quantum unique ergodicity for sl2 z h we eliminate the possibility of escape of mass for hecke maass forms of large eigenvalue for the modular group combined with the work of lindenstrauss this establishes the quantum unique ergodicity conjecture of rudnick and sarnak for hecke maass forms on the modular surface sl 2

graph characterization via ihara coefficients the novel contributions of this paper are twofold first we demonstrate how to characterize unweighted graphs in a permutation invariant manner using the polynomial coefficients from the ihara zeta function i e the ihara coefficients second we generalize the definition of the ihara coefficients to edge weighted graphs for an unweighted graph the ihara zeta function is the reciprocal of a quasi characteristic polynomial of the adjacency matrix of the associated oriented line graph since the ihara zeta function has poles that give rise to infinities the most convenient numerically stable representation is to work with the coefficients of the quasi characteristic polynomial moreover the polynomial coefficients are invariant to vertex order permutations and also convey information concerning the cycle structure of the graph to generalize the representation to edge weighted graphs we make use of the reduced bartholdi zeta function we prove that the computation of the ihara coefficients for unweighted graphs is a special case of our proposed method for unit edge weights we also present a spectral analysis of the ihara coefficients and indicate their advantages over other graph spectral methods we apply the proposed graph characterization method to capturing graph class structure and clustering graphs experimental results reveal that the ihara coefficients are more effective than methods based on laplacian spectra

diffusion wavelet embedding a multi resolution approach for graph embedding in vector space abstract in this article we propose a multiscale method of embedding a graph into a vector space using diffusion wavelets at each scale we extract a detail subspace and a corresponding lower scale approximation subspace to represent the graph representative features are then extracted at each scale to provide a scale space description of the graph the lower scale is constructed using a super node merging strategy based on nearest neighbor or maximum participation and the new adjacency matrix is generated using vertex identification this approach allows the comparison of graphs where the important structural differences may be present at varying scales additionally this method can improve the differentiating power of the embedded vectors and this property reduces the possibility of cospectrality typical in spectral methods substantially the experimental results show that augmenting the features of abstract levels to the graph features increases the graph classification accuracies in different datasets

quantum variance and ergodicity for the baker s map we prove an egorov theorem or quantum classical correspondence for the quantised baker s map valid up to the ehrenfest time this yields a logarithmic upper bound for the decay of the quantum variance and as a corollary a quantum ergodic theorem for this map

efficient computation of ihara coefficients using the bell polynomial recursion abstract the ihara zeta function has proved to be a powerful tool in the analysis of graph structures it is determined by the prime cycles of a finite graph g v e and can be characterized in terms of a quasi characteristic polynomial of the adjacency matrix t of the oriented line graph associated to g the coefficients of this polynomial referred to as ihara coefficients have been used to characterize graphs in a permutation invariant manner and allow for an efficient evaluation of the ihara zeta function in this paper we present a novel method for computing the ihara coefficients we first establish a characterization of the ihara coefficients in terms of complete bell polynomials and by exploiting a recursive relation for the latter we show how the ihara coefficients can be efficiently computed in o e 2 provided that the eigenvalues of t are known

theories for mutagenicity a study in first order and feature based induction abstract a classic problem from chemistry is used to test a conjecture that in domains for which data are most naturally represented by graphs theories constructed with inductive logic programming ilp will significantly outperform those using simpler feature based methods one area that has long been associated with graph based or structural representation and reasoning is organic chemistry in this field we consider the problem of predicting the mutagenic activity of small molecules a property that is related to carcinogenicity and an important consideration in developing less hazardous drugs by providing an ilp system with progressively more structural information concerning the molecules we compare the predictive power of the logical theories constructed against benchmarks set by regression neural and tree based methods

