scale sensitive dimensions uniform convergence and learnability learnability in valiant s pac learning model has been shown to be strongly related to the existence of uniform laws of large numbers these laws define a distribution free convergence property of means to expectations uniformly over classes of random variables classes of real valued functions enjoying such a property are also known as uniform glivenko cantelli classes in this paper we prove through a generalization of sauer s lemma that may be interesting in its own right a new characterization of uniform glivenko cantelli classes our characterization yields dudley gine and zinn s previous characterization as a corollary furthermore it is the first based on a gine and zinn s previous characterization as a corollary furthermore it is the first based on a simple combinatorial quantity generalizing the vapnik chervonenkis dimension we apply this result to obtain the weakest combinatorial condition known to imply pac learnability in the statistical regression or agnostic framework furthermore we find a characterization of learnability in the probabilistic concept model solving an open problem posed by kearns and schapire these results show that the accuracy parameter plays a crucial role in determining the effective complexity of the learner s hypothesis class

multiplicativity of lp norms for matrices abstract the l p norm and the l p operator norm of an m n complex matrix a ij are given by a p i j ij p 1 p and a p max ax p xe c n x p 1 respectively the main purpose of this paper is to investigate the multipicativity of the l p norms and their relation to the l p operator norms

chernoff bound of trellis coded modulation over correlated fading channels we derive a chernoff upper bound for the pairwise error probability in the presence of an additive white gaussian noise and a rayleigh or rice correlated fading the bound is useful for situations where perfect interleaving cannot be achieved we use it to determine some indications in the design of optimum trellis coded modulation for correlated fading channels

subdifferentials faces and dual matrices abstract a characterization of the dual matrices for the unitarily invariant norms is given moreover the connection between the dual matrices the subdifferentials of matrix norms and the faces of the unit ball of matrices is presented

a unified snr dependent analysis of space time codes in fading channels the pairwise error probability pep for multiple input multiple output mimo radio interfaces is investigated by means of a novel formulation based on compound matrices the proposed approach is suitable for any mimo system having average upper bounded pep written as det i gamma a zeta where a is a hermitian matrix zeta an integer number and gamma the signal to noise ratio snr that bound frequently results in mimo single and multicarrier transmissions it is shown that the minimization of the bounded pep should consider the whole set of nonzero compound matrices of a in particular the snr of interest marks the compound matrix that mainly drives the system performance both diversity advantage and coding gain are given as continuous functions of the variable gamma hence their asymptotic behaviors are taken as important case of studies the interaction effects between channel code and propagation environment are also discussed it is shown how the eigenvectors and eigenvalues of the autocorrelation channel matrix may be considered for code design it is also proved the maximization of the code rank is not always a necessary requirement for performance improvement being its optimal value fixed by the channel structure and snr of interest finally the analysis is applied to space time trellis coded transmissions over spatially correlated slow rayleigh fading channels

estimating the matrixp norm the holderp norm of anm n matrix has no explicit representation unlessp 1 2 or it is shown here that thep norm can be estimated reliably ino mn operations a generalization of the power method is used with a starting vector determined by a technique with a condition estimation flavour the algorithm nearly always computes ap norm estimate correct to the specified accuracy and the estimate is always within a factorn 1 1 p of a p as a by product a new way is obtained to estimate the 2 norm of a rectangular matrix this method is more general and produces better estimates in practice than a similar technique of cline conn and van loan

exact pairwise error probability of space time codes we describe a simple technique for the numerical calculation within any desired degree of accuracy of the pairwise error probability pep of space time codes over fading channels this method applies also to the calculation of e q spl radic spl xi for any nonnegative random variable spl xi whose moment generating function spl phi sub spl xi s e exp s spl xi is known its application to the multiple antenna independent rayleigh fading channel and to the rayleigh block fading channel is discussed and illustrated by two simple examples

