sufficient dimension reduction via squared loss mutual information estimation the goal of sufficient dimension reduction in supervised learning is to find the low dimensional subspace of input features that contains all of the information about the output values that the input features possess in this letter we propose a novel sufficient dimension reduction method using a squared loss variant of mutual information as a dependency measure we apply a density ratio estimator for approximating squared loss mutual information that is formulated as a minimum contrast estimator on parametric or nonparametric models since cross validation is available for choosing an appropriate model our method does not require any prespecified structure on the underlying distributions we elucidate the asymptotic bias of our estimator on parametric models and the asymptotic convergence rate on nonparametric models the convergence analysis utilizes the uniform tail bound of a u process and the convergence rate is characterized by the bracketing entropy of the model we then develop a natural gradient algorithm on the grassmann manifold for sufficient subspace search the analytic formula of our estimator allows us to compute the gradient efficiently numerical experiments show that the proposed method compares favorably with existing dimension reduction approaches on artificial and benchmark data sets

inseparable local uniformization abstract it is known since the works of zariski in the early 40s that desingularization of varieties along valuations called local uniformization of valuations can be considered as the local part of the desingularization problem it is still an open problem if local uniformization exists in positive characteristic and dimension larger than three in this paper we prove that zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions and therefore any valuation can be uniformized by a purely inseparable alteration

desingularization of quasi excellent schemes in characteristic zero grothendieck proved in ega iv that if any integral scheme of finite type over a locally noetherian scheme x admits a desingularization then x is quasi excellent and conjectured that the converse is probably true we prove this conjecture for noetherian schemes of characteristic zero namely starting with the resolution of singularities for algebraic varieties of characteristic zero we prove the resolution of singularities for noetherian quasi excellent q schemes

dimension reduction for nonelliptically distributed predictors sufficient dimension reduction methods often require stringent conditions on the joint distribution of the predictor or when such conditions are not satisfied rely on marginal transformation or reweighting to fulfill them approximately for example a typical dimension reduction method would require the predictor to have elliptical or even multivariate normal distribution in this paper we reformulate the commonly used dimension reduction methods via the notion of central solution space so as to circumvent the requirements of such strong assumptions while at the same time preserve the desirable properties of the classical methods such as n consistency and asymptotic normality imposing elliptical distributions or even stronger assumptions on predictors is often considered as the necessary tradeoff for overcoming the curse of dimensionality but the development of this paper shows that this need not be the case the new methods will be compared with existing methods by simulation and applied to a data set

continuity of the radius of convergence of differential equations on p adic analytic curves this paper deals with connections on non archimedean especially p adic analytic curves in the sense of berkovich the curves must be compact but the connections are allowed to have a finite number of meromorphic singularities on them for any choice of a semistable formal model of the curve we define a geometric intrinsic notion of normalized radius of convergence of a full set of local solutions as a function on the curve with values in 0 1 for a sufficiently refined choice of the semistable model we prove continuity logarithmic concavity and logarithmic piece wise linearity of that function we introduce and characterize robba connections that is connections whose sheaf of solutions is constant on any open disk contained in the curve precisely as it happens in the classical case

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat

using the bootstrap to select one of a new class of dimension reduction methods dimension reduction in a regression analysis of response y given a p dimensional vector of predictors x reduces the dimension of x by replacing it with a lower dimensional linear combination x of the x s without specifying a parametric model and without loss of information about the conditional distribution of y given x we unify three existing methods sliced inverse regression sir sliced average variance estimate save and principal hessian directions phd into a larger class of methods each method estimates a particular candidate matrix essentially a matrix of parameters we introduce broad classes of dimension reduction candidate matrices and we distinguish estimators of the matrices from the matrices themselves given these classes of methods and several ways to estimate any matrix we now have the problem of selecting a particular matrix and estimation method we propose bootstrap methodology to select among candidate matrices estimators and dimension and in particular we investigate l

nearest neighbor inverse regression sliced inverse regression sir formally introduced by li is a very general procedure for performing dimension reduction in nonparametric regression this paper considers a version of sir in which the slices are determined by nearest neighbors and the response variable takes value possibly in a multidimensional space it is shown under general conditions that the effective dimension reduction space can be estimated with rate n 1 2 where n is the sample size

multiple loci mapping via model free variable selection summary despite recent flourish of proposals o nv ariable selection genome wide multiple loci mapping remains to be challenging the majority of existing variable selection methods impose a model and often the homoscedastic linear model prior to selection however the true association between the phenotypical trait and the genetic markers is rarely known ap riori and the presence of epistatic interactions makes the association more complex than al inear relation model free variable selection o ers a useful alternative in th is context but the fact that the number of markers p often far exceeds the number of experimental units n renders all the existing model free solutions that require n p inapplicable in this article we examine a number of model free variable selection methods for small n large p regressions in the context of genome wide multiple loci mapping we propose and advocate a multivariate group wise adaptive penalization solution which requires no model prespecification and thus works for complex trait marker association and handles one variable at a time so that works for n p e ctiveness of the new method is demonstrated through bo th intensive simulations and a comprehensive real data analysis across 6100 gene expression traits

