learning eigenfunctions links spectral embedding and kernel pca in this letter we show a direct relation between spectral embedding methods and kernel principal components analysis and how both are special cases of a more general learning problem learning the principal eigenfunctions of an operator defined from a kernel and the unknown data generating density whereas spectral embedding methods provided only coordinates for the training points the analysis justifies a simple extension to out of sample examples the nystrom formula for multidimensional scaling mds spectral clustering laplacian eigenmaps locally linear embedding lle and isomap the analysis provides for all such spectral embedding methods the definition of a loss function whose empirical average is minimized by the traditional algorithms the asymptotic expected value of that loss defines a generalization performance and clarifies what these algorithms are trying to learn experiments with lle isomap spectral clustering and mds show that this out of sample embedding formula generalizes well with a level of error comparable to the effect of small perturbations of the training set on the embedding

block recombination approach for subquadratic space complexity binary field multiplication based on toeplitz matrix vector product in this paper we present a new method for parallel binary finite field multiplication which results in subquadratic space complexity the method is based on decomposing the building blocks of the fan hasan subquadratic toeplitz matrix vector multiplier we reduce the space complexity of their architecture by recombining the building blocks in comparison to other similar schemes available in the literature our proposal presents a better space complexity while having the same time complexity we also show that block recombination can be used for efficient implementation of the ghash function of galois counter mode gcm

concurrent error detection in finite field arithmetic operations using pipelined and systolic architectures in this work we consider detection of errors in polynomial dual and normal bases arithmetic operations error detection is performed by recomputing with the shifted operand method while the operation unit is in use this scheme is efficient for pipelined architectures particularly systolic arrays additionally one semisystolic multiplier for each of the polynomial dual type i and type ii optimal normal bases is presented the results show that for having better or similar space and time overheads compared to a number of related previous work the multipliers have generally a higher error detection capability e g the error detection capability of the reso based scheme for single and multiple stuck at faults in a polynomial basis multiplier is 100 percent finally we also comment on how reso can be used for concurrent error correction to deal with transient faults

embeddings of submanifolds and normal bundles this paper studies the embeddings of a complex submanifold s inside a complex manifold m in particular we are interested in comparing the embedding of s in m with the embedding of s as the zero section in the total space of the normal bundle ns of s in m we explicitly describe some cohomological classes allowing to measure the difference between the two embeddings in the spirit of the work by grauert griffiths and camacho movasati and sad we are also able to explain the geometrical meaning of the separate vanishing of these classes our results hold for any codimension but even for curves in a surface we generalize previous results due to laufert and camacho movasati and sad

an efficient optimal normal basis type ii multiplier this paper presents a new parallel multiplier for the galois field gf 2 sup m whose elements are represented using the optimal normal basis of type ii the proposed multiplier requires 1 5 m sup 2 m xor gates as compared to 2 m sup 2 m xor gates required by the massey omura multiplier the time complexities of the proposed and the massey omura multipliers are similar

a new approach to subquadratic space complexity parallel multipliers for extended binary fields based on toeplitz matrix vector products and coordinate transformation techniques we present a new scheme for subquadratic space complexity parallel multiplication in gf 2n using the shifted polynomial basis both the space complexity and the asymptotic gate delay of the proposed multiplier are better than those of the best existing subquadratic space complexity parallel multipliers for example with n being a power of 2 the space complexity is about 8 percent better while the asymptotic gate delay is about 33 percent better respectively another advantage of the proposed matrix vector product approach is that it can also be used to design subquadratic space complexity polynomial dual weakly dual and triangular basis parallel multipliers to the best of our knowledge this is the first time that subquadratic space complexity parallel multipliers are proposed for dual weakly dual and triangular bases a recursive design algorithm is also proposed for efficient construction of the proposed subquadratic space complexity multipliers this design algorithm can be modified for the construction of most of the subquadratic space complexity multipliers previously reported in the literature