on approximation problems with zero trace matrices abstract we consider some approximation problems in the linear space of complex matrices with respect to unitarily invariant norms we deal with special cases of approximation of a matrix by zero trace matrices moreover some characterizations of zero trace matrices are given by means of matrix approximation problems

use of geometric algebra compound matrices and the determinant of the sum of two matrices in this paper we demonstrate the capabilities of geometric algebra by the derivation of a formula for the determinant of the sum of two matrices in which both matrices are separated in the sense that the resulting expression consists of a sum of traces of products of their compound matrices for the derivation we introduce a vector of grassmann elements associated with an arbitrary square matrix we recall the concept of compound matrices and summarize some of their properties this paper introduces a new derivation and interpretation of the relationship between p forms and the pth compound matrix and it demonstrates the use of geometric algebra which has the potential to be applied to a wide range of problems

on the extremal structure of least upper bound norms and their dual abstract given finite dimensional real or complex banach spaces e and f with norms e r and f r we denote by n the least upper bound norm induced on l e f some results are given on the extremal structures of b the unit ball of n of its polar b and of b which is the polar of the unit ball of the least upper bound norm n the exposed faces the extreme points and a large family of other faces of b and b are presented it turns out that b is a subset of b the set of tangency points of the surfaces of b and b is completely determined and represented as the union of the exposed faces of b which are normal to rank one mappings we determine sharp bounds on the ranks of mappings in these exposed faces

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

certain isometries on rn abstract let r n be the linear space of all real column vectors with n coordinates given x x 1 x n t r n its p k norm is defined by x p k max x i 1 p x i k p 1 p 1 i 1 k n where k is a positive integer satisfying 1 k n and p satisfies 1 p whereas its c norm is defined by x c max c t px p is a generalized permutation matrix for any nonzero c r n this paper characterizes the isometries for c norms and p k norms some related results are also considered

operator norms multiplicativity factors and c numerical radii abstract let v be a normed vector space over c let b v denote the algebra of linear bounded operators on v and let n be an arbitrary seminorm or norm on b v in this paper we discuss multiplicativity factors for n i e constants 0 for which n n is submultiplicative we find that while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors in the infinite dimensional case n may or may not have such factors our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius called c numerical radii this is done with the help of a combinatorial inequality which seems to be of independent interest

characterization of the subdifferential of some matrix norms abstract a characterization is given of the subdifferential of matrix norms from two classes orthogonally invariant norms and operator or subordinate norms specific results are derived for some special cases

duality between some linear preserver problems ii isometries with respect to c special norms and matrices with fixed singular values let fm n m n denote the linear space of all m n complex or real matrices according as f c or r let c c1 cm 0 be such that c1 cm 0 the c spectral norm of a matrix a fm n is the quantity a c i imci i a where 1 a m a are the singular values of a let d d1 dm 0 where d1 dm 0 we consider the linear isometries between the normed spaces fm n c and fm n d and prove that they are dual transformations of the linear operators which map l d onto l c where l c x fm n x has singular values c1 cm

covering numbers for real valued function classes we find tight upper and lower bounds on the growth rate for the covering numbers of functions of bounded variation in the spl lscr sub 1 metric in terms of all the relevant constants we also find upper and lower bounds on covering numbers for general function classes over the family of spl lscr sub 1 dp metrics in terms of a scale sensitive combinatorial dimension of the function class

bounds on the number of examples needed for learning functions we prove general lower bounds on the number of examples needed for learning function classes within different natural learning models which are related to pac learning and coincide with the pac learning model of valiant in the case of 0 1 valued functions the lower bounds are obtained by showing that all nontrivial function classes contain a hard binary valued subproblem although at first glance it seems to be likely that real valued function classes are much harder to learn than their hardest binary valued subproblem we show that these general lower bounds cannot be improved by more than a logarithmic factor this is done by discussing some natural function classes like nondecreasing functions or piecewise smooth functions the function classes that were discussed in m j kearns and r e schapire proc 31st annual symposium on the foundations of computer science ieee computer society press los alamitos ca 1990 pp 382 392 full version j comput system sci 48 1994 pp 464 497 d kimber and p m long proc 5th annual workshop on computational learning theory acm new york 1992 pp 153 160 with certain restrictions concerning their slope