dimension reduction and semiparametric estimation of survival models in this paper we propose a new dimension reduction method by introducing a nominal regression model with the hazard function as the conditional mean which naturally retrieves information from complete data and censored data as well moreover without requiring the linearity condition the new method can estimate the entire central subspace consistently and exhaustively the method also provides an alternative approach for the analysis of censored data assuming neither the link function nor the distribution hence it exhibits superior robustness properties numerical studies show that the method can indeed be readily used to efficiently estimate survival models explore the data structures and identify important variables

estimating smooth monotone functions many situations call for a smooth strictly monotone function f of arbitrary flexibility the family of functions defined by the differential equation d 2 f w df where w is an unconstrained coefficient function comprises the strictly monotone twice differentiable functions the solution to this equation is f c 0 c 1 d 1 exp d 1 w where c 0 and c 1 are arbitrary constants and d 1 is the partial integration operator a basis for expanding w is suggested that permits explicit integration in the expression of f in fitting data it is also useful to regularize f by penalizing the integral of w 2 since this is a measure of the relative curvature in f applications are discussed to monotone nonparametric regression to the transformation of the dependent variable in non linear regression and to density estimation

a partial carry save on the fly correction multispeculative multiplier functional units that are designed to receive inputs and produce outputs using a non redundant format typically exhibit an inferior performance in order to overcome this limitation the carry save and partial carry save formats have been proposed both approaches are very suitable when implementing addition trees nevertheless if there are multiplications in the datapath the inputs to the multiplier must be reduced to a non redundant form to avoid applying the distributive property in this paper we present a multiplier able to receive two numbers in partial carry save format and produce a result in partial carry save format as well this is done by modifying the booth encoder and leveraging the generate and propagate group signals that are available because of the partial carry save format hence this can allow to fully implement datapaths without additional penalty cycles due to reductions to non redundant forms experiments show that our proposed multiplier has 15 percent shorter delay with respect to a conventional booth radix 4 multiplier moreover when combining it with partial carry save adders it is possible to reduce 36 percent execution time on average for several benchmarks achieving a 32 7 percent reduction in the energy delay product at the same time

reweighting to achieve elliptically contoured covariates in regression abstract we investigate a method of constructing weights to induce elliptically contoured covariates in regression analyses much recent work in regression has identified various data analytic and model robustness advantages associated with such covariates in particular new estimation methods like sir sirii save and phd have been built around the assumption of elliptically contoured covariates finite samples of regression covariates may deviate from this ideal in practice and the method developed here termed voronoi weighting can be used to induce elliptical symmetry in such samples in a number of examples we show that reweighting cases by the voronoi method can substantially enhance various procedures for covariates that deviate from elliptical symmetry we show that voronoi weighting in conjunction with some trimming via the minimum volume ellipsoid method can be effective

likelihood based sufficient dimension reduction we obtain the maximum likelihood estimator of the central subspace under conditional normality of the predictors given the response analytically and in simulations we found that our new estimator can preform much better than sliced inverse regression sliced average variance estimation and directional regression and that it seems quite robust to deviations from normality

carry save multiplication schemes without final addition carry save multipliers require an adder at the last step to convert the carry sum representation of the most significant half of the result into a non redundant form this paper presents n spl times n multiplication schemes where this conversion is performed with a circuit operating in parallel with the carry save array the most relevant feature of the proposed multipliers is that the full 2n bit result is produced unlike similar multiplication schemes presented in the literature

precision aware self quantizing hardware architectures for the discrete wavelet transform this paper presents designs for both bit parallel bp and digit serial ds precision optimized implementations of the discrete wavelet transform dwt with specific consideration given to the impact of depth the number of levels of dwt on the overall computational accuracy these methods thus allow customizing the precision of a multilevel dwt to a given error tolerance requirement and ensuring an energy minimal implementation which increases the applicability of dwt based algorithms such as jpeg 2000 to energy constrained platforms and environments additionally quantization of dwt coefficients to a specific target step size is performed as an inherent part of the dwt computation thereby eliminating the need to have a separate downstream quantization step in applications such as jpeg 2000 experimental measurements of design performance in terms of area speed and power for 90 nm complementary metal oxide semiconductor implementation are presented results indicate that while bp designs exhibit inherent speed advantages ds designs require significantly fewer hardware resources with increasing precision and dwt level a four level dwt with medium precision for example while the bp design is four times faster than the digital serial design occupies twice the area in addition to the bp and ds designs a novel flexible dwt processor is presented which supports run time configurable dwt parameters