entropic bounds on semiclassical measures for quantized one dimensional maps quantum ergodicity asserts that almost all infinite sequences of eigenstates of quantized ergodic hamiltonian systems are equidistributed in phase space this however does not prohibit existence of exceptional sequences which might converge to different non liouville classical invariant measures it has been recently shown by n anantharaman and s nonnenmacher in 20 21 with h koch that for anosov geodesic flows the metric entropy of any semiclassical measure must satisfy a certain bound this remarkable result seems to be optimal for manifolds of constant negative curvature but not in the general case where it might become even trivial if the negative curvature of the riemannian manifold varies a lot it has been conjectured by the same authors that in fact a stronger bound valid in the general case should hold

weight restrictions and free production in data envelopment analysis it is known that the incorporation of weight restrictions in models of data envelopment analysis may result in their infeasibility in our paper we investigate this effect in detail we show that the infeasibility is only one of several possible outcomes that point to a particular problem with weight restrictions for example the use of weight restrictions may also lead to zero or negative efficiency scores of some units removing problematic units from the data set does not necessarily remove the underlying problem caused by the weight restrictions and only makes it undetected we prove that all such problems arise when weight restrictions induce free or unlimited production of outputs in the underlying technology this is unacceptable from the production theory point of view and indicates that the weight restrictions need reassessing we develop analytical criteria and computational methods that allow us to identify the above problematic situations

quantum ergodicity and symmetry reduction abstract we study the spectral and ergodic properties of schrodinger operators on a compact connected riemannian manifold m without boundary in case that the examined system possesses certain symmetries more precisely if m carries an isometric and effective action of a compact connected lie group g we prove a generalized equivariant version of the semiclassical weyl law with an estimate for the remainder using a semiclassical functional calculus for h dependent functions and relying on recent results on singular equivariant asymptotics we then deduce an equivariant quantum ergodicity theorem under the assumption that the symmetry reduced hamiltonian flow on the principal stratum of the singular symplectic reduction of m is ergodic in particular we obtain an equivariant version of the shnirelman zelditch colin de verdiere theorem as well as a representation theoretic equidistribution theorem if m g is an orbifold similar results were recently obtained by kordyukov when g is trivial one recovers the classical results

the eigenvector moment flow and local quantum unique ergodicity we prove that the distribution of eigenvectors of generalized wigner matrices is universal both in the bulk and at the edge this includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries the proof relies on analyzing the eigenvector flow under the dyson brownian motion the key new ideas are 1 the introduction of the eigenvector moment flow a multi particle random walk in a random environment 2 an effective estimate on the regularity of this flow based on maximum principle and 3 optimal finite speed of propagation holds for the eigenvector moment flow with very high probability

hybrid subconvexity bounds for l left frac 1 2 hbox sym 2 f otimes g right fix an integer kappa geqslant 2 let p be prime and let k kappa be an even integer for f a holomorphic cusp form of weight k and full level and g a primitive holomorphic cusp form of weight 2 kappa and level p we prove hybrid subconvexity bounds for l left frac 1 2 text sym 2 f otimes g right in the k and p aspects when p frac 13 64 delta k p frac 3 8 delta for any 0 delta frac 11 128 these bounds are achieved through a first moment method with amplification when p frac 13 64 k leqslant p frac 4 13

scarred eigenstates for quantum cat maps of minimal periods in this paper we construct a sequence of eigenfunctions of the quantum arnold s cat map that in the semiclassical limit shows a strong scarring phenomenon on the periodic orbits of the dynamics more precisely those states have a semiclassical limit measure that is the sum of 1 2 the normalized lebesgue measure on the torus plus 1 2 the normalized dirac measure concentrated on any a priori given periodic orbit of the dynamics it is known the schnirelman theorem that most sequences of eigenfunctions equidistribute on the torus the sequences we construct therefore provide an example of an exception to this general rule our method of construction and proof exploits the existence of special values of h for which the quantum period of the map is relatively short and a sharp control on the evolution of coherent states up to this time scale we also provide a pointwise description of these states in phase space which uncovers their hyperbolic structure in the vicinity of the fixed points and yields more precise localization estimates