on the characterization of the extremal points of the unit sphere of matrices abstract we give a characterization of the extremal points of the unit sphere of matrices for the unitarily invariant norms we also investigate the properties of the dual matrices moreover we show that a result given by lau and riha is not correct

multiplicativity of lp norms for matrices ii abstract for 1 p let a p i 1 m j 1 n ij p 1 p be the l p norm of an m n complex a ij c m n the main purpose of this paper is to find for any p q 1 the best smallest possible constants m k n p q and m k n p q for which inequalities of the form ab p m k n p q a p b q ab p m k n p q a q b p hold for all a c m k b c k n this leads to upper bounds for inner products on c k and for ordinary l p operator norms on c m n

faces of the unit ball of a unitarily invariant norm abstract let be a unitarily invariant norm on the space of all n m complex matrices and let g r n r be the symmetric gauge function associated with that is to say we have a g s a for any a where s a is the nonincreasing sequence of singular values of a in this article we consider the relationship between the facial structures of the closed unit balls of g and which we denote by bg and b our main result gives a complete characterization of the faces of b in terms of the faces of bg for with each face e of b we associate a standard face f e of bg i e f e is a face of bg whose barycenter is a nonnegative nonincreasing n vector conversely each standard face f of bg originates a set of n m matrices say the set m f which is proven to be a face of b our main result also asserts that any face e of b is of the form q m fr where q and r are unitary matrices of appropriate orders and f f e we then explore the main result and draw some of its consequences for example we show that the singular values of the barycenter of a face e of b have the same multiplicities as the barycenter of f e formulas are given which relate the dimensions of e and f e the relative interior and the relative boundary of e are characterized in terms of the relative interior and the relative boundary of f e we completely describe the face of b generated by a matrix a our results also cover the case of real matrices

g invariant norms and g c radii abstract let v be a finite dimensional inner product space over f r or c and let g be a closed subgroup of the group of unitary operators on v a norm or a seminorm on v is said to be g invariant if g x x for all g g x v the concept of g invariant norm specializes to many interesting particular cases such as the absolute norms on f n symmetric gauge functions on r n unitarily invariant norms on f m n etc which are of wide research interest in this paper we study the general properties of g invariant norms our main strategy is to study g invariant norms via the g c radius rg c on v defined by r g c x max x g c g g where c v it is shown that the g c radii are very important g invariant seminorms because every g invariant norm or seminorm admits a representation in terms of them as a result one may focus attention on g c radii in order to get results on g invariant norms we study the norm properties of g c radii and obtain various results relating g invariant norms and g c radii the linear operators on v that preserve g invariant norms g invariant seminorms or g c radii are also investigated several open questions are mentioned

multiplicativity factors for c numerical radii abstract for matrices a c c n x n the c numerical radius of a is the nonnegative quantity r c a max tr cu au uunitary this generalizes the classical numerical radius r a it is known that r c constitutes a norm on c n x n if and only if c is nonscalar and tr c 0 for all such c we obtain multiplicativity factors for r c i e constant 0 for which r c is submultiplicative on c n x n

schur products and matrix completions we prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain schur product map defined on a certain subspace of matrices is a positive map by analyzing the positive elements of this subspace we obtain new proofs of the results of h dym and i gohberg and grone johnson sa and wolkowitz linear algebra appl 58 1984 109 124 linear algebra appl 36 1981 1 24 we also obtain a new proof of a result of u haagerup decomposition of completely bounded maps on operation algebras preprint characterizing the norm of schur product maps and a new hahn banach type extension theorem for these maps finally we obtain generalizations of many of these results to matrices of operators which we apply to the study of representations of certain subalgebras of the n n matrices