algorithm and architecture for logarithm exponential and powering computation an architecture for the computation of logarithm exponential and powering operations is presented in this paper based on a high radix composite algorithm for the computation of the powering function x sup y the algorithm consists of a sequence of overlapped operations 1 digit recurrence logarithm 2 left to right carry free lrcf multiplication and 3 online exponential a redundant number system is used and the selection in 1 and 3 is done by rounding except from the first iteration when selection by table look up is necessary to guarantee the convergence of the recurrences a sequential implementation of the algorithm with a control unit which allows the independent computation of logarithm and exponential is proposed and the execution times and hardware requirements are estimated for single and double precision floating point computations these estimates are obtained for radices from r 8 to r 1 024 according to an approximate model for the delay and area of the main logic blocks and help determining the radix values which lead to the most efficient implementations r 32 and r 128

exact mean integrated squared error an exact and easily computable expression for the mean integrated squared error mise for the kernel estimator of a general normal mixture density is given for gaussian kernels of arbitrary order this provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis the family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density shapes a number of applications of this method giving important new insights into kernel density estimation are presented among these is the discovery that the usual asymptotic approximations to the mise can be quite inaccurate especially when the underlying density contains substantial fine structure and also strong evidence that the practical importance of higher order kernels is surprisingly small for moderate sample sizes 1 introduction substantial research has been devoted to kernel density estimation this is because it provides a simple yet appealing context in which to study problems and issues that arise in all types of nonparametric curve estimation this includes regression spectral density and hazard estimation and also a variety of other estimators including histograms splines and orthogonal series three important and useful tools for understanding the behavior of nonparametric curve estimators are asymptotic analysis simulation and numerical calculation of error criteria each of these methods provides many useful insights into the complicated structure present in the study of curve estimation however each has its limitations as well rhe strength of asymptotic analysis is that it frequently allows simultaneous study of many different specific examples through general results applying to entire classes of settings the weakness of asymptotics is that they only describe behavior in the limit this is still very useful in many situations because the asymptotics describe the actual situation quite well however it is less useful when the asymptotics have not yet kicked in that is in studying situations where the asymptotically dominant effect has not taken over yet perhaps the biggest drawback to asymptotics is that it is very difficult to determine in a given situation which of these possibilities is occurring

over redundant digit sets and the design of digit by digit division units over redundant digit sets are defined as those ranging from spl minus s to s with s spl ges b b being the radix this paper presents new techniques for the direct computation of division that use an over redundant digit set for representing the quotient instead of simply redundant ones used previously in particular general criteria for synthesizing the digit selection rules and remainder updating are given for any radix and index of redundancy a methodology combining the use of over redundant digit sets with the prescaling of the divisor is also studied in order to achieve radix b division units with trivial digit selection functions it is also shown for the specific case of radix 4 that using a prescaling slightly wider than in a radix 4 unit by m d ercegovac and t lang 1990 possible to avoid the digit selection table the paper also presents a modified algorithm for on the fly conversion of the result into the irredundant form the proposed methodology can be considered as an alternative to existing division techniques

asymptotics for kernel estimate of sliced inverse regression to explore nonlinear structures hidden in high dimensional data and to estimate the effective dimension reduction directions in multivariate nonparametric regression li and duan proposed the sliced inverse regression sir method which is simple to use in this paper the asymptotic properties of the kernel estimate of sliced inverse regression are investigated it turns out that regardless of the kernel function the asymptotic distribution remains the same for a wide range of smoothing parameters

dimension reduction for the conditional mean in regressions with categorical predictors consider the regression of a response y on a vector of quantitative predictors x and a categorical predictor w in this article we describe a first method for reducing the dimension of x without loss of information on the conditional mean mathrm e y x w and without requiring a prespecified parametric model the method which allows for but does not require parametric versions of the subpopulation mean functions mathrm e y x w w includes a procedure for inference about the dimension of x after reduction this work integrates previous studies on dimension reduction for the conditional mean mathrm e y x in the absence of categorical predictors and dimension reduction for the full conditional distribution of y x w the methodology we describe may be particularly useful for constructing low dimensional summary plots to aid in model building at the outset of an analysis our proposals provide an often parsimonious alternative to the standard technique of modeling with interaction terms to adapt a mean function for different subpopulations determined by the levels of w examples illustrating this and other aspects of the development are presented

identifying regression outliers and mixtures graphically regressions in practice can include outliers and other unknown subpopulation structures for example mixtures of regressions occur if there is an omitted categorical predictor like gender or location and different regressions occur within each category the theory of regression graphics based on central subspaces can be used to construct graphical solutions to long standing problems of this type it is argued that in practice the central subspace automatically expands to incorporate outliers and regression mixtures thus methods of estimating the central subspace can be used to identify these phenomena without specifying a model examples illustrating the power of the theory are presented

