scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

principal components and orthogonal regression based on robust scales both principal components analysis pca and orthogonal regression deal with finding a p dimensional linear manifold minimizing a scale of the orthogonal distances of the m dimensional data points to the manifold the main conceptual difference is that in pca p is estimated from the data to attain a small proportion of unexplained variability whereas in orthogonal regression p equals m 1 the two main approaches to robust pca are using the eigenvectors of a robust covariance matrix and searching for the projections that maximize or minimize a robust univariate dispersion measure this article is more akin to second approach but rather than finding the components one by one we directly undertake the problem of finding for a given p a p dimensional linear manifold minimizing a robust scale of the orthogonal distances of the data points to the manifold the scale may be either a smooth m scale or a trimmed scale an iterative algorithm is developed that is shown to converge to a local minimum a

critical points and supersymmetric vacua ii asymptotics and extremal metrics motivated by the vacuum selection problem of string m theory we study a new geometric invariant of a positive hermitian line bundle l h m over a compact kahler manifold the expected distribution k crit z of critical points dlog s z h 0 of a gaussian random holomorphic section s h 0 m l with respect to h it is a measure on m whose total mass is the average number n crit h of critical points of a random holomorphic section we are interested in the metric dependence of n crit h especially metrics h which minimize n crit h we concentrate on the asymptotic minimization problem for the sequence of tensor powers l n h n m of the line bundle and their critical point densities k critn z we prove that k critn z has a complete asymptotic expansion in n whose coefficients are curvature invariants of h the first two terms in the expansion of n crit hn are topological invariants of l m the third term is a topological invariant plus a constantm 2 depending only on the dimension m of m times the calabi functional r m 2 dv olh whereis the scalar curvature of the curvature form of h we give an integral formula form 2 and show by a computer assisted calculation thatm 2 0 for m 3 hence that n crit hn is asymptotically minimized by the calabi extremal metric when one exists we conjecture thatm 2 0 in all dimensions i e that the calabi extremal metric is always the asymptotic minimizer

distribution of zeros of random and quantum chaotic sections of positive line bundles we study the limit distribution of zeros of certain sequences of holomorphic sections of high powers m n of a positive holomorphic hermitian line bundle l over a compact complex manifold m our first result concerns random sequences of sections using the natural probability measure on the space of sequences of orthonormal bases s n j of h 0 m l n we show that for almost every sequence s n j the associated sequence of zero currents 1 n z s n j tends to the curvature form of l thus the zeros of a sequence of sections s n h 0 m l n chosen independently and at random become uniformly distributed our second result concerns the zeros of quantum ergodic eigenfunctions where the relevant orthonormal bases s n j of h 0 m l n consist of eigensections of a quantum ergodic map we show that also in this case the zeros become uniformly distributed

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

asymptotic behaviour of s estimates of multivariate location parameters and dispersion matrices it is shown under appropriate conditions that rousseeuw s minimum volume estimator and other s estimators of multivariate location and dispersion parameters are consistent under certain differentiability conditions the estimates are asymptotically normally distributed with a norming factor of n 1 2

on the lower order terms of the asymptotic expansion of tian yau zelditch we compute the first three coefficients of the asymptotic expansion of tian yau zelditch we also prove that in general the fcth coefficient is a polynomial of the curvature and its derivative of weight k

a fast algorithm for the minimum covariance determinant estimator the minimum covariance determinant mcd method of rousseeuw is a highly robust estimator of multivariate location and scatter its objective is to find h observations out of n whose covariance matrix has the lowest determinant until now applications of the mcd were hampered by the computation time of existing algorithms which were limited to a few hundred objects in a few dimensions we discuss two important applications of larger size one about a production process at philips with n 677 objects and p 9 variables and a dataset from astronomy with n 137 256 objects and p 27 variables to deal with such problems we have developed a new algorithm for the mcd called fast mcd the basic ideas are an inequality involving order statistics and determinants and techniques which we call selective iteration and nested extensions for small datasets fast mcd typically finds the exact mcd whereas for larger datasets it gives more accurate results than existing algorithms and is faster by orders

