the novelty of our approach resides in its use of topological persistence to guide the merging of clusters our algorithm provides additional feedback in the form of a set of points in the plane called a persistence diagram pd which provably reflects the prominences of the modes of the density in practice this feedback enables the user to choose relevant parameter values so that under mild sampling conditions the algorithm will output the correct number of clusters a notion that can be made formally sound within persistence theory in addition the output clusters have the property that their spatial locations are bound to the ones of the basins of attraction of the peaks of the density the algorithm only requires rough estimates of the density at the data points and knowledge of approximate pairwise distances between them it is therefore applicable in any metric space meanwhile its complexity remains practical although the size of the input distance matrix may be up to quadratic in the number of data points a careful implementation only uses a linear amount of memory and takes barely more time to run than to read through the input

vector diffusion maps and the connection laplacian we introduce vector diffusion maps vdm a new mathematical framework for organizing and analyzing massive high dimensional data sets images and shapes vdmis a mathematical and algorithmic generalization of diffusion maps and other nonlinear dimensionality reduction methods such as lle isomap and laplacian eigenmaps while existing methods are either directly or indirectly related to the heat kernel for functions over the data vdm is based on the heat kernel for vector fields vdm provides tools for organizing complex data sets embedding them in a low dimensional space and interpolating and regressing vector fields over the data in particular it equips the data with a metric which we refer to as the vector diffusion distance in the manifold learning setup where the data set is distributed on a low dimensional manifold embedded in we prove the relation between vdm and the connection laplacian operator for vector fields over the manifold 2012 wiley periodicals inc

string submodular functions with curvature constraints consider the problem of choosing a string of actions to optimize an objective function that is string submodular it was shown in previous papers that the greedy strategy consisting of a string of actions that only locally maximizes the step wise gain in the objective function achieves at least a 1 e 1 approximation to the optimal strategy this paper improves this approximation by introducing additional constraints on curvature namely total backward curvature total forward curvature and elemental forward curvature we show that if the objective function has total backward curvature sigma then the greedy strategy achieves at least a 1 sigma 1 e sigma approximation of the optimal strategy if the objective function has total forward curvature epsilon then the greedy strategy achieves at least a 1 epsilon approximation of the optimal strategy moreover we consider a generalization of the diminishing return property by defining the elemental forward curvature we also introduce the notion of string matroid and consider the problem of maximizing the objective function subject to a string matroid constraint we investigate two applications of string submodular functions with curvature constraints 1 choosing a string of actions to maximize the expected fraction of accomplished tasks and 2 designing a string of measurement matrices such that the information gain is maximized

an approximation algorithm for the generalized assignment problem the generalized assignment problem can be viewed as the following problem of scheduling parallel machines with costs each job is to be processed by exactly one machine processing jobj on machinei requires timep ij and incurs a cost ofc ij each machinei is available fort i time units and the objective is to minimize the total cost incurred our main result is as follows there is a polynomial time algorithm that given a valuec either proves that no feasible schedule of costc exists or else finds a schedule of cost at mostc where each machinei is used for at most 2t i time units we also extend this result to a variant of the problem where instead of a fixed processing timep ij there is a range of possible processing times for each machine job pair and the cost linearly increases as the processing time decreases we show that these results imply a polynomial time 2 approximation algorithm to minimize a weighted sum of the cost and the makespan i e the maximum job completion time we also consider the objective of minimizing the mean job completion time we show that there is a polynomial time algorithm that given valuesm andt either proves that no schedule of mean job completion timem and makespant exists or else finds a schedule of mean job completion time at mostm and makespan at most 2t

brown peterson and ordinary cohomology theories of classifying spaces for compact lie groups the steenrod algebra structures of h bg z p for compact lie groups are studied using these brown peterson cohomology and morava k theory are computed for many concrete cases all these cases have properties similar as torsion free lie groups or finite groups e g bp odd bg 0