computational experience with confidence regions and confidence intervals for nonlinear least squares we present the results of a monte carlo study of the leading methods for constructing approximate confidence regions and confidence intervals for parameters estimated by nonlinear least squares we examine three variants of the linearization method the likelihood method and the lack of fit method the linearization method is computationally inexpensive produces easily understandable results and is widely used in practice the likelihood and lack of fit methods are much more expensive and more difficult to report in our tests both the likelihood and lack of fit methods perform very reliably all three variants of the linearization method however often grossly underestimate confidence regions and sometimes significantly underestimate confidence intervals the linearization method variant based solely on the jacobian matrix appears preferable to the two variants that use the full hessian matrix because it is less expensive more numerically stable and at least as accurate the bates and watts curvat

nonlinear confounding in high dimensional regression it is not uncommon to find nonlinear patterns in the scatterplots of regressor variables but how such findings affect standard regression analysis remains largely unexplored this article offers a theory on nonlinear confounding a term for describing the situation where a certain nonlinear relationship in regressors leads to difficulties in modeling and related analysis of the data the theory begins with a measure of nonlinearity between two regressor variables it is then used to assess nonlinearity between any two projections from the high dimensional regressor and a method of finding most nonlinear projections is given nonlinear confounding is addressed by taking a fresh new look at fundamental issues such as the validity of prediction and inference diagnostics regression surface approximation model uncertainty and fisher information loss

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

misspecified nonconvex statistical optimization for sparse phase retrieval existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying true statistical models to address this issue we take a first step towards taming model misspecification by studying the high dimensional sparse phase retrieval problem with misspecified link functions in particular we propose a simple variant of the thresholded wirtinger flow algorithm that given a proper initialization linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions we further provide extensive numerical experiments to support our theoretical findings

sufficient dimension reduction based on an ensemble of minimum average variance estimators we introduce a class of dimension reduction estimators based on an ensemble of the minimum average variance estimates of functions that characterize the central subspace such as the characteristic functions the box cox transformations and wavelet basis the ensemble estimators exhaustively estimate the central subspace without imposing restrictive conditions on the predictors and have the same convergence rate as the minimum average variance estimates they are flexible and easy to implement and allow repeated use of the available sample which enhances accuracy they are applicable to both univariate and multivariate responses in a unified form we establish the consistency and convergence rate of these estimators and the consistency of a cross validation criterion for order determination we compare the ensemble estimators with other estimators in a wide variety of models and establish their competent performance

on the fly rounding computing arithmetic in implementations of operations based on digit recurrence algorithms such as division left to right multiplication and square root the result is obtained in digit serial form from most significant digit to least significant to reduce the complexity of the result digit selection and allow the use of redundant addition the result digit has values from a signed digit set as a consequence the result has to be converted to conventional representation which can be done on the fly as the digits are produced without the use of a carry propagate adder the authors describe three ways to modify this conversion process so that the result is rounded the resulting operation is fast because no carry propagate addition is needed the schemes described apply also to online arithmetic operations

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

fast multiplication without carry propagate addition conventional schemes for fast multiplication accumulate the partial products in redundant form carry save or signed digit and convert the result to conventional representation in the last step this step requires a carry propagate adder which is comparatively slow and occupies a significant area of the chip in a vlsi implementation a report is presented on a multiplication scheme left to right carry free lrcf that does not require this carry propagate step the lrcf scheme performs the multiplication most significant bit first and produces a conventional sign and magnitude product most significant n bits by means of an on the fly conversion the resulting implementation is fast and regular and is very well suited for vlsi the lrcf scheme for general radix r and a radix 4 signed digit implementation are presented

determining the dimensionality in sliced inverse regression abstract a general regression problem is one in which a response variable can be expressed as some function of one or more different linear combinations of a set of explanatory variables as well as a random error term sliced inverse regression is a method for determining these linear combinations in this article we address the problem of determining how many linear combinations are involved procedures based on conditional means and conditional covariance matrices as well as a procedure combining the two approaches are considered in each case we develop a test that has an asymptotic chi squared distribution when the vector of explanatory variables is sampled from an elliptically symmetric distribution

on sliced inverse regression with high dimensional covariates sliced inverse regression is a promising method for the estimation of the central dimension reduction subspace cdr space in semiparametric regression models it is particularly useful in tackling cases with high dimensional covariates in this article we study the asymptotic behavior of the estimate of the cdr space with high dimensional covariates that is when the dimension of the covariates goes to infinity as the sample size goes to infinity strong and weak convergence are obtained we also suggest an estimation procedure of the bayes information criterion type to ascertain the dimension of the cdr space and derive the consistency a simulation study is conducted

a note on sliced inverse regression with regularizations in li and yin 2008 biometrics 64 124 131 a ridge sir estimator is introduced as the solution of a minimization problem and computed thanks to an alternating least squares algorithm this methodology reveals good performance in practice in this note we focus on the theoretical properties of the estimator it is shown that the minimization problem is degenerated in the sense that only two situations can occur either the ridge sir estimator does not exist or it is zero