mixed multiplicativity and lp norms for matrices abstract let cm n denote the class of m n complex matrices and let n1 n2 and n3 be arbitrary norms on cm n cm k and ck n respectively in this paper we discuss the best least positive constant minwhich satisfies n 1 ab min n 2 a n 3 b a c m k b c k n in particular for 1 p let a p i 1 m j 1 n ij p 1 p be the lp norm of a matrix a ij cm n then for arbitrary p q r such that 1 p q r we determine explicitly the best constant min for which ab p min a q b r a c m k b c k n

performance and design of space time coding in fading channels the pairwise error probability upper bounds of space time codes stcs in independent rician fading channels are derived based on the performance analysis novel code design criteria for slow and fast rayleigh fading channels are developed it is found that in fading channels the stc design criteria depend on the value of the possible diversity gain of the system in slow fading channels when the diversity gain is smaller than four the code error performance is dominated by the minimum rank and the minimum determinant of the codeword distance matrix however when the diversity gain is larger than or equal to four the performance is dominated by the minimum squared euclidean distance based on the proposed design criteria new codes are designed and evaluated by simulation

characterization of best approximations in normed linear spaces of matrices by elements of finite dimensional linear subspaces abstract a general characterization theorem of best approximations in normed linear spaces is specialized to the linear space of real n n matrices endowed with the spectral norm

permutation invariant norms abstract a norm or seminorm on on r n is permutation invariant if px x for all n n permutation matrices p and for all x r n we present a systematic study of permutation invariant seminorms their relations with other types of norms on matrices are discussed in addition we consider a special class of permutation invariant seminorms the c radii defined and denoted by r c x max c t px p a permutation for any given c r n it is shown that r c are the building blocks of all permutation invariant seminorms if the entries of c are not all equal and their sum is not 0 then r c is a norm on r n for such r c we study their norm properties and characterize their isometry groups

unitary invariance and spectral variation abstract we call a norm on operators or matrices weakly unitarily invariant if its value at operator a is not changed by replacing a by u au provided only that u is unitary this class includes such norms as the numerical radius we extend to all such norms an inequality that bounds the spectral variation when a normal operator a is replaced by another normal b in terms of the arclength of any normal path from a to b computed using the norm in question related results treat the local metric geometry of the manifold of normal operators we introduce a representation for weakly unitarily invariant matrix norms in terms of function norms over the unit ball and identify this correspondence explicitly in certain cases

the importance of convexity in learning with squared loss we show that if the closure of a function class f under the metric induced by some probability distribution is not convex then the sample complexity for agnostically learning f with squared loss using only hypotheses in f is spl omega ln 1 spl delta spl epsiv sup 2 where 1 spl delta is the probability of success and spl epsiv is the required accuracy in comparison if the class f is convex and has finite pseudodimension then the sample complexity is o 1 spl epsiv ln 1 spl epsiv ln 1 b if a nonconvex class f has finite pseudodimension then the sample complexity for agnostically learning the closure of the convex hull of f is o 1 spl epsiv 1 spl epsiv ln 1 spl epsiv ln 1 spl delta hence for agnostic learning learning the convex hull provides better approximation capabilities with little sample complexity penalty

on the complexity of learning from drifting distributions abstract we consider two models of on line learning of binary valued functions from drifting distributions due to bartlett we show that if each example is drawn from a joint distribution which changes in total variation distance by at mosto e3 d log 1 e between trials then an algorithm can achieve a probability of a mistake at mosteworse than the best function in a class of vc dimensiond we prove a corresponding necessary condition ofo e3 d finally in the case that a fixed function is to be learned from noise free examples we show that if the distributions on the domain generating the examples change by at mosto e2 d log 1 e then any consistent algorithm learns to within accuracye