coherence pursuit fast simple and robust principal component analysis this paper presents a remarkably simple yet powerful algorithm termed coherence pursuit cop to robust principal component analysis pca as inliers lie in a low dimensional subspace and are mostly correlated an inlier is likely to have strong mutual coherence with a large number of data points by contrast outliers either do not admit low dimensional structures or form small clusters in either case an outlier is unlikely to bear strong resemblance to a large number of data points given that cop sets an outlier apart from an inlier by comparing their coherence with the rest of the data points the mutual coherences are computed by forming the gram matrix of the normalized data points subsequently the sought subspace is recovered from the span of the subset of the data points that exhibit strong coherence with the rest of the data as cop only involves one simple matrix multiplication it is significantly faster than the state of the art robust pca algorithms we derive analytical performance guarantees for cop under different models for the distributions of inliers and outliers in both noise free and noisy settings cop is the first robust pca algorithm that is simultaneously non iterative provably robust to both unstructured and structured outliers and can tolerate a large number of unstructured outliers

robust recovery of corrupted low rankmatrix by implicit regularizers low rank matrix recovery algorithms aim to recover a corrupted low rank matrix with sparse errors however corrupted errors may not be sparse in real world problems and the relationship between l1 regularizer on noise and robust m estimators is still unknown this paper proposes a general robust framework for low rank matrix recovery via implicit regularizers of robust m estimators which are derived from convex conjugacy and can be used to model arbitrarily corrupted errors based on the additive form of half quadratic optimization proximity operators of implicit regularizers are developed such that both low rank structure and corrupted errors can be alternately recovered in particular the dual relationship between the absolute function in l1 regularizer and huber m estimator is studied which establishes a connection between robust low rank matrix recovery methods and m estimators based robust principal component analysis methods extensive experiments on synthetic and real world data sets corroborate our claims and verify the robustness of the proposed framework

macropca an all in one pca method allowing for missing values as well as cellwise and rowwise outliers multivariate data are typically represented by a rectangular matrix table in which the rows are the objects cases and the columns are the variables measurements when there are many variables

principal component analysis based on robust estimators of the covariance or correlation matrix influence functions and efficiencies a robust principal component analysis can be easily performed by computing the eigenvalues and eigenvectors of a robust estimator of the covariance or correlation matrix in this paper we derive the influence functions and the corresponding asymptotic variances for these robust estimators of eigenvalues and eigenvectors the behaviour of several of these estimators is investigated by a simulation study it turns out that the theoretical results and simulations favour the use of s estimators since they combine a high efficiency with appealing robustness properties 2000 biometrika trust

trimmed comparison of distributions this article introduces an analysis of similarity of distributions based on the l2 wasserstein distance between trimmed distributions our main innovation is the use of the impartial trimming methodology already considered in robust statistics which we adapt to this setup instead of simply removing data at the tails to provide some robustness to the similarity analysis we develop a data driven trimming method aimed at maximizing similarity between distributions dissimilarity is then measured in terms of the distance between the optimally trimmed distributions we provide illustrative examples showing the improvements over previous approaches and give the relevant asymptotic results to justify the use of this methodology in applications

judgments of sound volume effects of signal duration level and interaural characteristics on the perceived extensity of broadband noise while auditory stimuli are often described in terms of their apparent extensity such descriptions have usually not been collected systematically moreover the few deliberate attempts to evaluate image size have rarely gone beyond the classic parameters of stimulus frequency and intensity in the present study a direct magnitude estimation procedure was employed seventeen subjects numerically estimated the apparent size of images produced by broadband noise stimuli under earphone listening conditions signals were presented either dichotically uncorrelated noise diotically correlated noise or monaurally noise led to a single earphone the signals in each of these modes varied in duration 100 300 1000 and 3000 msec and intensity level 60 75 and 90 db a weighted size estimates were plotted as power functions and analyzed with a repeated measures design analysis of variance consistent with previous research the main effects of duration and intensity were both highly significant p 0 001 in addition a highly significant effect for mode of presentation was found p 0 001 across conditions dichotic stimulation produced the largest images and monaural stimulation the smallest about half the size of the diotic images this last result is the first quantification of previous anecdotal observations general implications of these results were discussed