design of testable structures defined by simple loops a methodology is given for generating combinational structures from high level descriptions using assignment statements if statements and single nested loops of register transfer rt level operators the generated structures are cellular and are interconnected in a tree structure a general algorithm is given to test cellular tree structures with a test length which grows only linearly with the size of the tree it is proved that this test length is optimal to within a constant factor ways of making the structures self checking are also indicated

principal support vector machines for linear and nonlinear sufficient dimension reduction we introduce a principal support vector machine psvm approach that can be used for both linear and nonlinear sufficient dimension reduction the basic idea is to divide the response variables into slices and use a modified form of support vector machine to find the optimal hyperplanes that separate them these optimal hyperplanes are then aligned by the principal components of their normal vectors it is proved that the aligned normal vectors provide an unbiased n consistent and asymptotically normal estimator of the sufficient dimension reduction space the method is then generalized to nonlinear sufficient dimension reduction using the reproducing kernel hilbert space in that context the aligned normal vectors become functions and it is proved that they are unbiased in the sense that they are functions of the true nonlinear sufficient predictors we compare psvm with other sufficient dimension reduction methods by simulation and in real data analysis and through both comparisons firmly establish its practical advantages

on the interpretation of regression plots abstract a framework is developed for the interpretation of regression plots including plots of the response against selected covariates residual plots added variable plots and detrended added variable plots it is shown that many of the common interpretations associated with these plots can be misleading the framework also allows for the generalization of standard plots and the development of new plotting paradigms a paradigm for graphical exploration of regression problems is sketched

on almost linearity of low dimensional projections from high dimensional data this paper studies the shapes of low dimensional projections from high dimensional data after standardization let mathbf x be a p dimensional random variable with mean zero and identity covariance for a projection beta mathbf x beta 1 find another direction b so that the regression curve of b mathbf x against beta mathbf x is as nonlinear as possible we show that when the dimension of mathbf x is large for most directions beta even the most nonlinear regression is still nearly linear our method depends on the construction of a pair of p dimensional random variables mathbf w 1 mathbf w 2 called the rotational twin and its density function with respect to the standard normal density with this we are able to obtain closed form expressions for measuring deviation from normality and deviation from linearity in a suitable sense of average as an interesting by product from a given set of data we can find simple unbiased estimates of e f beta mathbf x t phi 1 t 1 2 and e lbrack e mathbf x mid beta beta mathbf x t 2 t 2 f 2 beta mathbf x t phi 2 1 t rbrack where phi 1 is the standard normal density f beta mathbf x is the density for beta mathbf x and the e is taken with respect to the uniformly distributed beta this is achieved without any smoothing and without resorting to any laborious projection procedures such as grand tours our result is related to the work of diaconis and freedman the impact of our result on several fronts of data analysis is discussed for example it helps establish the validity of regression analysis when the link function of the regression model may be grossly wrong a further generalization which replaces beta mathbf x by b mathbf x with b beta 1 ldots beta k for k randomly selected orthonormal vectors beta i i 1 ldots k helps broaden the scope of application of sliced inverse regression sir

principal hessian directions revisited abstract li has suggested the method of principal hessian directions for estimating plotting directions that capture curvature in regression functions this article revisits li s proposal offering a number of suggestions for improved application

inverse regression based uncertainty quantification algorithms for high dimensional models many uncertainty quantification uq approaches suffer from the curse of dimensionality that is their computational costs become intractable for problems involving a large number of uncertainty parameters in these situations the classic monte carlo often remains the preferred method of choice because its convergence rate o n 1 2 where n is the required number of model simulations does not depend on the dimension of the problem however many high dimensional uq problems are intrinsically low dimensional because the variation of the quantity of interest qoi is often caused by only a few latent parameters varying within a low dimensional subspace known as the sufficient dimension reduction sdr subspace in the statistics literature motivated by this observation we propose two inverse regression based uq algorithms iruq for high dimensional problems both algorithms use inverse regression to convert the original high dimensional problem to a low dimensional one which is then efficiently solved by building a response surface for the reduced model for example via the polynomial chaos expansion the first algorithm which is for the situations where an exact sdr subspace exists is proved to converge at rate o n 1 hence much faster than mc the second algorithm which doesn t require an exact sdr employs the reduced model as a control variate to reduce the error of the mc estimate the accuracy gain could still be significant depending on how well the reduced model approximates the original high dimensional one iruq also provides several additional practical advantages it is non intrusive it does not require computing the high dimensional gradient of the qoi and it reports an error bar so the user knows how reliable the result is