convex analysis on cartan subspaces in 1937 von neumann 31 gave a famous characterization of unitarily invariant matrix norms that is norms f on cp q satisfying f uxv f x for all unitary matrices u and v and matrices x in cp q his result states that such norms are those functions of the form g where the map x cp q 7 x r has components the singular values 1 x 2 x p x of x assuming p q and g is a norm on rp invariant under sign changes and permutations of components furthermore he showed the respective dual norms satisfy

a space time code design for cpm diversity order and coding gain sufficient conditions are derived under which m ary partial and full response continuous phase modulation cpm space time st codes will attain both full spatial diversity and optimal coding gain general code construction rules are desirable due to the nonlinearity and inherent memory of the cpm signals which makes manual design or computer search difficult using a linear decomposition of cpm signals with tilted phase we identify a rank criterion for m ary partial and full response cpm that specifies the set of allowable modulation indices we also propose a coding gain design criterion optimization of the coding gain for cpm st codes is shown to depend on the cpm frequency phase shaping pulse modulation index and codewords the modulation indices and phase shaping functions that improve the coding gain are specified finally optimization of coding gain for st cpm and orthogonal st cpm codewords is discussed

on the higher numerical radius and spectral norm abstract let c n n be the vector space of n n complex matrices and denote by u n the group of n n unitary matrices in c n n for 1 m n set jm im 0m n whereim is the m m identity matrix and 0k denotes a k k zero matrix with this for ain c n n and 1 m n we define r m a max tr j m uau u u n and a m max tr j m uav u v u n as the m numerical radius and m spectral norm of a respectively when m 1 these quantities are reduced to the classical numerical radius and spectral norm in this note we obtain inequalities concerning m and rm for 1 m n examples are given to show that the inequalities obtained cannot be improved we also study multiplicativity factors for rm 1 m n

properties of the approximate generalized inverses of a class of matrices abstract we investigate the properties of the approximate inverses g of an n 1 by n real matrix d of rank n using any arbitrary vector norm we give an explicit formula for g and we formulate the condition on d under which g is unique the main result is an extension of the research by flores de chela one consider also the approximate lp inverses 1 p of the matrices having the block form in a bt im where denotes the kronecker product m r 1 n s 1 in is the identity matrix of order n a r m r b r s n rank a r and rank b s

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

some aspects of the theory of norms abstract let v be a finite dimensional vector space motivated by theory or applications one might want to consider different kinds of norms on v in this paper we discuss some results and problems involving different classes of norms on a vector space studied by this author in the past few years the paper consists of five sections section 1 concerns the conditions on two vectors x y v satisfying x y for all in a certain class of norms section 2 concerns the isometry groups of g invariant norms i e norms that satisfy g x x for all x v and for all g g where g is a group of unitary orthogonal operators on v section 3 concerns g invariant norms that satisfy some special properties section 4 concerns the best approximation s x 0 t of y where y t v with respect to different kinds of norms additional open problems topics and references are mentioned in section 5

exposed faces and duality for symmetric and unitarily invariant norms abstract let be a unitarily invariant norm on the space of real or complex n m matrices and g the associated symmetric gauge function thus a g s a where s a is the decreasing sequence of singular values of a denote by b and bg the closed unit balls of and g in a previous paper we showed a close relationship between the faces of b and those of bg in particular to each face of b we associated a standard face of bg and we used this association to completely describe the matrices that are members of an individual face of in the present paper we consider the duality operator that transforms each exposed face of b into an exposed face of the unit ball of s dual and the duality operator that does the same with exposed faces of bg we show that these two operators are very nicely related among other results we prove the following given an exposed face e of b the standard face associated with the dual of e is precisely the dual of the standard face of bg associated with e e is an exposed face of b if and only if its associated standard face is an exposed face of bg as a by product we completely determine the subdifferential of in terms of the subdifferential of g and we completely characterize the matrices that are dual to a given matrix a with respect to

