construction of robust frames in erasure recovery abstract erasures occur often in signal transmission in 10 the authors introduced robust frames with which one can recover erased frame coefficients even if the locations are unknown in this paper we give some explicit construction of robust frames

multiplicative lidskii s inequalities and optimal perturbations of frames abstract in this paper we study two design problems in frame theory on the one hand given a fixed finite frame f f j j i n for c d we compute those dual frames g of f that are optimal perturbations of the canonical dual frame for f under certain restrictions on the norms of the elements of g on the other hand we compute those v f v f j j i n for invertible operators v which are close to the identity that are optimal perturbations of f that is we compute the optimal perturbations of f among frames g g j j i n that have the same linear relations as f in both cases optimality is measured with respect to submajorization of the eigenvalues of the frame operators hence our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error the key tool for these results is a multiplicative analogue of lidskii s inequality in terms of log majorization and a characterization of the case of equality

numerically and spectrally optimal dual frames in hilbert spaces abstract finding a best dual frame that minimizes the reconstruction errors when erasures occur is a deep rooted problem in frame theory the primary purpose of this paper is to introduce a new measurement for constructing optimal duals we consider the numerical radius of the error operator which has some advantages over the previous measures then we give several equivalent conditions for which the canonical dual and alternate duals are numerically optimal dual frames moreover we obtain some relations between optimal duals spectrally optimal duals and numerically optimal duals for erasures finally we survey the effect of operator perturbation on spectrally optimal duals this leads to finding some equivalent conditions under which the canonical dual is a spectrally optimal dual for any r erasures

sparsity and spectral properties of dual frames we study sparsity and spectral properties of dual frames of a given finite frame we show that any finite frame has a dual with no more than n 2 non vanishing entries where n denotes the ambient dimension and that for most frames no sparser dual is possible moreover we derive an expression for the exact sparsity level of the sparsest dual for any given finite frame using a general ized notion of spark we then study the spectral properties of dual frames in terms of singular values of the synthesis operator we provide a complete characteriza tion for which spectral patterns of dual frames are possible for a fixed frame for many cases we provide simple explicit constructions for dual frames with a given spectrum in particular if the constraint on the dual is that it be tight

combined 5 2 cv f test for comparing supervised classification learning algorithms dietterich 1998 reviews five statistical tests and proposes the 5 2 cv t test for determining whether there is a significant difference between the error rates of two classifiers in our experiments we noticed that the 5 2 cv t test result may vary depending on factors that should not affect the test and we propose a variant the combined 5 2 cv f test that combines multiple statistics to get a more robust test simulation results show that this combined version of the test has lower type i error and higher power than 5 2 cv proper

markov cross validation for time series model evaluations cross validation cv is a simple and universal tool to estimate generalization ability however existing cvs do not work well for periodicity overlapping or correlation of series the corresponding three criteria aimed at describing these properties are presented based on them we put forward a novel markov cross validation m cv whose data partition can be seen as a markov process the partition ensures that samples in each subset are neither too close nor too far in doing so overfitting model or information loss of series which may result in underestimation or overestimation of the error can be avoided furthermore subsets from m cv partition could well represent the original series and it may be extended to time series or stream data sampling theoretical analysis shows that m cv is the unique one which meets all of above criteria among current cvs in addition the error estimation on subsets is proved to have less variance than that on original series therefore it ensures the stability of m cv experimental results demonstrate that the proposed m cv has lower bias variance and time consumption than other cvs

deconvolution by successive approximations although various methods of deconvolution have been known for many years they are not generally regarded as being routinely usable the successive approximation method described in this article should be an improvement in that respect it operates in the time domain and is based on a simple procedure i first discuss the mathematics involved and then give some examples to illustrate how the method works and some of the things it can do