critical points and supersymmetric vacua i supersymmetric vacua universes of string m theory may be identified with certain critical points of a holomorphic section the superpotential of a hermitian holomorphic line bundle over a complex manifold an important physical problem is to determine how many vacua there are and how they are distributed as the superpotential varies over physically relevant ensembles in several papers over the last few years m r douglas and co workers have studied such vacuum statistics problems for a variety of physical models at the physics level of rigor do ad dd the present paper is the first of a series by the present authors giving a rigorous mathematical foundation for the vacuum statistics problem it sets down basic results on the statistics of critical points s 0 of random holomorphic sections of hermitian holomorphic line bundles with respect to a metric connection when the sections are endowed with a gaussian measure the principal results give formulas for the expected density and number of critical points of fixed morse index of gaussian random sections relative to they are particularly concrete for riemann surfaces in our subsequent work the results will be applied to the vacuum statistics problem and to the purely geometric problem of studying metrics which minimize the expected number of critical points

robust line estimation with errors in both variables abstract the estimator holding the central place in the theory of the multivariate errors in the variables ev model results from performing orthogonal regression on variables rescaled according to the covariance matrix of the errors our simulations on the univariate model essentially relegate this estimator to use only in large samples on data with no trace of outlier contamination a modification requiring a robust preliminary slope is proposed that essentially sets out the generalization to ev of the robust w estimator in regression it is demonstrated that the modification is robust to outlier contamination even in small samples given a sufficiently good preliminary estimator candidates for a preliminary slope estimator based on the data are discussed and the performance of one under simulation is examined

robust estimates of location and dispersion for high dimensional datasets the computing times of high breakdown point estimates of multivariate location and scatter increase rapidly with the number of variables which makes them impractical for high dimensional datasets such as those used in data mining we propose an estimator of location and scatter based on a modified version of the gnanadesikan kettenring robust covariance estimate we compare its behavior with that of the stahel donoho sd and rousseeuw and van driessen s fast mcd fmcd estimates in simulations with contaminated multivariate normal data our estimate is almost as good as sd and clearly better than fmcd it is much faster than both especially for large dimension we give examples with real data with dimensions between 5 and 93 in which the proposed estimate is as good as or better than sd and fmcd at detecting outliers and other structures with much shorter computing times

interaural correlation and the binaural summation of loudness the effect of interaural correlation on the loudness for noisebands was measured using a loudness matching task in naive listeners the task involved a sequence of loudness comparisons for which the intensity of one stimulus in a given comparison was varied using a one up one down adaptive rule the task provided an estimate of the level difference in decibels for which two stimulus conditions have equal loudness giving measures of loudness difference in equivalent decibel units dbequiv concurrent adaptive tracks measured loudness differences between 1 0 and 1 and between these binaural stimuli and the monaural case for various noisebands for all noisebands monaural stimuli required approximately 6db higher levels than 1 for equal loudness for most noisebands 1 and 1 were almost equal in loudness with 1 being slightly louder in the majority of measurements while 0 was about 2dbequiv louder than 1 or 1 however noisebands with significant high frequency energy showed smaller differences for 3745 4245hz 0 was only about 0 85dbequiv louder than 1 and for 100 5000hz it was non significantly louder perhaps 0 7dbequiv

universality and scaling of correlations between zeros on complex manifolds we study the limit as n of the correlations between simultaneous zeros of random sections of the powers l n of a positive holomorphic line bundle l over a compact complex manifold m when distances are rescaled so that the average density of zeros is independent of n we show that the limit correlation is independent of the line bundle and depends only on the dimension of m and the codimension of the zero sets we also provide some explicit formulas for pair correlations in particular we prove that hannay s limit pair correlation function for su 2 polynomials holds for all compact riemann surfaces