duality between some linear preserver problems iii c spectral norms and skew symmetric matrices with fixed singular values abstract let f denote either the complex field c or the real field r let v be sn f or kn f the vector spaces of all n n symmetric and skew symmetric matrices respectively over f for c c1 cn 0 with c1 cn 0 the c spectral norm of a matrix a v is the quantity a c i l n c i i a where 1 a n a are the singular values of a let d d1 dn 0 with d1 dn 0 we study the linear isometries between the normed spaces v c and v d by using the fact that they are dual transformations of the linear operators which map d onto c where c x v x has singular values c 1 c n it is shown that such isometries and hence their dual transformations exist if and only if c and d are scalar multiples of each other in such case we completely determine the structure of such isometries and prove that they and their dual transformations belong to a same class of operators in the proof we obtain characterizations of the extreme points of the unit ball in v for different cases with respect to c which is of independent interest

isometries of norm ideals of compact operators abstract let j be a symmetric norm ideal of compact operators on hilbert space h and assume that the finite rank operators are dense in j and that j is not the ideal of hilbert schmidt operators a linear transformation on j is an isometry of j onto itself if and only if there are unitary operators u and v on h such that either x uxv or x uxtv where xt denotes the transpose of x with respect to a fixed orthonormal basis of h

efficient algorithms for learning functions with bounded variation we show that the class fbv of 0 1 valued functions with total variation at most 1 can be agnostically learned with respect to the absolute loss in polynomial time from o 1 e2 log 1 examples matching a known lower bound to within a constant factor we establish a bound of o 1 m on the expected error of a polynomial time algorithm for learning fbv in the prediction model also matching a known lower bound to within a constant factor applying a known algorithm transformation to our prediction algorithm we obtain a polynomial time pac learning algorithm for fbv with a sample complexity bound of o 1 e log 1 this also matches a known lower bound to within a constant factor

chernoff bounds on pairwise error probabilities of space time codes we derive chernoff bounds on pairwise error probabilities of coherent and noncoherent space time signaling schemes first general chernoff bound expressions are derived for a correlated ricean fading channel and correlated additive gaussian noise then we specialize the obtained results to the cases of space time separable noise white noise and uncorrelated fading we derive approximate chernoff bounds for high and low signal to noise ratios snrs and propose optimal signaling schemes we also compute the optimal number of transmitter antennas for noncoherent signaling with unitary mutually orthogonal space time codes

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

equivalence of models for polynomial learnability abstract in this paper we consider several variants of valiant s learnability model that have appeared in the literature we give conditions under which these models are equivalent in terms of the polynomially learnable concept classes they define these equivalences allow comparisons of most of the existing theorems in valiant style learnability and show that several simplifying assumptions on polynomial learning algorithms can be made without loss of generality we also give a useful reduction of learning problems to the problem of finding consistent hypotheses and give comparisons and equivalences between valiant s model and the prediction learning models of haussler littlestone and warmuth in 29th annual ieee symposium on foundations of computer science 1988

faces and traces of the unit ball of a symmetric gauge function abstract we investigate the faces of the closed unit ball bg x g x 1 of a symmetric gauge function g r n r firstly we describe the group of symmetries of a face f of bg in terms of its barycenter we introduce the concept of standard face of bg i e a face whose relative interior meets the set d x r n x 1 x n 0 called here the standard cone of r n and the concept of trace of a face f which is the intersection of f with d characterizations of standard faces are given and we show that the trace of a face if nonempty is the trace of a standard face it is proved that a face f and its trace have the same dimension iff f is a standard face a relation is found between the dimension of f and the dimension of its symmetry core i e the set of points of f that are fixed by all symmetries of f other results of this type are given the main result has a long and delicate proof it completely characterizes in several different ways the subsets of bg that are traces of faces of bg