tight approximation algorithms for maximum separable assignment problems a separable assignment problem sap is defined by a set of bins and a set of items to pack in each bin a value fij for assigning item j to bin i and a separate packing constraint for each bin i e for each bin a family of subsets of items that fit in to that bin the goal is to pack items into bins to maximize the aggregate value this class of problems includes the maximum generalized assignment problem gap 1 and a distributed caching problem dcp described in this paper given a approximation algorithm for finding the highest value packing of a single bin we give a polynomial time lp rounding based 1 1 e approximation algorithm a simple polynomial time local search 1 e approximation algorithm for any e 0 therefore for all examples of sap that admit an approximation scheme for the single bin problem we obtain an lp based algorithm with 1 1 e e approximation and a local search algorithm with e approximation guarantee furthermore for cases in which the subproblem admits a fully polynomial approximation scheme such as for gap the lp based algorithm analysis can be strengthened to give a guarantee of 1 1 e the best previously known approximation algorithm for gap is a approximation by shmoys and tardos and chekuri and khanna our lp algorithm is based on rounding a new linear programming relaxation with a provably better integrality gap to complement these results we show that sap and dcp cannot be approximated within a factor better than 1 1 e unless np dtime no log log n even if there exists a polynomial time exact algorithm for the single bin problem we extend the 1 1 e approximation algorithm to a constant factor approximation algorithms for a nonseparable assignment problem with applications in maximizing revenue for budget constrained combinatorial auctions and the adwords assignment problem we generalize the local search algorithm to yield a e approximation algorithm for the maximum k median problem with hard capacities

torsion algebraic cycles and complex cobordism atiyah and hirzebruch gave the first counterexamples to the hodge conjecture with integral coefficients 3 that conjecture predicted that every integral cohomology class of hodge type p p on a smooth projective variety should be the class of an algebraic cycle but atiyah and hirzebruch found additional topological properties which must be satisfied by the integral cohomology class of an algebraic cycle here we provide a more systematic explanation for their results by showing that the classical cycle map from algebraic cycles modulo algebraic equivalence to integral cohomology factors naturally through a topologically defined ring which is richer than integral cohomology the new ring is based on complex cobordism a well developed topological theory which has been used only rarely in algebraic geometry since hirzebruch used it to prove the riemann roch theorem 17 this factorization of the classical cycle map implies the topological restrictions on algebraic cycles found by atiyah and hirzebruch it goes beyond their work by giving a topological method to show that the classical cycle map can be noninjective as well as nonsurjective the kernel of the classical cycle map is called the griffiths group and the topological proof here that the griffiths group can be nonzero is the first proof of this fact which does not use hodge theory the proof here gives nonzero torsion elements in the griffiths group whereas griffiths s hodge theoretic proof gives nontorsion elements 13 this topological argument also gives examples of algebraic cycles in the kernel of various related cycle maps where few or no examples were known before thus answering some questions posed by colliot thelene and schoen 8 p 14 37 p 13 colliot thelene asked in particular whether the map ch2 x n h4 x z n is injective for all smooth complex projective varieties x here ch x is the group of codimension i algebraic cycles modulo rational equivalence the first examples where colliot thelene s map is not injective were found by kollar and van geemen see 4 p 135 very recently bloch and esnault found examples defined over number fields 7 over nonalgebraically closed fields k there are other examples of smooth projective varieties xk with ch2 xk n hg4t xk z n not injective due to colliot thelene and sansuc as reinterpreted by salberger see 9 and 8 remark 7 6 1 and parimala and suresh 31 these elements of ch2 xk n are not shown to remain nonzero in ch2 xc n however here our topological method

a klein bottle based dictionary for texture representation a natural object of study in texture representation and material classification is the probability density function in pixel value space underlying the set of small patches from the given image inspired by the fact that small n times n n n high contrast patches from natural images in gray scale accumulate with high density around a surface fancyscript k subset mathbb r n 2 k r n 2 with the topology of a klein bottle carlsson et al international journal of computer vision 76 1 1 12 2008 we present in this paper a novel framework for the estimation and representation of distributions around fancyscript k k of patches from texture images more specifically we show that most n times n n n patches from a given image can be projected onto fancyscript k k yielding a finite sample s subset fancyscript k s k whose underlying probability density function can be represented in terms of fourier like coefficients which in turn can be estimated from s s we show that image rotation acts as a linear transformation at the level of the estimated coefficients and use this to define a multi scale rotation invariant descriptor we test it by classifying the materials in three popular data sets the curet uiuctex and kth tips texture databases