l infty norms of eigenfunctions of arithmetic surfaces here and elsewhere a l1o0 as a xc is a basic problem of quantum chaos s in any case almost nothing beyond 0 2 is known about 110 joc when the curvature is negative one can push the standard techniques and replace a1 4 by a1 4 log a in this case in this paper we use arithmetic techniques in particular modular correspondences to obtain the first improvement in the exponent 0 2 we also obtain lower bounds on the l norms in these arithmetic cases in more detail let a b be a quaternion division algebra over q a is linearly generated by 1 w q wq over q and w2 a q2 b wq qw 0 here a b e z are square free and we will assume that a 0 as usual the norm and trace are defined by n a a d tr a a oy where if a xo x1w x2q x3wq

the quantum unique ergodicity conjecture for thin sets abstract we consider some analogs of the quantum unique ergodicity conjecture for geodesics and shrinking families of sets in particular we prove the analog of the que conjecture for eisenstein series restricted to the infinite geodesic connecting 0 and inside the modular surface

unfolding kernel embeddings of graphs in this paper we investigate the use of manifold learning techniques to enhance the separation properties of standard graph kernels the idea stems from the observation that when we perform multidimensional scaling on the distance matrices extracted from the kernels the resulting data tends to be clustered along a curve that wraps around the embedding space a behavior that suggests that long range distances are not estimated accurately resulting in an increased curvature of the embedding space hence we propose to use a number of manifold learning techniques to compute a low dimensional embedding of the graphs in an attempt to unfold the embedding manifold and increase the class separation we perform an extensive experimental evaluation on a number of standard graph datasets using the shortest path borgwardt and kriegel 2005 graphlet shervashidze et al 2009 random walk kashima et al 2003 and weisfeiler lehman shervashidze et al 2011 kernels we observe the most significant improvement in the case of the graphlet kernel which fits with the observation that neglecting the locational information of the substructures leads to a stronger curvature of the embedding manifold on the other hand the weisfeiler lehman kernel partially mitigates the locality problem by using the node labels information and thus does not clearly benefit from the manifold learning interestingly our experiments also show that the unfolding of the space seems to reduce the performance gap between the examined kernels highlightswe use manifold learning techniques to enhance the performance of graph kernels we try to increase the class separation by unfolding the embedding manifold we propose a procedure to find the optimal embedding of the data the kernels which neglect the locational information show the largest improvement the unfolding of the space helps to reduce the performance gap between the kernels

estimates for rankin selberg l functions and quantum unique ergodicity abstract subconvex bounds in the weight aspect for rankin selberg l functions associated to two cusp forms are established these bounds are applied to prove the equidistribution of mass conjecture for cm forms

entropy of semiclassical measures in dimension 2 we study the high energy asymptotic properties of eigenfunctions of the laplacian in the case of a compact riemannian surfacem of anosov type to do this we look at families of distributions associated to them on the cotangent bundle t m and we derive entropic properties on their accumulation points in the high energy limit the so called semiclassical measures we show that the kolmogorov sinai entropy of a semiclassical measure for the geodesic ow gt is bounded from below by half of the ruelle upper bound i e hks g 1 2 s m d where is the upper lyapunov exponent at point

fractal uncertainty principle with explicit exponent we prove an explicit formula for the dependence of the exponent beta in the fractal uncertainty principle of bourgain dyatlov ann math 187 1 43 2018 on the dimension delta and on the regularity constant c r for the regular set in particular this implies an explicit essential spectral gap for convex co compact hyperbolic surfaces when the hausdorff dimension of the limit set is close to 1

enumeration of graphs with the same ihara zeta function abstract we enumerate all connected graphs with minimal vertex degree 2 on at most 11 vertices and determine their ihara zeta functions we also count the number of such graphs for which there is another graph with the same zeta function we then use these graphs to conjecture properties of the graphs determined by the zeta function in addition we study switching constructions proposed by godsil and mckay to determine whether they preserve the zeta function we show that gm switching is not strong enough to preserve the zeta function but gm switching following the notation of haemers and spence does preserve the zeta function