improved area time tradeoffs for field multiplication using optimal normal bases in this paper we propose new schemes for subquadratic arithmetic complexity multiplication in binary fields using optimal normal bases the schemes are based on a recently proposed method known as block recombination which efficiently computes the sum of two products of toeplitz matrices and vectors specifically here we take advantage of some structural properties of the matrices and vectors involved in the formulation of field multiplication using optimal normal bases this yields new space and time complexity results for corresponding bit parallel multipliers

low complexity bit parallel canonical and normal basis multipliers for a class of finite fields we present a new low complexity bit parallel canonical basis multiplier for the field gf 2m generated by an all one polynomial the proposed canonical basis multiplier requires m sup 2 1 xor gates and m sup 2 and gates we also extend this canonical basis multiplier to obtain a new bit parallel normal basis multiplier

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

bit serial and bit parallel montgomery multiplication and squaring over gf 2 m multiplication and squaring are main finite field operations in cryptographic computations and designing efficient multipliers and squarers affect the performance of cryptosystems in this paper we consider the montgomery multiplication in the binary extension fields and study different structures of bit serial and bit parallel multipliers for each of these structures we study the role of the montgomery factor and then by using appropriate factors propose new architectures specifically we propose two bit serial multipliers for general irreducible polynomials and then derive bit parallel montgomery multipliers for two important classes of irreducible polynomials in this regard first we consider trinomials and provide a way for finding efficient montgomery factors which results in a low time complexity then we consider type ii irreducible pentanomials and design two bit parallel multipliers which are comparable to the best finite field multipliers reported in the literature moreover we consider squaring using this family of irreducible polynomials and show that this operation can be performed very fast with the time complexity of two xor gates

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

concurrent error detection and correction in gaussian normal basis multiplier over gf 2 m fault based cryptanalysis has been developed to effectively break both private key and public key cryptosystems making robust finite field multiplication a very important research topic in recent years however no robust normal basis multiplier has been proposed in the literature therefore this investigation presents a semisystolic gaussian normal basis multiplier based on the proposed gaussian normal basis multiplier both concurrent error detection and correction capabilities can be easily achieved using time redundancy technology with no hardware modification

normal bases via general gauss periods gauss periods have been used successfully as a tool for constructing normal bases in finite fields starting from a primitive r th root of unity one obtains under certain conditions a normal basis for mathbb f q n over f q where r is a prime and nk r 1 for some integer k we generalize this construction by allowing arbitrary integers r with nk varphi r and find in many cases smaller values of k than is possible with the previously known approach

a new architecture for a parallel finite field multiplier with low complexity based on composite fields a bit parallel structure for a multiplier with low complexity in galois fields is introduced the multiplier operates over composite fields gf 2 sup n sup m with k nm the karatsuba ofman algorithm a karatsuba and y ofmanis 1963 is investigated and applied to the multiplication of polynomials over gf 2 sup n it is shown that this operation has a complexity of order o k sup log23 under certain constraints regarding k a complete set of primitive field polynomials for composite fields is provided which perform module reduction with low complexity as a result multipliers for fields gf 2 sup k up to k 32 with low gate counts and low delays are listed the architectures are highly modular and thus well suited for vlsi implementation

a generalized method for constructing subquadratic complexity gf 2 sup k multipliers we introduce a generalized method for constructing subquadratic complexity multipliers for even characteristic field extensions the construction is obtained by recursively extending short convolution algorithms and nesting them to obtain the short convolution algorithms the winograd short convolution algorithm is reintroduced and analyzed in the context of polynomial multiplication we present a recursive construction technique that extends any d point multiplier into an n d sup k point multiplier with area that is subquadratic and delay that is logarithmic in the bit length n we present a thorough analysis that establishes the exact space and time complexities of these multipliers using the recursive construction method we obtain six new constructions among which one turns out to be identical to the karatsuba multiplier all six algorithms have subquadratic space complexities and two of the algorithms have significantly better time complexities than the karatsuba algorithm