projection pursuit approach to robust dispersion matrices and principal components primary theory and monte carlo abstract this article proposes and discusses a type of new robust estimators for covariance correlation matrices and principal components via projection pursuit techniques the most attractive advantage of the new procedures is that they are of both rotational equivariance and high breakdown point besides they are qualitatively robust and consistent at elliptic underlying distributions the monte carlo study shows that the best of the new estimators compare favorably with other robust methods they provide as good a performance as m estimators and somewhat better empirical breakdown properties

robust low rank approximation of data matrices with elementwise contamination we propose a robust method to approximate an n p data matrix with one of rank q the method is based on yohai s regression mm estimates it is intended to be resistant against the existence of both atypical rows and of scattered atypical cells and also to be able to cope with missing values we propose an algorithm based on alternating m regressions and a starting estimate based on successive rank one fits which involves o npq operations simulations show that our estimate outperforms competing estimates in terms of both efficiency and resistance three high dimensional real data sets are analyzed the running time of our estimate for large data sets is shown to be less than that of its competitors

robust principal component analysis for power transformed compositional data geochemical surveys collect sediment or rock samples measure the concentration of chemical elements and report these typically either in weight percent or in parts per million ppm there are usually a large number of elements measured and the distributions are often skewed containing many potential outliers we present a new robust principal component analysis pca method for geochemical survey data that involves first transforming the compositional data onto a manifold using a relative power transformation a flexible set of moment assumptions are made which take the special geometry of the manifold into account the kent distribution moment structure arises as a special case when the chosen manifold is the hypersphere we derive simple moment and robust estimators ro of the parameters which are also applicable in high dimensional settings the resulting pca based on these estimators is done in the tangent space and is related to the power transformation method used in correspondence analysis t

critical points and supersymmetric vacua iii string m models a fundamental problem in contemporary string m theory is to count the number of inequivalent vacua satisfying constraints in a string theory model this article contains the first rigorous results on the number and distribution of supersymmetric vacua of type iib string theories compactified on a calabi yau 3 fold x with flux in particular complete proofs of the counting formulas in ashok douglas ad and denef douglas dd1 are given together with van der corput style remainder estimates

generalized dynamic principal components abstractbrillinger defined dynamic principal components dpc for time series based on a reconstruction criterion he gave a very elegant theoretical solution and proposed an estimator which is consistent under stationarity here we propose a new enterally empirical approach to dpc the main differences with the existing methods mainly brillinger procedure are 1 the dpc we propose need not be a linear combination of the observations and 2 it can be based on a variety of loss functions including robust ones unlike brillinger we do not establish any consistency results however contrary to brillinger s which has a very strong stationarity flavor our concept aims at a better adaptation to possible nonstationary features of the series we also present a robust version of our procedure that allows to estimate the dpc when the series have outlier contamination we give iterative algorithms to compute the proposed procedures that can be used with a large number of variables our nonrobust and robust pr

sparse principal component analysis based on least trimmed squares sparse principal component analysis pca is used to obtain stable and interpretable principal components pcs from high dimensional data a robust sparse pca method is proposed to handle potentia

generating partially correlated noise a comparison of methods there are three standard methods for generating two channels of partially correlated noise the two generator method the three generator method and the symmetric generator method these methods allow an experimenter to specify a target cross correlation between the two channels but actual generated noises show statistical variability around the target value numerical experiments were done to compare the variability for those methods as a function of the number of degrees of freedom the results of the experiments quantify the stimulus uncertainty in diverse binaural psychoacoustical experiments incoherence detection perceived auditory source width envelopment noise localization lateralization and the masking level difference the numerical experiments found that when the elemental generators have unequal powers the different methods all have similar variability when the powers are constrained to be equal the symmetric generator method has much smaller variability than the other two