on ip1 p2 antinorms of nonnegative matrices let 0 p p1 p2 1 x xi rn a aik rm n i e the vector x is elementwise positive and so is the matrix a the lp antinorm of x is defined by x p ixpi1p the lp1 p2 antinorm of a by a p1 p2 k iap1ikp2p11p2 and the lp1 p2 operator antinorm of a by a minp1 p2 infx rn ax p1 x p2 several properties of the antinorms p p1 p2 and minp1 p2 especially the supermultiplicativity of p1 p2 are studied and generalized to hold for nonnegative matrices the results are analogous with the well known submultiplicativity properties of lq1 q2 norms and lq1 q2 operator norms q1 q2 1 a generalization is outlined

properties of linear approximations of matrices in the spectral norm abstract the properties of linear approximations of a matrix are presented with respect to the spectrum norm we describe the connection between all spectral approximations of a given matrix and we give a sufficient condition under which the spectral approximation is unique

overgroups of some classical linear groups with applications to linear preserver problems abstract the study of linear operators on a matrix space that leave invariant certain functions subsets or relations is commonly referred to as the study of linear preserver problems and has attracted the attention of many mathematicians in the last few decades dynkin in his classic paper studied maximal subgroups of the classical groups and showed how his results may be used to study preserver problems the purpose of this paper is to further exploit this very powerful approach the subgroups g of the general linear group that we deal with are not necessarily maximal for the applications that we have in mind we obtained a description of all possible overgroups of g the results are then applied to various linear preserver problems shorter alternative proofs for various existing results are given and some open questions are answered

design of a class of multirate systems using a maximum relative spl lscr sup 2 error criterion a criterion for designing the class of multirate systems for rate changing is presented this criterion arises from a model matching perspective with maximum relative spl lscr sup 2 error over a general class of inputs and is a natural extension of the standard chebyshev method for filter design using multirate and convex analysis techniques the criterion is shown to lead to a convex matrix valued function approximation problem an algorithm using convex optimization is proposed to solve the problem an example illustrates the use of the algorithm and effectiveness over standard filter design techniques

norms of certain matrices with applications to variations of the spectra of matrices and matrix pencils abstract we start by proving a lower bound for the lp operator norm of a submatrix with sufficiently large dimensions and some relations between the lp norm and the lp operator norm we then study the lp norm and the lp operator norm of certain matrices arising in estimating the eigenvalue generalized eigenvalue singular value and generalized singular value variations of matrices matrix pencils and matrix pairs finally as applications of our bounds upon the lp norm and lp operator norm of certain matrices we give several new bounds for the variations of the spectra of matrices and matrix pencils these new bounds are generalizations of the weyl lidskii theorem for hermitian matrices and the bhatia davis theorem for unitary matrices and of some results obtained by the author recently for diagonalizable pencils with real spectra

joint transmitter receiver optimization for multi input multi output systems with decision feedback the joint transmit receive optimization problem for multiuser communication systems with decision feedback is investigated it is shown that minimization of the geometric mean squared error defined as the determinant of the error covariance matrix leads to a tractable transmitter optimization problem for general multi input multi output decision feedback systems several computational results are included that highlight system performance for a variety of useful transmission scenarios

multiplicativity factors and mixed multiplicativity abstract a number of authors have discussed multiplicativity factors associated with a single norm on an operator algebra we extend this concept to multiplicativity factors associated with three norms on spaces not necessarily algebras of bounded linear operators we generalize previous results and give several finite and infinite dimensional examples

a generalization of spectral radius numerical radius and spectral norm abstract let c n be the linear space of complex column vectors with n coordinates associated with the usual inner product denote by k the set of all k tuples of orthonormal vectors in c n given a real number p 1and an n n complex matrix a with eigenvalues 1 n we define p k max j 1 k i j p 1 p 1 i 1 k n r p k max i 1 k x i ax i p 1 p x 1 x k k and a p k max i 1 k x i ay i p 1 p x 1 x k y 1 y k k as the p k spectral radius p k numerical radius and p k spectral norm of a respectively for k 1 they are the classical spectral radius numerical radius and spectral norm of a in this note we study the matrices for which p k a rp k a p k a a p k or rp k a a p k moreover the norm properties of rp k and p k are investigated

