z4 kerdock codes orthogonal spreads and extremal euclidean line sets when m is odd spreads in an orthogonal vector space of type omega 2m 2 2 are related to binary kerdock codes and extremal line sets in rr 2 m 1 with prescribed angles spreads in a 2m dimensional binary symplectic vector space are related to kerdock codes over zz 4 and extremal line sets in cc 2 m with prescribed angles these connections involve binary real and complex geometry associated with extraspecial 2 groups a geometric map from symplectic to orthogonal spreads is shown to induce the gray map from a corresponding zz 4 kerdock code to its binary image these geometric considerations lead to the construction for any odd composite m of large numbers of zz 4 kerdock codes they also produce new zz 4 linear kerdock and preparata codes 1991 mathematics subject classification primary 94b60 secondary 51m15 20c99

minimization of a closed loop response to a fixed input for siso systems it is shown that the problem of minimizing a regulated response of a single input single output system due to a fixed bounded input can be converted via polynomial techniques to a linear infinite dimensional chebyshev data fitting problem approximating feasible solutions within any specified degree of accuracy can be obtained by converting the original problem into a sequence of increasingly large finite dimensional chebyshev approximation problems for which solution stable and efficient numerical methods exist a direct formula for calculating tight upper bounds to the approximation error is provided the link between the present algebraic approach and the dahleh and pearson functional analytic one 1988 is also discussed

minimization of a regulated response to a fixed input the problem of minimizing a regulated output due to a specific bounded input is solved in the siso single input single output discrete time case it is shown that in most situations the optimal solution will not exist but given any epsilon 0 it is possible to find a suboptimal solution with performance within epsilon of the optimal performance the proposed approach is extended to the problem of shaping the error transient response it is shown how time domain templates can be used in this problem in the same way that frequency weighting functions are used in the h sup infinity and h sup 2 problems

evenly distributed unitaries on the structure of unitary designs we clarify the mathematical structure underlying unitary t designs these are sets of unitary matrices evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group we present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design lower bounds for the number of elements of 2 designs are derived we show how to turn mutually unbiased bases into approximate 2 designs whose cardinality is optimal in leading order designs of higher order are discussed and an example of a unitary 5 design is presented we comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups further we sketch connections to problems in linear optics and questions regarding typical entanglement

emergence of phase and shift invariant features by decomposition of natural images into independent feature subspaces olshausen and field 1996 applied the principle of independence maximization by sparse coding to extract features from natural images this leads to the emergence of oriented linear filters that have simultaneous localization in space and in frequency thus resembling gabor functions and simple cell receptive fields in this article we show that the same principle of independence maximization can explain the emergence of phase and shift invariant features similar to those found in complex cells this new kind of emergence is obtained by maximizing the independence between norms of projections on linear subspaces instead of the independence of simple linear filter outputs the norms of the projections on such independent feature subspaces then indicate the values of invariant features

kerdock codes determine unitary 2 designs the non linear binary kerdock codes are known to be gray images of certain extended cyclic codes of length n 2 m over mathbb z 4 we show that exponentiating these mathbb z 4 valued codewords by i triangleq sqrt 1 produces stabilizer states that are quantum states obtained using only clifford unitaries these states are also the common eigenvectors of commuting hermitian matrices forming maximal commutative subgroups mcs of the pauli group we use this quantum description to simplify the derivation of the classical weight distribution of kerdock codes next we organize the stabilizer states to form n 1 mutually unbiased bases and prove that automorphisms of the kerdock code permute their corresponding mcs thereby forming a subgroup of the clifford group when represented as symplectic matrices this subgroup is isomorphic to the projective special linear group psl 2 n we show that this automorphism group acts transitively on the pauli matrices which implies that the ensemble is pauli mixing and hence forms a unitary 2 design the kerdock design described here was originally discovered by cleve et al 2016 but the connection to classical codes is new which simplifies its description and translation to circuits significantly sampling from the design is straightforward the translation to circuits uses only clifford gates and the process does not require ancillary qubits finally we also develop algorithms for optimizing the synthesis of unitary 2 designs on encoded qubits i e to construct logical unitary 2 designs software implementations are available at https github com nrenga symplectic arxiv18a which we use to provide empirical gate complexities for up to 16 qubits