cleanliness and log characteristic cycles for vector bundles with flat connections let x be a proper smooth algebraic variety over a field k of characteristic zero and let d be a divisor with simple normal crossings let m be a vector bundle over x d equipped with a flat connection with possible irregular singularities along d we define a cleanliness condition which roughly says that the singularities of the connection are controlled by the singularities at the generic points of d when this condition is satisfied we compute explicitly the associated log characteristic cycle and relate it to the so called refined irregularities as a corollary of a log variant of kashiwara dubson formula we obtain the euler characteristic of the de rham cohomology of the vector bundle under a mild technical hypothesis on m

dimension reduction for conditional mean in regression in many situations regression analysis is mostly concerned with inferring about the conditional mean of the response given the predictors and less concerned with the other aspects of the conditional distribution in this paper we develop dimension reduction methods that incorporate this consideration we introduce the notion of the central mean subspace cms a natural inferential object for dimension reduction when the mean function is of interest we study properties of the cms and develop methods to estimate it these methods include a new class of estimators which requires fewer conditions than phd and which displays a clear advantage when one of the conditions for phd is violated cms also reveals a transparent distinction among the existing methods for dimension reduction ols phd sir and save we apply the new methods to a data set involving recumbent cows

transformed sufficient dimension reduction we propose a general framework for dimension reduction in regression to fill the gap between linear and fully nonlinear dimension reduction the main idea is to first transform each of the raw predictors monotonically and then search for a low dimensional projection in the space defined by the transformed variables both user specified and data driven transformations are suggested in each case the methodology is first discussed in generality and then a representative method is proposed and evaluated by simulation the proposed methods are applied to a real dataset

l 1 regularized least squares for support recovery of high dimensional single index models with gaussian designs it is known that for a certain class of single index models sims y f xp 1t 0 e support recovery is impossible when x n 0 ip p and a model complexity adjusted sample size is below a critical threshold recently optimal algorithms based on sliced inverse regression sir were suggested these algorithms work provably under the assumption that the design x comes from an i i d gaussian distribution in the present paper we analyze algorithms based on covariance screening and least squares with l1 penalization i e lasso and demonstrate that they can also enjoy optimal up to a scalar rescaled sample size in terms of support recovery albeit under slightly different assumptions on f and e compared to the sir based algorithms furthermore we show more generally that lasso succeeds in recovering the signed support of 0 if x n 0 and the covariance satisfies the irrepresentable condition our work extends existing results on the support recovery of lasso for the linear model to a more general class of sims

functorial desingularization of quasi excellent schemes in characteristic zero the nonembedded case we prove that any reduced noetherian quasi excellent scheme of characteristic zero admits a strong desingularization which is functorial with respect to all regular morphisms as a main application we deduce that any reduced formal variety of characteristic zero admits a strong functorial desin gularization also we show that as an easy formal consequence of our main result one obtains strong functorial desingularization for many other spaces of characteristic zero including quasi excellent stacks and formal schemes and complex or non archimedean analytic spaces moreover these functors easily generalize to non compact setting by use of generalized convergent blow up sequences with regular centers

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

a modified partial product generator for redundant binary multipliers due to its high modularity and carry free addition a redundant binary rb representation can be used when designing high performance multipliers the conventional rb multiplier requires an additional rb partial product rbpp row because an error correcting word ecw is generated by both the radix 4 modified booth encoding mbe and the rb encoding this incurs in an additional rbpp accumulation stage for the mbe multiplier in this paper a new rb modified partial product generator rbmppg is proposed it removes the extra ecw and hence it saves one rbpp accumulation stage therefore the proposed rbmppg generates fewer partial product rows than a conventional rb mbe multiplier simulation results show that the proposed rbmppg based designs significantly improve the area and power consumption when the word length of each operand in the multiplier is at least 32 bits these reductions over previous nb multiplier designs incur in a modest delay increase approximately 5 percent the power delay product can be reduced by up to 59 percent using the proposed rb multipliers when compared with existing rb multipliers

a new family of power transformations to improve normality or symmetry summary we introduce a new power transformation family which is well defined on the whole real line and which is appropriate for reducing skewness and to approximate normality it has properties similar to those of the box cox transformation for positive variables the large sample properties of the transformation are investigated in the contect of a single random sample

sliced regression for dimension reduction a new dimension reduction method involving slicing the region of the response and applying local kernel regression to each slice is proposed compared with the traditional inverse regression methods e g sliced inverse regression sir the new method is free of the linearity condition and has much better estimation accuracy compared with the direct estimation methods e g mave the new method is much more robust against extreme values and can capture the entire central subspace cs exhaustively to determine the cs dimension a consistent cross validation criterion is developed extensive numerical studies including a real example confirm our theoretical findings

on the fly conversion of redundant into conventional representations an algorithm to convert redundant number representations into conventional representations is presented the algorithm is performed concurrently with the digit by digit generation of redundant forms by schemes such as srt division it has a step delay roughly equivalent to the delay of a carry save adder and simple implementation the conversion scheme is applicable in arithmetic algorithms such as nonrestoring division square root and on line operations in which redundantly represented results are generated in a digit by digit manner from most significant to least significant