low complexity word level sequential normal basis multipliers for efficient hardware implementation of finite field arithmetic units the use of a normal basis is advantageous in this paper two classes of architectures for multipliers over the finite field gf 2 sup m are proposed these multipliers are of sequential type i e after receiving the coordinates of the two input field elements they go through k 1 spl les k spl les m iterations i e clock cycles to finally yield all the coordinates of the product in parallel the value of k depends on the word size w spl lceil m k spl rceil for w 1 these multipliers are highly area efficient and require fewer number of logic gates even when compared with the most area efficient multipliers available in the open literature this makes the proposed multipliers suitable for applications where the value of m is large but space is of concern e g resource constrained cryptographic systems additionally if the field dimension m is composite i e m kn then the extension of one class of the architectures yields a highly efficient multiplier over composite fields

structure of parallel multipliers for a class of fields gf 2 m abstract this paper presents a configuration of parallel multipliers for gf 2 m based on canonical bases the possible parallel multipliers by the proposed configuration are limited to a class of fields gf 2 m however they can be constructed by o m 2 and gates and o m 2 eor gates with the structural modularity this is a desirable feature for the hardware implementation and their operation time is about log m t where m is the dimension of gf 2 m and t is the delay time of an eor gate in order to construct such parallel multipliers we define two types of polynomials of special form over gf 2 one is called all one polynomial denoted by aop and the other is called equally spaced polynomial denoted by esp furthermore we show a necessary and sufficient condition for esps to be irreducible over gf 2 and the uniqueness of the irreducible esps over gf 2 finally we propose the configuration of parallel multipliers for a class of fields gf 2 m based on irreducible aops and esps over gf 2

mixtures of probabilistic principal component analyzers principal component analysis pca is one of the most popular techniques for processing compressing and visualizing data although its effectiveness is limited by its global linearity while nonlinear variants of pca have been proposed an alternative paradigm is to capture data complexity by a combination of local linear pca projections however conventional pca does not correspond to a probability density and so there is no unique way to combine pca models therefore previous attempts to formulate mixture models for pca have been ad hoc to some extent in this article pca is formulated within a maximum likelihood framework based on a specific form of gaussian latent variable model this leads to a well defined mixture model for probabilistic principal component analyzers whose parameters can be determined using an expectationmaximization algorithm we discuss the advantages of this model in the context of clustering density modeling and local dimensionality reduction and we demonstrate its application to image compression and handwritten digit recognition

stringing high dimensional data for functional analysis we propose stringing a class of methods where one views high dimensional observations as functional data stringing takes advantage of the high dimension by representing such data as discretized and noisy observations that originate from a hidden smooth stochastic process assuming that the observations result from scrambling the original ordering of the observations of the process stringing reorders the components of the high dimensional vectors followed by transforming the high dimensional vector observations into functional data established techniques from functional data analysis can be applied for further statistical analysis once an underlying stochastic process and the corresponding random trajectory for each subject have been identified stringing of high dimensional data is implemented with distance based metric multidimensional scaling mapping high dimensional data to locations on a real interval such that predictors that are close in a suitable sample metric also are located close to each

subquadratic computational complexity schemes for extended binary field multiplication using optimal normal bases based on a recently proposed toeplitz matrix vector product approach a subquadratic computational complexity scheme is presented for multiplications in binary extended finite fields using type i and ii optimal normal bases

efficient montgomery based semi systolic multiplier for even type gnb of gf 2 m efficient finite field multiplication is crucial for implementing public key cryptosystem to achieve this multipliers using gaussian normal basis have been widely explored in previous works in this paper based on proposed gaussian normal basis montgomery gnbm representation a semi systolic even type gnbm multiplier is developed analysis shows that the proposed architecture saves about 57 percent space complexity and 50 percent time complexity when compared with the only existing semi systolic even type gnb multiplier moreover due to properties of regularity and modularity the proposed multiplier is very suitable for vlsi implementation