quantum data hiding we expand on our work on quantum data hiding hiding classical data among parties who are restricted to performing only local quantum operations and classical communication locc we review our scheme that hides one bit between two parties using bell 1964 states and we derive upper and lower bounds on the secrecy of the hiding scheme we provide an explicit bound showing that multiple bits can be hidden bitwise with our scheme we give a preparation of the hiding states as an efficient quantum computation that uses at most one ebit of entanglement a candidate data hiding scheme that does not use entanglement is presented we show how our scheme for quantum data hiding can be used in a conditionally secure quantum bit commitment scheme

noise contrastive estimation of unnormalized statistical models with applications to natural image statistics we consider the task of estimating from observed data a probabilistic model that is parameterized by a finite number of parameters in particular we are considering the situation where the model probability density function is unnormalized that is the model is only specified up to the partition function the partition function normalizes a model so that it integrates to one for any choice of the parameters however it is often impossible to obtain it in closed form gibbs distributions markov and multi layer networks are examples of models where analytical normalization is often impossible maximum likelihood estimation can then not be used without resorting to numerical approximations which are often computationally expensive we propose here a new objective function for the estimation of both normalized and unnormalized models the basic idea is to perform nonlinear logistic regression to discriminate between the observed data and some artificially generated noise with this approach the normalizing partition function can be estimated like any other parameter we prove that the new estimation method leads to a consistent convergent estimator of the parameters for large noise sample sizes the new estimator is furthermore shown to behave like the maximum likelihood estimator in the estimation of unnormalized models there is a trade off between statistical and computational performance we show that the new method strikes a competitive trade off in comparison to other estimation methods for unnormalized models as an application to real data we estimate novel two layer models of natural image statistics with spline nonlinearities

design of spl lscr sub 1 optimal controllers with flexible disturbance rejection level this note presents a new design methodology that allows for flexible management of the tradeoff between the ability of a system to attenuate disturbance signals versus its expected worst peak to peak amplification the proposed strategy applies to linear time invariant systems which are subject to persistent disturbance signals and combines a recently developed quasi robust linear programming concept with a well known spl lscr sub 1 optimal controller synthesis approach the benefit of the resulting technique is demonstrated using an example

estimation of non normalized statistical models by score matching one often wants to estimate statistical models where the probability density function is known only up to a multiplicative normalization constant typically one then has to resort to markov chain monte carlo methods or approximations of the normalization constant here we propose that such models can be estimated by minimizing the expected squared distance between the gradient of the log density given by the model and the gradient of the log density of the observed data while the estimation of the gradient of log density function is in principle a very difficult non parametric problem we prove a surprising result that gives a simple formula for this objective function the density function of the observed data does not appear in this formula which simplifies to a sample average of a sum of some derivatives of the log density given by the model the validity of the method is demonstrated on multivariate gaussian and independent component analysis models and by estimating an overcomplete filter set for natural image data

a generalized gaussian image model for edge preserving map estimation the authors present a markov random field model which allows realistic edge modeling while providing stable maximum a posterior map solutions the model referred to as a generalized gaussian markov random field ggmrf is named for its similarity to the generalized gaussian distribution used in robust detection and estimation the model satisfies several desirable analytical and computational properties for map estimation including continuous dependence of the estimate on the data invariance of the character of solutions to scaling of data and a solution which lies at the unique global minimum of the a posteriori log likelihood function the ggmrf is demonstrated to be useful for image reconstruction in low dosage transmission tomography