optimal sufficient dimension reduction in regressions with categorical predictors though partial sliced inverse regression partial sir chiaromonte et al 2002 sufficient dimension reduction in regressions with categorical predictors ann statist 30 475 497 extended the scope of sufficient dimension reduction to regressions with both continuous and categorical predictors its requirement of homogeneous predictor covariances across the subpopulations restricts its application in practice when this condition fails partial sir may provide misleading results in this article we propose a new estimation method via a minimum discrepancy approach without this restriction our method is optimal in terms of asymptotic efficiency and its test statistic for testing the dimension of the partial central subspace always has an asymptotic chi squared distribution it also gives us the ability to test predictor effects an asymptotic chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of the continuous predictors given the remaining predictors is obtained

efficient integration of sufficient dimension reduction and prediction in discriminant analysis abstractsufficient dimension reduction sdr methods are popular model free tools for preprocessing and data visualization in regression problems where the number of variables is large unfortunate

assessing the number of linear components in a general regression problem abstract this article presents a method for testing the dimension of a general regression model without assuming any specific model for the distribution of the regressors examples of application and comparisons with existing methods are also studied

on directional regression for dimension reduction we introduce directional regression dr as a method for dimension reduction like contour regression dr is derived from empirical directions but achieves higher accuracy and requires substantially less computation dr naturally synthesizes the dimension reduction estimators based on conditional moments such as sliced inverse regression and sliced average variance estimation and in doing so combines the advantages of these methods under mild conditions it provides exhaustive and consistent estimate of the dimension reduction space we develop the asymptotic distribution of the dr estimator and from that a sequential test procedure to determine the dimension of the central space we compare the performance of dr with that of existing methods by simulation and find strong evidence of its advantage over a wide range of models finally we apply dr to analyze a data set concerning the identification of hand written digits

iterative algorithm and architecture for exponential logarithm powering and root extraction an algorithm and architecture for powering computation and root extraction with fixed point and floating point exponents is presented in this paper the algorithm is based on an optimized iterative sequence of parallel and or overlapped operations 1 reciprocal 2 high radix digit recurrence logarithm 3 left to right carry free multiplication and 4 high radix online exponential a redundant number system is used to allow for the overlapping of the different operations of the algorithm as the logarithm and exponential are part of the sequence of operations some minor changes are made to allow for the independent computation of the logarithm and exponential functions a sequential implementation of the algorithm is proposed and the execution times and hardware requirements are estimated for single and double precision floating point computations these estimates are obtained for several radices according to an approximate model for the delay and area of the main logic blocks and help to determine the radix values which lead to the most efficient implementations

diagnostic studies in sufficient dimension reduction sufficient dimension reduction in regression aims to reduce the predictor dimension by replacing the original predictors with some set of linear combinations of them without loss of information numerous dimension reduction methods have been developed based on this paradigm however little effort has been devoted to diagnostic studies within the context of dimension reduction in this paper we introduce methods to check goodness of fit for a given dimension reduction subspace the key idea is to extend the so called distance correlation to measure the conditional dependence relationship between the covariates and the response given a reduction subspace our methods require minimal assumptions which are usually much less restrictive than the conditions needed to justify the original methods asymptotic properties of the test statistic are studied numerical examples demonstrate the effectiveness of the proposed approach

asymptotic theory of overparameterized structural models abstract a theory of overparameterized structural models is presented in such a model some redundant parameters are involved the parameter vector is not identified and the information matrix is not nonsingular the minimum discrepancy function mdf test statistic is shown to have an asymptotic chi squared distribution almost everywhere for a wide class of discrepancy functions asymptotic distribution properties of the mdf estimators are investigated the factor analysis model is discussed as an example

hybrid signed digit number systems a unified framework for redundant number representations with bounded carry propagation chains a novel hybrid number representation is proposed it includes the two s complement representation and the signed digit representation as special cases the hybrid number representations proposed are capable of bounding the maximum length of carry propagation chains during addition to any desired value between 1 and the entire word length the framework reveals a continuum of number representations between the two extremes of two s complement and signed digit number systems and allows a unified performance analysis of the entire spectrum of implementations of adders multipliers and alike we present several static cmos implementations of a two operand adder which employ the proposed representations we then derive quantitative estimates of area in terms of the required number of transistors and the maximum carry propagation delay for such an adder the analysis clearly illustrates the trade offs between area and execution time associated with each of the possible representations we also discuss adder trees for parallel multipliers and show that the proposed representations lead to compact adder trees with fast execution times in practice the area available to a designer is often limited in such cases the designer can select the particular hybrid representation that yields the most suitable implementation fastest lowest power consumption etc while satisfying the area constraint similarly if the worst case delay is predetermined the designer can select a hybrid representation that minimizes area or power under the delay constraint