full space local topology extraction for cross modal retrieval with the ever increasing availability of various kinds of multimedia data cross modal retrieval which enables information retrieval from various types of data given various types of query has become a research hotspot hashing based techniques have been developed to solve this problem however most previous works cannot capture the shared underlying structure of real world multimodal data which degrades their retrieval performances in this paper we propose a novel hashing method based on the extraction of the common manifold structure shared among different feature spaces to faithfully represent the common structure two kinds of local topology information are exploited in our method local angles are incorporated within the extraction of local topology of each feature space which is then used to learn a common intermediate subspace after heterogeneous features being embedded into this subspace local similarities are exploited to extract the local topology between different feature spaces and learn compact hamming embeddings to facilitate cross modal retrieval the proposed method is referred to as full space local topology extraction for hashing extensive comparisons with other state of the art methods on three benchmark multimedia data sets demonstrate the superiority of our proposed method in terms of retrieval recall and search accuracy

parallel multipliers based on special irreducible pentanomials the state of the art galois field gf 2 sup m multipliers offer advantageous space and time complexities when the field is generated by so special irreducible polynomial to date the best complexity results have been obtained when the irreducible polynomial is either a trinomial or an equally spaced polynomial esp unfortunately there exist only a few irreducible esps in the range of interest for most of the applications e g error correcting codes computer algebra and elliptic curve cryptography furthermore it is not always possible to find an irreducible trinomial of degree m in this range for those cases where neither an irreducible trinomial nor an irreducible esp exists the use of irreducible pentanomials has been suggested irreducible pentanomials are abundant and there are several eligible candidates for a given m we promote the use of two special types of irreducible pentanomials we propose new mastrovito and dual basis multiplier architectures based on these special irreducible pentanomials and give rigorous analyses of their space and time complexity

mass in kahler geometry we prove a simple explicit formula for the mass of any asymptotically locally euclidean ale kahler manifold assuming only the sort of weak fall off conditions required for the mass to actually be well defined for ale scalar flat kahler manifolds the mass turns out to be a topological invariant depending only on the underlying smooth manifold the first chern class of the complex structure and the kahler class of the metric when the metric is actually ae asymptotically euclidean our formula not only implies a positive mass theorem for kahler metrics but also yields a penrose type inequality for the mass

a new construction of massey omura parallel multiplier over gf 2 sup m the massey omura multiplier of gf 2 sup m uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are cyclically shifted from one another in the past it was shown that for a class of finite fields defined by irreducible all one polynomials the parallel massey omura multiplier had redundancy and a modified architecture of lower circuit complexity was proposed in this article it is shown that not only does this type of multiplier contain redundancy in that special class of finite fields but it also has redundancy in fields gf 2 sup m defined by any irreducible polynomial by removing the redundancy we propose a new architecture for the normal basis parallel multiplier which is applicable to any arbitrary finite field and has significantly lower circuit complexity compared to the original massey omura normal basis parallel multiplier the proposed multiplier structure is also modular and hence suitable for vlsi realization when applied to fields defined by the irreducible all one polynomials the multiplier s circuit complexity matches the best result available in the open literature

subquadratic space complexity multiplier using even type gnb based on efficient toeplitz matrix vector product multiplication schemes based on toeplitz matrix vector product tmvp have been proposed by many researchers tmvp can be computed using the recursive two way and three way split methods which are composed of four blocks among them we improve the space complexity of the component matrix formation cmf block this result derives the improvements of multiplication schemes based on tmvp also we present a subquadratic space complexity gf 2 m multiplier with even type gaussian normal basis gnb in order to design the multiplier we formulate field multiplication as a sum of two tmvps and efficiently compute the sum as a result for type 2 and 4 gnbs the proposed multipliers outperform other similar schemes the proposed type 6 gnb is the first subquadrtic space complexity multiplier with its explicit complexity formula

algebraicity of formal varieties and positivity of vector bundles we propose a positivity condition for vector bundles on a projective variety and prove an algebraicity criterion for formal schemes then we apply the algebraicity criterion to the study of formal principle in algebraic geometry