equidistribution of cusp forms in the level aspect let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing square free level q infinity we prove that the pushforward of the mass off to the modular curve of level 1 equidistributes with respect to the poincare measure

equidistribution of eisenstein series in the level aspect we prove an equidistribution property of the eisenstein series for congruence subgroups as the level goes to infinity this is an analogy of the phenomenon called quantum ergodicity

arithmetic and equidistribution of measures on the sphere motivated by problems of mathematical physics quantum chaos questions of equidistribution of eigenfunctions of the laplace operator on a riemannian manifold have been studied by several authors we consider here in analogy with arithmetic hyperbolic surfaces orthonormal bases of eigenfunctions of the laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of hecke operators which act on these spherical harmonics we formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moments tend to moments of the gaussian as the eigenvalue increases for such orthonormal bases we show that these conjectures are related to the analytic properties of degree eight arithmetic l functions associated to triples of eigenfunctions moreover we establish the conjecture for the third moments and give a conditional on standard analytic conjectures about these arithmetic l functions proof of the equidistribution of mass conjecture

quantum unique ergodicity of degenerate eisenstein series on gl n in the area of quantum chaos it is of great interest to study the distribution of the l 2 mass of eigenfunctions of the laplacian as eigenvalues tend to infinity luo and sarnak first formulated and proved arithmetic quantum unique ergodicity for the continuous spectrum spanned by eisenstein series of the hyperbolic laplacian on sl 2 mathbb z backslash mathbb h by utilizing the sub convexity bounds of l functions associated to maass cusp forms in this paper we build on luo and sarnak s method and explore the structure properties of the constant terms of gl n eisenstein series and extend their results to higher ranks we prove quantum unique ergodicity for a subspace of the continuous spectrum spanned by the degenerate eisenstein series on gl n

equidistribution of zeros of holomorphic sections in the non compact setting we consider tensor powers l n of a positive hermitian line bundle l h l over a non compact complex manifold x in the compact case b shiffman and s zelditch proved that the zeros of random sections become asymptotically uniformly distributed as n with respect to the natural measure coming from the curvature of l under certain boundedness assumptions on the curvature of the canonical line bundle of x and on the chern form of l we prove a non compact version of this result we give various applications including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of sl 2 and to the hyperbolic measure the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity we also give estimates for the speed of convergence of the currents of integration on the zero divisors

hecke theory and equidistribution for the quantization of linear maps of the torus 1 1 background one of the key issues of quantum chaos is the nature of the semiclassical limit of eigenstates of classically chaotic systems when the classical system is given by the geodesic flow on a compact riemannian manifoldm or rather on its cotangent bundle one can formulate the problem as follows the quantum hamiltonian is in suitable units represented by the positive laplacian on m to measure the distribution of its eigenstates we start with a smooth classical observable that is a smooth function on the unit cotangent bundle s m via some choice of quantization from symbols to pseudodifferential operators we form its quantization op f this is a zero order pseudodifferential operator with principal symbol f the expectation value of op f in the eigenstate is op f let j be a sequence of normalized eigenfunctions j j j 0 m j 2 1 the problem then is to understand the possible limits as j of the distributions f c s m op f j j 1 1

a hybrid reproducing graph kernel based on information entropy abstract a number of graph kernel based methods have been developed with great success in many fields but very little research has been published that is concerned with a graph kernel in reproducing kernel hilbert space rkhs in this paper we firstly start with a derived expression for two forms of information entropy of an undirected graph they are approximated von neumann entropy and shannon entropy and depend on vertex degree statistics secondly we show the basic solution of a generalized differential operator this solution is a specific reproducing kernel called the h1 reproducing kernel in h1 space and then it is proven to satisfy the condition of mercer kernel thirdly based on the two aforementioned forms of information entropy and h1 reproducing kernel we define two reproducing graph kernels one is approximated von neumann entropy reproducing graph kernel avnergk the other is shannon entropy reproducing graph kernel sergk and then we prove that they satisfy the condition of mercer kernel finally to obtain better classification results we further propose a hybrid reproducing graph kernel hrgk based on the two reproducing graph kernels we use the hrgk as a means to establish the similarity between a pair of graphs experimental results reveal that our method gives better classification performance on graphs extracted from several graph datasets