frames graphs and erasures two uniform frames and their use for the coding of vectors are the main subject of this paper these frames are known to be optimal for handling up to two erasures in the sense that they minimize the largest possible error when up to two frame coefficients are set to zero here we consider various numerical measures for the reconstruction error associated with a frame when an arbitrary number of the frame coefficients of a vector are lost we derive general error bounds for two uniform frames when more than two erasures occur and apply these to concrete examples we show that among the 227 known equivalence classes of two uniform 36 15 frames arising from hadamard matrices there are 5 that give smallest error bounds for up to 8 erasures

fitting a bivariate additive model by local polynomial regression while the additive model is a popular nonparametric regression method many of its theoretical properties are not well understood especially when the backfitting algorithm is used for computation of the estimators this article explores those properties when the additive model is fitted by local polynomial regression sufficient conditions guaranteeing the asymptotic existence of unique estimators for the bivariate additive model are given asymptotic approximations to the bias and the variance of a homoscedastic bivariate additive model with local polynomial terms of odd and even degree are computed this model is shown to have the same rate of convergence as that of univariate local polynomial regression

probability modelled optimal frames for erasures abstract we establish a probability model for constructing optimal parseval frames when erasures occur during the transmission process of the frame coefficient data set such frames are called probability modelled pm optimal frames for erasures while pm optimal frames exist for all erasures it is usually difficult to construct them we characterize all the pm optimal frames for one and two erasures and propose an algorithm to construct these frames examples are given to demonstrate the construction and to compare the decoding reconstruction effects when both pm optimal parseval frames and uniform length parseval frames are used for encoding

optimal frames for erasures we study frames from the viewpoint of coding theory we introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting

a general framework for the statistical analysis of the sources of variance for classification error estimators estimating the prediction error of classifiers induced by supervised learning algorithms is important not only to predict its future error but also to choose a classifier from a given set model selection if the goal is to estimate the prediction error of a particular classifier the desired estimator should have low bias and low variance however if the goal is the model selection in order to make fair comparisons the chosen estimator should have low variance assuming that the bias term is independent from the considered classifier this paper follows the analysis proposed in 1 about the statistical properties of k fold cross validation estimators and extends it to the most popular error estimators resubstitution holdout repeated holdout simple bootstrap and 0 632 bootstrap estimators without and with stratification we present a general framework to analyze the decomposition of the variance of different error estimators considering the nature of the variance irreducible reducible variance and the different sources of sensitivity internal external sensitivity an extensive empirical study has been performed for the previously mentioned estimators with naive bayes and c4 5 classifiers over training sets obtained from assorted probability distributions the empirical analysis consists of decomposing the variances following the proposed framework and checking the independence assumption between the bias and the considered classifier based on the obtained results we propose the most appropriate error estimations for model selection under different experimental conditions

frame representations and parseval duals with applications to gabor frames let x n be a frame for a hilbert space h we investigate the conditions under which there exists a dual frame for x n which is also a parseval or tight frame we show that the existence of a parseval dual is equivalent to the problem whether x n can be dilated to an orthonormal basis under an oblique projection a necessary and sufficient condition for the existence of parseval duals is obtained in terms of the frame excess for a frame g g g induced by a projective unitary representation of a group g it is possible that g g g can have a parseval dual but does not have a parseval dual of the same type the primary aim of this paper is to present a complete characterization for all the projective unitary representations such that every frame g g g with a necessary lower frame bound condition has a parseval dual of the same type as an application of this characterization together with a result about lattice tiling we prove that every gabor frame g g l k again with the same necessary lower frame bound condition has a parseval dual of the same type if and only if the volume of the fundamental domain of l k is less than or equal to 1 2