shorter stabilizer circuits via bruhat decomposition and quantum circuit transformations in this paper we improve the layered implementation of arbitrary stabilizer circuits introduced by aaronson and gottesman in phys rev a 70 052328 2004 to implement a general stabilizer circuit we reduce their 11 stage computation h c p c p c h p c p c over the gate set consisting of hadamard controlled not and phase gates into a 7 stage computation of the form c cz p h p cz c we show arguments in support of using cz stages over the c stages not only the use of cz stages allows a shorter layered expression but cz stages are simpler and appear to be easier to implement compared to the c stages based on this decomposition we develop a two qubit gate depth 14n 4 implementation of stabilizer circuits over the gate library mathrm h mathrm p mathrm cnot executable in the linear nearest neighbor lnn architecture improving best previously known depth 25n circuit also executable in the lnn architecture our constructions rely on bruhat decomposition of the symplectic group and on folding arbitrarily long sequences of the form p c m into a three stage computation p cz c our results include the reduction of the 11 stage decomposition h c p c p c h p c p c into a 9 stage decomposition of the form c p c p h c p c p this reduction is based on the bruhat decomposition of the symplectic group this result also implies a new normal form for stabilizer circuits we show that a circuit in this normal form is optimal in the number of hadamard gates used we also show that the normal form has an asymptotically optimal number of parameters

minimum overshoot design for siso discrete time systems the problem of designing a controller to minimize the maximum negative error due to a specific bounded input is studied it is found that the solution can be obtained by solving an infinite linear programming problem in the special case of a step input a simple analytical formula for minimum overshoot is derived for a plant with one unstable pole and one nonminimum phase zero the solution technique has potential application to other control problems which seek to minimize quantities not directly representable as a norm

young diagrammatic methods for the representation theory of the classical groups of type bn cn dn since the appearance of the celebrated article the classical groups of weyl 7 numerous influences have occurred in the representation theory of the classical groups especially it has become folklore that the facts in this field should be expressed only in terms of young diagrams the aim of this paper is one of the realizations of this principle a classical model of our approach is the results for gl n which are known as littlewood richardson coefficients and the kostka coefficients these two coefficients can be stated in the combinatorial manner using only young diagrams independently of the rank n of gl n we generalize these results to the other classical groups over c namely we give explicit formulas to decompose the tensor product of two irreducible representations into irreducible constituents also we have obtained an interesting relationship linking the irreducible characters of sp and so which we call duality relation between the representations of sp and so as for explicit formulas for the multiplicities of weights in the form of polynomials in the rank n we refer to 3

technical communique optimization based design of fixed order controllers for command following for discrete time scalar systems we propose an approach for designing feedback controllers of fixed order to minimize an upper bound on the peak magnitude of the tracking error to a given command input the work makes use of linear programming to design over a class of closed loop systems recently proposed for the rejection of non zero initial conditions and bounded disturbances we incorporate performance robustness in the form of a guaranteed upper bound on the peak magnitude of the tracking error under plant coprime factor uncertainty

bayesian model comparison with the hyvarinen score computation and consistency the bayes factor is a widely used criterion in model comparison and its logarithm is a difference of out of sample predictive scores under the logarithmic scoring rule however when some of the ca

unitary 2 designs from random x and z diagonal unitaries unitary 2 designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries often referred to as haar random unitaries they are used in a wide variety of theoretical and practical quantum information protocols and also have been used to model the dynamics in complex quantum many body systems here we show that unitary 2 designs can be approximately implemented by alternately repeating random unitaries diagonal in the pauli z basis and pauli x basis we also provide a converse about the number of repetitions needed to achieve unitary 2 designs these results imply that the process after l repetitions achieves a d l approximate unitary 2 design based on the construction we further provide quantum circuits that efficiently implement approximate unitary 2 designs although a more efficient implementation of unitary 2 designs is known our quantum circuit has its own merit that it is divided into a constant number of commuting parts which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time we finally interpret the result in terms of the dynamics generated by time dependent hamiltonians and provide for the first time a random disordered time dependent hamiltonian that generates a unitary 2 design after switching interactions only a few times unitary 2 designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries often referred to as haar random unitaries they are used in a wide variety of theoretical and practical quantum information protocols and also have been used to model the dynamics in complex quantum many body systems here we show that unitary 2 designs can be approximately implemented by alternately repeating random unitaries diagonal in the pauli z basis and pauli x basis we also provide a converse about the number of repetitions needed to achieve unitary 2 designs these results imply that the process after l repetitions achieves a d l approximate unitary 2 design based on the construction we further provide quantum circuits that efficiently implement approximate unitary 2 designs although a more efficient implementation of unitary 2 designs is known our quantum circuit has its own merit that it is divided into a constant number of commuting parts which