backtrackless walks on a graph the aim of this paper is to explore the use of backtrackless walks and prime cycles for characterizing both labeled and unlabeled graphs the reason for using backtrackless walks and prime cycles is that they avoid tottering and can increase the discriminative power of the resulting graph representation however the use of such methods is limited in practice because of their computational cost in this paper we present efficient methods for computing graph kernels which are based on backtrackless walks in a labeled graph and whose worst case running time is the same as that of kernels based on random walks for clustering unlabeled graphs we construct feature vectors using ihara coefficients since these coefficients are related to the frequencies of prime cycles in the graph to efficiently compute the low order coefficients we present an o v 3 algorithm which is better than the o v 6 worst case running time of previously known algorithms in the experimental evaluation we apply the proposed method to clustering both labeled and unlabeled graphs the results show that using backtrackless walks and prime cycles instead of random walks can increase the accuracy of recognition

semiclassical limits for the hyperbolic plane we study the concentration properties of high energy eigenfunctions for the laplace beltrami operator of the hyperbolic plane with special consideration of automorphic eigenfunctions at the center of our investigation is the microlocal lift of an eigenfunction to sl 2 introduced by s zelditch the microlocal lift is based on s helgason s fourier transform and has a straightforward description in terms of the lie algebra sl 2 we begin with an elementary demonstration of zelditch s exact differential equation for the microlocal lift further for a sequence of suitably bounded eigenfunctions with eigenvalues tending to infinity we show that the limit of microlocal lifts is a geodesic flow invariant positive measure our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the macdonald bessel functions we find that with a scaling of auxiliary parameters the corresponding high energy limit converges to the positive dirac measure for the lift to sl 2 of a single geodesic on the upper half plane in particular at high energy the microlocal lift of a macdonald bessel function is concentrated along the lift of a single geodesic we further find that the convergence is uniform in parameters and thus our considerations can be applied to study automorphic eigenfunctions in particular the maass cusp forms for certain cofinite subgroups of sl 2 a formula is presented for the microlocal lift of a cusp form as an integral of a family of dirac measures and a measure given directly by sums of products of the classical fourier coefficients of the cusp form the formula establishes that questions on quantum chaos for automorphic eigenfunctions are equivalent to classically stated questions on twisted sums of coefficients in particular we find that the uniform distribution for the microlocal limit is equivalent to a uniform distribution for the limit of twisted coefficient sums we extend the considerations to the case of the nonholomorphic eisenstein series for the modular group we find that a result of w luo and p sarnak is equivalent to a limit sum formula involving the riemann zeta function and the elementary divisor function the formula is suggestive of a formula of s ramanujan

local semicircle law for random regular graphs we consider random d regular graphs on n vertices with degree d at least log n 4 we prove that the green s function of the adjacency matrix and the stieltjes transform of its empirical spectral measure are well approximated by wigner s semicircle law down to the optimal scale given by the typical eigenvalue spacing up to a logarithmic correction aside from well known consequences for the local eigenvalue distribution this result implies the complete isotropic delocalization of all eigenvectors and a probabilistic version of quantum unique ergodicity 2017 wiley periodicals inc

random weighted sobolev inequalities and application to quantum ergodicity this paper is a continuation of poiret et al ann henri poincare 16 651 689 2015 where we studied a randomisation method based on the laplacian with harmonic potential here we extend our previous results to the case of any polynomial and confining potential v on mathbb r d we construct measures under concentration type assumptions on the support of which we prove optimal weighted sobolev estimates on mathbb r d this construction relies on accurate estimates on the spectral function in a non compact configuration space then we prove random quantum ergodicity results without specific assumption on the classical dynamics finally we prove that almost all bases of hermite functions are quantum uniquely ergodic