on the convergence of formal equivalence between embeddings let j x w denote the morphism embedding the compact complex manifold x into the manifold w the formal principle holds for j if given any other embedding j x w together with a formal isomorphism between j and j there is an actual isomorphism between j and j for definitions see 2 1 for instance the formal principle holds for exceptional curves of the first kind on surfaces on the other hand arnold 1 recently gave the example of an embedding with a topologically trivial normal bundle of an elliptic curve into a surface for which the formal principle fails a number of articles dealing with the formal principle in algebraic as well as in analytic geometry have been published nirenberg spencer 19 grauert 7 hironaka rossi 16 griffiths 9 10 hironaka 14 hironaka matsumura 15 hartshorne 11 artin 2 gieseker 6 moreover a few weeks after the present work was written commichaugrauert 23 proved that the formal principle holds under an assumption one positivity which seems much weaker than ours we hope that the present article retains some interest because of its method which may work in the algebraic and or singular case we are interested in the complex analytic case in the situation where the normal bundle is fairly positive or non negative this means there are enough submanifolds in w around x we propose a conjecture and then give our result we let d denote the germ of the space of submanifolds near x an open subset in the douady space ud denote the universal submanifold of d x w and lid denote the projection from ud into w

nonlocal estimation of manifold structure we claim and present arguments to the effect that a large class of manifold learning algorithms that are essentially local and can be framed as kernel learning algorithms will suffer from the curse of dimensionality at the dimension of the true underlying manifold this observation invites an exploration of nonlocal manifold learning algorithms that attempt to discover shared structure in the tangent planes at different positions a training criterion for such an algorithm is proposed and experiments estimating a tangent plane prediction function are presented showing its advantages with respect to local manifold learning algorithms it is able to generalize very far from training data on learning handwritten character image rotations where local nonparametric methods fail

modular multiplication without trial division let n 1 we present a method for multiplying two integers called n residues modulo n while avoiding division by n n residues are represented in a nonstandard way so this method is useful only if several computations are done modulo one n the addition and subtraction algorithms are unchanged 1 description some algorithms 1 2 4 5 require extensive modular arith metic we propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms other recent algorithms for modular arithmetic appear in 3 6 fix n 1 define an a residue to be a residue class modulo n select a radix r coprime to n possibly the machine word size or a power thereof such that r n and such that computations modulo r are inexpensive to process let r l and n be integers satisfying 0 n then return t n else return t to validate redc observe mn tn n tmod r so t is an integer also tr tmod n so t tr x mod n thirdly 0 t mn rn rn so 0 t 2n if r and n are large then t mn may exceed the largest double precision value one can circumvent this by adjusting m so r m 0 given two numbers x and y between 0 and n 1 inclusive let z redc xy then z xy r x mod n so xr l yr x zrx mod n also 0 z n so z is the product of x and y in this representation other algorithms for operating on n residues in this representation can be derived from the algorithms normally used the addition algorithm is unchanged since xr x yr x zr x mod n if and only if x y z mod n also unchanged are

graph optimized locality preserving projections locality preserving projections lpp is a typical graph based dimensionality reduction dr method and has been successfully applied in many practical problems such as face recognition however lpp depends mainly on its underlying neighborhood graph whose construction suffers from the following issues 1 such neighborhood graph is artificially defined in advance and thus does not necessary benefit subsequent dr task 2 such graph is constructed using the nearest neighbor criterion which tends to work poorly due to the high dimensionality of original space 3 it is generally uneasy to assign appropriate values for the neighborhood size and heat kernel parameter involved in graph construction to address these problems we develop a novel dr algorithm called graph optimized locality preserving projections golpp the idea is to integrate graph construction with specific dr process into a unified framework which results in an optimized graph rather than predefined one moreover an entropy regularization term is incorporated into the objective function for controlling the uniformity level of the edge weights in graph so that a principled graph updating formula naturally corresponding to conventional heat kernel weights can be obtained finally the experiments on several publicly available uci and face data sets show the feasibility and effectiveness of the proposed method with encouraging results