robust sparse principal component analysis a method for principal component analysis is proposed that is sparse and robust at the same time the sparsity delivers principal components that have loadings on a small number of variables making them easier to interpret the robustness makes the analysis resistant to outlying observations the principal components correspond to directions that maximize a robust measure of the variance with an additional penalty term to take sparseness into account we propose an algorithm to compute the sparse and robust principal components the algorithm computes the components sequentially and thus it can handle datasets with more variables than observations the method is applied on several real data examples and diagnostic plots for detecting outliers and for selecting the degree of sparsity are provided a simulation experiment studies the effect on statistical efficiency by requiring both robustness and sparsity supplementary materials are available online on the journal web site

robust estimation of dispersion matrices and principal components abstract this paper uses monte carlo methods to compare the performances of several robust procedures for estimating a correlation matrix and its principal components the estimators are formed either from separate bivariate analyses or by simultaneous manipulation of all variables by using techniques such as multivariate trimming and m estimation the m estimators stand up exceptionally well they and the multivariate trimming procedure are especially effective at estimating the principal components including a near singularity however the m estimators can break down relatively easily when the dimensionality is large and the outliers are asymmetric with missing data the element wise approach becomes more attractive

robust estimation in the errors in variables model summary orthogonal regression analogues of m estimates of regression called here orthogonal regression m estimates are defined these estimates are shown to be consistent at elliptical errors in variables models and robust if the corresponding loss function is bounded the orthogonal regression analogues of regression scale estimates called here orthogonal regression s estimates are considered as well in particular they provide a robust estimate for the scale of the orthogonal residuals a crucial quantity in the computation of orthogonal regression m estimates finally we present an algorithm for computing orthogonal regression s and m estimates and the results of a small monte carlo experiment

s estimators for functional principal component analysis principal component analysis is a widely used technique that provides an optimal lower dimensional approximation to multivariate or functional datasets these approximations can be very useful in identifying potential outliers among high dimensional or functional observations in this article we propose a new class of estimators for principal components based on robust scale estimators for a fixed dimension q we robustly estimate the q dimensional linear space that provides the best prediction for the data in the sense of minimizing the sum of robust scale estimators of the coordinates of the residuals we also study an extension to the infinite dimensional case our method is consistent for elliptical random vectors and is fisher consistent for elliptically distributed random elements on arbitrary hilbert spaces numerical experiments show that our proposal is highly competitive when compared with other methods we illustrate our approach on a real dataset where the robust estimator discovers atypi

a general trimming approach to robust cluster analysis we introduce a new method for performing clustering with the aim of fitting clusters with different scatters and weights it is de signed by allowing to handle a proportionof contaminating data to guarantee the robustness of the method as a characteristic fea ture restrictions on the ratio between the maximum and the mini mum eigenvalues of the groups scatter matrices are introduced this makes the problem to be well defined and guarantees the consistency of the sample solutions to the population ones the method covers a wide range of clustering approaches depend ing on the strength of the chosen restrictions our proposal includes an algorithm for approximately solving the sample problem

robust principal component analysis based on maximum correntropy criterion principal component analysis pca minimizes the mean square error mse and is sensitive to outliers in this paper we present a new rotational invariant pca based on maximum correntropy criterion mcc a half quadratic optimization algorithm is adopted to compute the correntropy objective at each iteration the complex optimization problem is reduced to a quadratic problem that can be efficiently solved by a standard optimization method the proposed method exhibits the following benefits 1 it is robust to outliers through the mechanism of mcc which can be more theoretically solid than a heuristic rule based on mse 2 it requires no assumption about the zero mean of data for processing and can estimate data mean during optimization and 3 its optimal solution consists of principal eigenvectors of a robust covariance matrix corresponding to the largest eigenvalues in addition kernel techniques are further introduced in the proposed method to deal with nonlinearly distributed data numerical results demonstrate that the proposed method can outperform robust rotational invariant pcas based on l1 norm when outliers occur