decision theoretic generalizations of the pac model for neural net and other learning applications we describe a generalization of the pac learning model that is based on statistical decision theory in this model the learner receives randomly drawn examples each example consisting of an instance x in x and an outcome y in y and tries to find a hypothesis h x a where h in h that specifies the appropriate action a in a to take for each instance x in order to minimize the expectation of a loss l y a here x y and a are arbitrary sets l is a real valued function and examples are generated according to an arbitrary joint distribution on x times y special cases include the problem of learning a function from x into y the problem of learning the conditional probability distribution on y given x regression and the problem of learning a distribution on x density estimation we give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces h and show how this implies learnability with bounded sample size disregarding computational complexity as an application we give distribution independent upper bounds on the sample size needed for learning with feedforward neural networks our theorems use a generalized notion of vc dimension that applies to classes of real valued functions adapted from pollard s work and a notion of capacity and metric dimension for classes of functions that map into a bounded metric space supersedes 89 30 and 90 52 also in information and computation vol 100 no 1 september 1992

on the jacopini technique the general concern of the jacopini technique is the question is it consistent to extend a given lambda calculus with certain equations the technique was introduced by jacopini in 1975 in his proof that in the untyped lambda calculus omega is easy i e omega can be assumed equal to any other closed term without violating the consistency of the lambda calculus the presentations of the jacopini technique that are known from the literature are difficult to understand and hard to generalise in this paper we generalise the jacopini technique for arbitrary lambda calculi we introduce the concept of proof replaceability by which the structure of the technique is simplified considerably we illustrate the simplicity and generality of our formulation of the technique with some examples we apply the jacopini technique to the lambda mu calculus and we prove a general theorem concerning the consistency of extensions of the lambda mu calculus of a certain form many well known examples e g the easi ness of omega are immediate consequences of this general theorem

predicting 0 1 functions on randomly drawn points we consider the problem of predicting 0 1 valued functions on rn and smaller domains based on their values on randomly drawn points our model is related to valiant s pac learning model but does not require the hypotheses used for prediction to be represented in any specified form in our main result we show how to construct prediction strategies that are optimal to within a constant factor for any reasonable class f of target functions this result is based on new combinatorial results about classes of functions of finite vc dimension we also discuss more computationally efficient algorithms for predicting indicator functions of axis parallel rectangles more general intersection closed concept classes and halfspaces in rn these are also optimal to within a constant factor finally we compare the general performance of prediction strategies derived by our method to that of those derived from methods in pac learning theory

geometrical relations between space time block code designs and complexity reduction in this work the geometric relation between space time block code design for the coherent channel and its noncoherent counterpart is exploited to get an analog of the information theoretic inequality i x s lesi x h s in terms of diversity it provides a lower bound on the performance of noncoherent codes when used in coherent scenarios this leads in turn to a code design decomposition result splitting coherent code design into two complexity reduced subtasks moreover a geometrical criterion for high performance space time code design is derived

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar

the design of trellis coded mpsk for fading channels performance criteria it has been well established that the appropriate criterion for optimum trellis coded modulation design on the additive white gaussian noise channel is maximization of the free euclidean distance it is shown that when the trellis coded modulation is used on a rician fading channel with interleaving deinterleaving the design of the code of optimum performance is guided by other factors in particular the length of the shortest error event path and the product of branch distances possibly normalized by the euclidean distance of the path along the path although maximum free distance d sub free is still an important consideration it plays a less significant role the more severe the fading is on the channel these considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over rician fading channels if no interleaving deinterleaving is used then once again the design of the trellis code is guided by maximizing d sub free