linearly connected sequences and spectrally optimal dual frames for erasures abstract in the case that a frame is prescribed for applications and erasures occur in the process of data transmissions we examine optimal dual frames for the recovery from single erasures in contrast to earlier papers we consider the spectral radius of the error operator instead of its operator norm as a measure of optimality this notion of optimality is natural when the neumann series is used to recover the original data in an iterative manner we obtain a complete characterization of spectrally one erasure optimal dual frames in terms of the redundancy distribution of the prescribed frame our characterization relies on the connection between erasure optimal frames and the linear connectivity property of the frame we prove that the linear connectivity property is equivalent to the intersection dependent property and is also closely related to the well known concept of a k independent set additionally we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iterative reconstruction work

p woven frames abstract the known family of woven collections of frames have interesting properties and applications yet they cannot be easily constructed the present paper introduces new collections of frames sharing many properties with the original ones and having simpler construction the frames i i i and i i i for a hilbert space h are called partition woven or briefly p woven if there exists a nontrivial subset of i such that the family g i i i represented as the triple i i and called a p weaven frame is a frame defined by g i i if i and g i i otherwise the paper also studies the class of cp woven collections of frames which is a subset of the p woven collections of frames and satisfy a stronger condition the set of all woven frames is a strict subset of the set of all cp woven frames which is further a strict subset of the set of all p woven frames several examples and counterexamples are studied to show whether or not a pair of frames is p woven or cp woven

precise deconvolution using the fermat number transform abstract a new algorithm to solve convolution systems of linear equations is described modular arithmetic and the fermat number theoretical transform are used the algorithm permits elimination of the rounding off errors of the solution of the deconvolution problem

the existence of tight gabor duals for gabor frames and subspace gabor frames abstract let k and l be two full rank lattices in r d we give a complete characterization for all the gabor frames that admit tight dual of the same type the characterization is given in terms of the center valued trace of the von neumann algebra generated by the left regular projective unitary representations associated with the time frequency lattice k l two applications of this characterization were obtained i we are able to prove that every gabor frame has a tight dual if and only if the volume of k l is less than or equal to 1 2 ii we are able to obtain sufficient or necessary conditions for the existence of tight gabor pseudo duals for subspace gabor frames in various cases in particular we prove that every subspace gabor frame has a tight gabor pseudo dual if either the volume v k l 1 2 or v k l 2 moreover if k z d l z d with 1 then a subspace gabor frame g g l k has a tight gabor pseudo dual only when g g l k itself is already tight

extreme points and identification of optimal alternate dual frames abstract the main purpose of this paper is to identify the set of optimal dual frames under erasures to this end we characterize extreme points in the set of all optimal duals for 1 erasure also we give some conditions under which an alternate dual frame is either not optimal or is a non unique optimal dual moreover we obtain a new characterization of frames when the canonical dual is the unique optimal dual frame for erasures

properties of finite dual fusion frames abstract a new notion of dual fusion frame has been recently introduced by the authors in this article that notion is further motivated and it is shown that it is suitable to deal with questions posed in a finite dimensional real or complex hilbert space reinforcing the idea that this concept of duality solves the question about an appropriate definition of dual fusion frames it is shown that for overcomplete fusion frames there always exist duals different from the canonical one conditions that assure the uniqueness of duals are given the relation of dual fusion frame systems with dual frames and dual projective reconstruction systems is established optimal dual fusion frames for the reconstruction in case of erasures of subspaces and optimal dual fusion frame systems for the reconstruction in case of erasures of local frame vectors are determined examples that illustrate the obtained results are exhibited

reconstruction of signals from frame coefficients with erasures at unknown locations we propose new approaches to the problems of recovering signals from the rearranged frame coefficients or frame coefficients with erasures at either known or unknown locations these problems naturally arise from applications where the encoded information needs to be transmitted for example in signal image processing information and coding theory and communications we show that with the appropriate choices of the frames that are used for encoding the signal with erasures occurring at known locations can be easily recovered without inverting the sub frame operators each time our new easy to implement and cost efficient algorithm provides perfect reconstruction of the original signal to address the problem of recovering erased coefficients from unknown locations we propose to use a class of frames that are almost robust with respect to m erasures we prove that every frame with uniform excess can be rescaled to an almost robust frame and the locations of erased data can be perfectly recovered for almost all the signals similar results are obtained for recovering the original order of a disordered rearranged set of frame coefficients numerical examples are presented to test the main results whenever the received data are noise free we can recover the original signal exactly from frame coefficients with erasures at unknown locations or from a disordered set of frame coefficients