number variance for arithmetic hyperbolic surfaces we prove that the number variance for the spectrum of an arithmetic surface is highly nonrigid in part of the universal range in fact it is close to having a poisson behavior this fact was discovered numerically by schmit bogomolny georgeot and giannoni it has its origin in the high degeneracy of the length spectrum first observed by selberg

quantisations of piecewise parabolic maps on the torus and their quantum limits for general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand the wignerfunctions of eigenstates converge weakly to invariant measures of the classical system the so called quantum limits and one would like to understand which invariant measures can occur that way thereby classifying the semiclassical behaviour of eigenfunctions

fast depth based subgraph kernels for unattributed graphs in this paper we investigate two fast subgraph kernels based on a depth based representation of graph structure both methods gauge depth information through a family of k layer expansion subgraphs rooted at a vertex 1 the first method commences by computing a centroid based complexity trace for each graph using a depth based representation rooted at the centroid vertex that has minimum shortest path length variance to the remaining vertices 2 this subgraph kernel is computed by measuring the jensen shannon divergence between centroid based complexity entropy traces the second method on the other hand computes a depth based representation around each vertex in turn the corresponding subgraph kernel is computed using isomorphisms tests to compare the depth based representation rooted at each vertex in turn for graphs with n vertices the time complexities for the two new kernels are o n 2 and o n 3 in contrast to o n 6 for the classic gartner graph kernel 3 key to achieving this efficiency is that we compute the required shannon entropy of the random walk for our kernels with o n 2 operations this computational strategy enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state of the art graph kernels experiments on standard bioinformatics and computer vision graph datasets demonstrate the effectiveness and efficiency of our new subgraph kernels highlightswe investigate two fast subgraph kernels based on a depth based representation our kernels gauge depth information rooted at a vertex for a graph the time complexities for the two kernels are o n 2 and o n 3 we evaluate the performance of our subgraph kernels on standard graph datasets we demonstrate the effectiveness of the proposed subgraph kernels

laplacian embedded regression for scalable manifold regularization semi supervised learning ssl as a powerful tool to learn from a limited number of labeled data and a large number of unlabeled data has been attracting increasing attention in the machine learning community in particular the manifold regularization framework has laid solid theoretical foundations for a large family of ssl algorithms such as laplacian support vector machine lapsvm and laplacian regularized least squares laprls however most of these algorithms are limited to small scale problems due to the high computational cost of the matrix inversion operation involved in the optimization problem in this paper we propose a novel framework called laplacian embedded regression by introducing an intermediate decision variable into the manifold regularization framework by using insensitive loss we obtain the laplacian embedded support vector regression lapesvr algorithm which inherits the sparse solution from svr also we derive laplacian embedded rls laperls corresponding to rls under the proposed framework both lapesvr and laperls posses a simpler form of a transformed kernel which is the summation of the original kernel and a graph kernel that captures the manifold structure the benefits of the transformed kernel are two fold 1 we can deal with the original kernel matrix and the graph laplacian matrix in the graph kernel separately and 2 if the graph laplacian matrix is sparse we only need to perform the inverse operation for a sparse matrix which is much more efficient when compared with that for a dense one inspired by kernel principal component analysis we further propose to project the introduced decision variable into a subspace spanned by a few eigenvectors of the graph laplacian matrix in order to better reflect the data manifold as well as accelerate the calculation of the graph kernel allowing our methods to efficiently and effectively cope with large scale ssl problems extensive experiments on both toy and real world data sets show the effectiveness and scalability of the proposed framework

the modulus of continuity for γ0 m semi classical limits we study the behavior of a large eigenvalue limit of eigenfunctions for the hyperbolic laplacian for the modular quotient sl 2 fejer summation and results of s zelditch are used to show that the microlocal lifts of eigenfunctions have large eigenvalue limit a geodesic flow invariant measure for the modular unit cotangent bundle the limit is studied for hecke maass forms joint eigenfunctions of the hecke operators and the hyperbolic laplacian the first modulus of continuity result is presented for the limit the singular concentration set of the limit cannot be a compact union of closed geodesics and measured geodesic laminations