a high speed reduced size adder under left to right input arrival an efficient parallel adder under left to right input arrival is proposed making full use of the delay of the input arrival it produces the sum within a small constant delay after the arrival of the final bits its amount of hardware is proportional to the operand length it can be applied to the quotient conversion in an array divider

sufficient dimension reduction via inverse regression a family of dimension reduction methods the inverse regression ir family is developed by minimizing a quadratic objective function an optimal member of this family the inverse regression estimator ire is proposed along with inference methods and a computational algorithm the ire has at least three desirable properties 1 its estimated basis of the central dimension reduction subspace is asymptotically efficient 2 its test statistic for dimension has an asymptotic chi squared distribution and 3 it provides a chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of predictors given the remaining predictors current methods like sliced inverse regression belong to a suboptimal class of the ir family comparisons of these methods are reported through simulation studies the approach developed here also allows a relatively straightforward derivation of the asymptotic null distribution of the test statistic for dimension used in slic

graphics for regressions with a binary response abstract central dimension reduction subspaces which characterize the dependence of a response variable on one or more predictors are developed and then used to guide the construction and interpretation of graphics for regression problems with a binary response variable graphical methods requiring neither a link function nor residuals are suggested for both development and criticism of model components implied by the central dimension reduction subspace

sufficient dimension reduction through discretization expectation estimation in the context of sufficient dimension reduction the goal is to parsimoniously recover the central subspace of a regression model many inverse regression methods use slicing estimation to recover the central subspace the efficacy of slicing estimation depends heavily upon the number of slices however the selection of the number of slices is an open and long standing problem in this paper we propose a discretization expectation estimation method which avoids selecting the number of slices while preserving the integrity of the central subspace this generic method assures root n consistency and asymptotic normality of slicing estimators for many inverse regression methods and can be applied to regressions with multivariate responses a bic type criterion for the dimension of the central subspace is proposed comprehensive simulations and an illustrative application show that our method compares favourably with existing estimators

partial least squares estimator for single index models the partial least squares pls approach first constructs new explanatory variables known as factors or components which are linear combinations of available predictor variables a small subset of these factors is then chosen and retained for prediction we study the performance of pls in estimating single index models especially when the predictor variables exhibit high collinearity we show that pls estimates are consistent up to a constant of proportionality we present three simulation studies that compare the performance of pls in estimating single index models with that of sliced inverse regression sir in the first two studies we find that pls performs better than sir when collinearity exists in the third study we learn that pls performs well even when there are multiple dependent variables the link function is non linear and the shape of the functional form is not known

bootstrap testing of the rank of a matrix via least squared constrained estimation to test if an unknown matrix m0 has a given rank null hypothesis noted h0 we consider a statistic that is a squared distance between an estimator and the submanifold of fixed rank matrix under h0 this statistic converges to a weighted chi squared distribution we introduce the constrained bootstrap cs bootstrap to estimate the law of this statistic under h0 an important point is that even if h0 fails the cs bootstrap reproduces the behavior of the statistic under h0 as a consequence the cs bootstrap is employed to estimate the nonasymptotic quantile for testing the rank we provide the consistency of the procedure and the simulations shed light on the accuracy of the cs bootstrap with respect to the traditional asymptotic comparison more generally the results are extended to test whether an unknown parameter belongs to a submanifold of the euclidean space finally the cs bootstrap is easy to compute it handles a large family of tests and it works under mild assumptions

high performance low power left to right array multiplier design we present a high performance low power design of linear array multipliers based on a combination of the following techniques signal flow optimization in 3 2 adder array for partial product reduction left to right leapfrog lrlf signal flow and splitting of the reduction array into upper lower parts the resulting upper lower lrlf ullrlf multiplier is compared with tree multipliers from automatic layout experiments we find that ullrlf multipliers have similar power delay and area as tree multipliers for n spl les 32 with more regularity and inherently shorter interconnects the ullrlf structure presents a competitive alternative to tree structures in the design of fast low power multipliers implemented in deep submicron vlsi technology

extending sliced inverse regression sliced inverse regression sir and an associated chi squared test for dimension have been introduced as a method for reducing the dimension of regression problems whose predictor variables are normal in this article the assumptions on the predictor distribution under which the chi squared test was proved to apply are relaxed and the result is extended a general weighted chi squared test that does not require normal regressors for the dimension of a regression is given simulations show that the weighted chi squared test is more reliable than the chi squared test when the regressor distribution digresses from normality significantly and that it compares well with the chi squared test when the regressors are normal

on irregularities of fourier transforms of regular holonomic d modules abstract we study fourier transforms of regular holonomic d modules by using the theory of fourier sato transforms of enhanced ind sheaves developed by kashiwara schapira and d agnolo kashiwara a formula for their enhanced solution complexes will be obtained moreover we show that some parts of their characteristic cycles and irregularities are expressed by the geometries of the original d modules