nilpotent bridging for unions of two bases in a recent article the authors discovered a new efficient method for perfect reconstruction from frame erasures called nilpotent bridging in order to perform the reconstruction it is necessary to invert a matrix called the bridge matrix which consists of inner products of the frame vectors indexed by the erasure set with vectors from the corresponding dual frame indexed by a set known as the bridge set which is disjoint from the erasure set if no search procedure is required to find a bridge set of the same size as the erasure set for which the bridge matrix is invertible the dual frame pair is said to satisfy the full skew spark property equivalently a frame satisfies the full skew spark property if every subset of the complement of the erasure set with the same cardinality as the erasure set yields an invertible bridge matrix in this paper we consider dual frame pairs which consist of a union of two bases and the union of their corresponding dual bases in finite dimensions it is shown that if we restrict the class of bridge sets to block bridge sets then a bridge set search is also unnecessary for most unions of two bases and most unions of two orthonormal bases by most we mean the set of unions of two bases resp orthonormal bases for which no bridge set search is necessary contains an open dense set in the set of all unions of two bases resp orthonormal bases so while not every bridge set of the same cardinality as the erasure set yields an invertible bridge matrix the bridge sets which have the special structure of block bridge sets are very likely to yield an invertible bridge matrix these unions of two bases are said to satisfy the block skew spark property in infinite dimensions we also show that the block skew spark property holds for a dense but not necessarily open set of unions of two riesz resp orthonormal bases lastly applications to shannon whittaker sampling theory are discussed and several open questions are posed pertaining to sampling theory

comparison of two bandwidth selectors with dependent errors for nonparametric regression in the case of dependent observations cross validation is known to be severely affected by dependence this effect is precisely quantified through a limiting distribution for the cross validated bandwidth the performance of two methods the leave 2e 1 out version of cross validation and partitioned cross validation which adjust for the dependence effect on bandwidth selection is investigated the bandwidths produced by these two methods are analyzed by further limiting distributions which reveal significantly different characteristics simulations demonstrate that the asymptotic effects hold for reasonable sample sizes

no unbiased estimator of the variance of k fold cross validation most machine learning researchers perform quantitative experiments to estimate generalization error and compare the performance of different algorithms in particular their proposed algorithm in order to be able to draw statistically convincing conclusions it is important to estimate the uncertainty of such estimates this paper studies the very commonly used k fold cross validation estimator of generalization performance the main theorem shows that there exists no universal valid under all distributions unbiased estimator of the variance of k fold cross validation the analysis that accompanies this result is based on the eigen decomposition of the covariance matrix of errors which has only three different eigenvalues corresponding to three degrees of freedom of the matrix and three components of the total variance this analysis helps to better understand the nature of the problem and how it can make naive estimators that don t take into account the error correlations due to the overlap between training and test sets grossly underestimate variance this is confirmed by numerical experiments in which the three components of the variance are compared when the difficulty of the learning problem and the number of folds are varied

consistent cross validatory model selection for dependent data hv block cross validation this paper considers the impact of shao s 1993 recent results regarding the asymptotic inconsistency of model selection via leave one out cross validation on h block cross validation a cross validatory method for dependent data proposed by burman chow and nolan 1994 journal of time series analysis 13 189 207 it is shown that h block cross validation is inconsistent in the sense of shao 1993 journal of american statistical association 88 422 486 495 and therefore is not asymptotically optimal a modification of the h block method dubbed hv block cross validation is proposed which is asymptotically optimal the proposed approach is consistent for general stationary observations in the sense that the probability of selecting the model with the best predictive ability converges to 1 as the total number of observations approaches infinity this extends existing results and yields a new approach which contains leave one out cross validation leave nv out cross validation and h block cross validation as special cases applications are considered