weak subconvexity for central values of l functions we describe a general method to obtain weak subconvexity bounds for many classes of l functions we give several examples of our bound and our work has applications to a conjecture of rudnick and sarnak for the mass equidistribution of hecke eigenforms

pattern vectors from algebraic graph theory graph structures have proven computationally cumbersome for pattern analysis the reason for this is that before graphs can be converted to pattern vectors correspondences must be established between the nodes of structures which are potentially of different size to overcome this problem in this paper we turn to the spectral decomposition of the laplacian matrix we show how the elements of the spectral matrix for the laplacian can be used to construct symmetric polynomials that are permutation invariants the coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner we extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a hermitian property matrix that can be viewed as a complex analogue of the laplacian to embed the graphs in a pattern space we explore whether the vectors of invariants can be embedded in a low dimensional space using a number of alternative strategies including principal components analysis pca multidimensional scaling mds and locality preserving projection lpp experimentally we demonstrate that the embeddings result in well defined graph clusters our experiments with the spectral representation involve both synthetic and real world data the experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure the real world experiments show that the method can be used to locate clusters of graphs

hierarchical graph embedding in vector space by graph pyramid abstract loss of information is the major challenge in graph embedding in vector space which reduces the impact of representational power of graphs in pattern recognition tasks the objective of this article is to present a hierarchical framework which can decrease this loss in a reasonable computational time inspired by multi resolution ideas in image processing a graph pyramid is formed based on a selected graph summarization algorithm which can provide the required information for classification all the pyramid levels or some of them are embedded into a vector through an available embedding method which constructs an informative description containing both local and global features the experiments are conducted on graphs with numerical and categorical attributes in the numerical case a proposed summarization algorithm is applied while in the categorical case k snap graph summarization is applied the results indicate that this new framework is efficient in terms of accuracy and time consumption in the context of classification problems it is observed that this improvement is achieved regardless of selected embedding techniques

a quantum jensen shannon graph kernel for unattributed graphs in this paper we use the quantum jensen shannon divergence as a means of measuring the information theoretic dissimilarity of graphs and thus develop a novel graph kernel in quantum mechanics the quantum jensen shannon divergence can be used to measure the dissimilarity of quantum systems specified in terms of their density matrices we commence by computing the density matrix associated with a continuous time quantum walk over each graph being compared in particular we adopt the closed form solution of the density matrix introduced in rossi et al 2013 27 28 to reduce the computational complexity and to avoid the cumbersome task of simulating the quantum walk evolution explicitly next we compare the mixed states represented by the density matrices using the quantum jensen shannon divergence with the quantum states for a pair of graphs described by their density matrices to hand the quantum graph kernel between the pair of graphs is defined using the quantum jensen shannon divergence between the graph density matrices we evaluate the performance of our kernel on several standard graph datasets from both bioinformatics and computer vision the experimental results demonstrate the effectiveness of the proposed quantum graph kernel highlightswe compute a density matrix for a graph using the continuous time quantum walk we compute the quantum jensen shannon divergence between graph density matrixes we define a quantum jensen shannon graph kernel using the quantum divergence we evaluate the performance of our quantum kernel on standard graph datasets we demonstrate the effectiveness of the proposed quantum kernel

energy level statistics of the hadamard gutzwiller ensemble abstract the statistical properties of the quantal energy levels of the hadamard gutzwiller ensemble whose classical members belong to the class of systems with hard chaos are investigated based on a sample of 4500 energy levels it is shown that the short range statistics as nearest neighbour spacing distributions are governed by the goe predictions of random matrix theory which was first surmised by wigner and by landau and smorodinsky for nuclear level statistics this result strengthens the hypothesis that quantum systems with chaotic classical counterpart display level repulsion as predicted by random matrix theory however the level statistics describing correlations over greater level distances deviate from the goe predictions which is explained as a simple consequrnce of the fact that the spectral rigidity 3 l introduced by dyson and mehta saturates non universally at a finite value for l in complete agreement with the semiclassical theory developed by berry

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus we look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribu tion in the semiclassical limit we construct super scars that are stable under the arithmetic symmetries of the system and localize on invariant manifolds we show that these super scars exist only when there are isotropic rational subspaces invariant under the linear map in the case where there are no such scars we com pute the variance of the fluctuations of the matrix elements for the desymmetrized system and present a conjecture for their limiting distributions

on the approximate normality of eigenfunctions of the laplacian the main result of this paper is a bound on the distance between the distribution of an eigenfunction of the laplacian on a compact riemannian manifold and the gaussian distribution if x is a random point on a manifold m and f is an eigenfunction of the laplacian with l 2 norm one and eigenvalue then d tv f x z 2 e f x 2 f x 2 this result is applied to construct specific examples of spherical harmonics of arbitrary odd degree which are close to gaussian in distribution a second application is given to random linear combinations of eigenfunctions on flat tori

discriminative graph embedding for label propagation in many applications the available information is encoded in graph structures this is a common problem in biological networks social networks web communities and document citations we investigate the problem of classifying nodes labels on a similarity graph given only a graph structure on the nodes conventional machine learning methods usually require data to reside in some euclidean spaces or to have a kernel representation applying these methods to nodes on graphs would require embedding the graphs into these spaces by embedding and then learning the nodes on graphs most methods are either flexible with different learning objectives or efficient enough for large scale applications we propose a method to embed a graph into a feature space for a discriminative purpose our idea is to include label information into the embedding process making the space representation tailored to the task we design embedding objective functions that the following learning formulations become spectral transforms we then reformulate these spectral transforms into multiple kernel learning problems our method while being tailored to the discriminative tasks is efficient and can scale to massive data sets we show the need of discriminative embedding on some simulations applying to biological network problems our method is shown to outperform baselines

a polynomial characterization of hypergraphs using the ihara zeta function the aim of this paper is to seek a compact characterization of irregular unweighted hypergraphs for the purposes of clustering to this end we develop a polynomial characterization for hypergraphs based on the ihara zeta function we investigate the flexibility of the polynomial coefficients for learning relational structures with different relational orders furthermore we develop an efficient method for computing the coefficient set our representation for hypergraphs takes into account not only the vertex connections but also the hyperedge cardinalities and thus can distinguish different relational orders which is prone to ambiguity if the hypergraph laplacian is used in our experimental evaluation we demonstrate the effectiveness of the proposed characterization for clustering irregular unweighted hypergraphs and its advantages over the spectral characterization of the hypergraph laplacian

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom

coined quantum walks lift the cospectrality of graphs and trees in this paper we explore how a spectral technique suggested by coined quantum walks can be used to distinguish between graphs that are cospectral with respect to standard matrix representations the algorithm runs in polynomial time and moreover can distinguish many graphs for which there is no subexponential time algorithm that is proven to be able to distinguish between them in the paper we give a description of the coined quantum walk from the field of quantum computing the evolution of the walk is governed by a unitary matrix we show how the spectrum of this matrix is related to the spectrum of the transition matrix of the classical random walk however despite this relationship the behaviour of the quantum walk is vastly different from the classical walk this leads us to define a new matrix based on the amplitudes of paths of the walk whose spectrum we use to characterise graphs we carry out three sets of experiments using this matrix representation firstly we test the ability of the spectrum to distinguish between sets of graphs that are cospectral with respect to standard matrix representation these include strongly regular graphs and incidence graphs of balanced incomplete block designs bibds secondly we test our method on all regular graphs on up to 14 vertices and all trees on up to 24 vertices this demonstrates that the problem of cospectrality is often encountered with conventional algorithms and tests the ability of our method to resolve this problem thirdly we use distances obtained from the spectra of s u 3 to cluster graphs derived from real world image data and these are qualitatively better than those obtained with the spectra of the adjacency matrix thus we provide a spectral representation of graphs that can be used in place of standard spectral representations far less prone to the problems of cospectrality