approximations for gabor and wavelet frames let be a frame vector under the action of a collection of unitary operators u motivated by the recent work of frank paulsen and tiballi and some application aspects of gabor and wavelet frames we consider the existence and uniqueness of the best approximation by normalized tight frame vectors we prove that for any frame induced by a projective unitary representation for a countable discrete group the best normalized tight frame ntf approximation exists and is unique therefore it applies to gabor frames including gabor frames for subspaces and frames induced by translation groups similar results hold for semi orthogonal wavelet frames

complex equiangular cyclic frames and erasures we derive various interesting properties of complex equiangular cyclic frames for many pairs n k using gauss sums and number theory we further use these results to study the random and burst errors of some special classes of complex equiangular cyclic n k frames

k dimensional error free deconvolution using the fermat number transform abstract the paper presents the generalization of the one dimensional deconvolution algorithm using the fermat number transform for the k dimensional deconvolution the problems of how to express a k dimensional convolution system in the form of special matrix equation how to calculate the precise value of the matrix determinant and how to remove rounding off errors in the process of deconvolution are solved in the paper the results are used in the algorithm derivation

optimal dual frames for erasures for any given frame for the purpose of encoding in a finite dimensional hilbert space we investigate its dual frames that are optimal for erasures for the purpose of decoding we show that in general the canonical dual is not necessarily optimal moreover optimal dual frames are not necessarily unique we present some sufficient conditions under which the canonical dual is the unique optimal dual frame for the erasure problem as an application we get that the canonical dual is the only optimal dual when either the frame is induced by a group representation or the frame is uniform tight

frames of uniform subframe bounds with applications to erasures abstract we introduce a class of frames called usb frames frames of uniform subframe bounds which mean that any subsequence of the frame is a frame for the subspace it spans and the frame bounds depend only on the cardinality of the subsequence we give several necessary and sufficient conditions for a frame to be a usb frame as an application we show that a tight usb frame is optimal for erasures in the sense that the reconstruction error is independent of the position of erasures we give several explicit construction of usb frames

optimal dual frames for communication coding with probabilistic erasures assume that a frame is preselected for encoding in a communication system we investigate the optimal dual frames for signal reconstruction decoding which minimize the maximal error when the probabilistic erasures occur in the transmission process from the perspective of mathematical theory of frames we set up a probability model under which we define the probability optimal po dual frames for a given frame when the frame based coding involves probabilistic erasures we obtain a sufficient and necessary condition under which the canonical dual frame is the unique po dual frame additionally we derive some general sufficient conditions for which the canonical dual frame is either not optimal or it is optimal but not the unique optimal one we present two simulation examples to compare the reconstruction effects when both the po dual frames and the general optimal go dual frames are used for reconstruction

geometrically uniform frames we introduce a new class of finite dimensional frames with strong symmetry properties called geometrically uniform gu frames that are defined over a finite abelian group of unitary matrices and are generated by a single generating vector the notion of gu frames is then extended to compound gu cgu frames which are generated by a finite abelian group of unitary matrices using multiple generating vectors the dual frame vectors and canonical tight frame vectors associated with gu frames are shown to be gu and therefore also generated by a single generating vector which can be computed very efficiently using a fourier transform ft defined over the generating group of the frame similarly the dual frame vectors and canonical tight frame vectors associated with cgu frames are shown to be cgu the impact of removing single or multiple elements from a gu frame is considered a systematic method for constructing optimal gu frames from a given set of frame vectors that are not gu is also developed finally the euclidean distance properties of gu frames are discussed and conditions are derived on the abelian group of unitary matrices to yield gu frames with strictly positive distance spectrum irrespective of the generating vector