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

how to efficiently select an arbitrary clifford group element we give an algorithm which produces a unique element of the clifford group on n qubits cn from an integer 0 i cn the number of elements in the group the algorithm involves o n 3 operations and provides in addition to a canonical mapping from the integers to group elements g a factorization of g into a sequence of at most 4n symplectic transvections the algorithm can be used to efficiently select random elements of cn which are often useful in quantum information theory and quantum computation we also give an algorithm for the inverse map indexing a group element in time o n 3

optimal symplectic householder transformations for sr decomposition the sr factorization is a key step for some important structure preserving eigenproblems in this work we introduce symplectic householder transformations and show their main features furthermore we construct a new algorithm for computing the sr factorization based on these transformations the new algorithm is the analogous of the classical qr factorization via householder transformations unlike the latter it involves free parameters we show how to choose these parameters in an optimal way and derive an algorithm implementing this some illustrating numerical experiments comparisons and computational aspects are given

energy based models for sparse overcomplete representations we present a new way of extending independent components analysis ica to overcomplete representations in contrast to the causal generative extensions of ica which maintain marginal independence of sources we define features as deterministic linear functions of the inputs this assumption results in marginal dependencies among the features but conditional independence of the features given the inputs by assigning energies to the features a probability distribution over the input states is defined through the boltzmann distribution free parameters of this model are trained using the contrastive divergence objective hinton 2002 when the number of features is equal to the number of input dimensions this energy based model reduces to noiseless ica and we show experimentally that the proposed learning algorithm is able to perform blind source separation on speech data in additional experiments we train overcomplete energy based models to extract features from various standard data sets containing speech natural images hand written digits and faces

fundamental limitations on the time domain shaping of response to a fixed input in this paper we examine time domain limitation of performance problems in feedback control system design the main result is a theorem which gives a dual formulation to the problem of determining absolute limits on the time domain shaping of the response to a fixed input the duals that arise are particularly illuminating they have a simple interpretation as optimizing the impulse response of a transfer function whose poles are readily constructed from the data much insight is obtainable from the dual for example in helping to classify classes of plants with the same quantitative behavior new results on classical performance measures such as overshoot and undershoot are presented we also examine minimization of the difference between the maximum and minimum values of the error response a quantity we term fluctuation

a novel parameter estimation method for boltzmann machines we propose a novel estimator for a specific class of probabilistic models on discrete spaces such as the boltzmann machine the proposed estimator is derived from minimization of a convex risk function and can be constructed without calculating the normalization constant whose computational cost is exponential order we investigate statistical properties of the proposed estimator such as consistency and asymptotic normality in the framework of the estimating function small experiments show that the proposed estimator can attain comparable performance to the maximum likelihood expectation at a much lower computational cost and is applicable to high dimensional data

a hierarchical bayesian model for learning nonlinear statistical regularities in nonstationary natural signals capturing statistical regularities in complex high dimensional data is an important problem in machine learning and signal processing models such as principal component analysis pca and independent component analysis ica make few assumptions about the structure in the data and have good scaling properties but they are limited to representing linear statistical regularities and assume that the distribution of the data is stationary for many natural complex signals the latent variables often exhibit residual dependencies as well as nonstationary statistics here we present a hierarchical bayesian model that is able to capture higher order nonlinear structure and represent nonstationary data distributions the model is a generalization of ica in which the basis function coefficients are no longer assumed to be independent instead the dependencies in their magnitudes are captured by a set of density components each density component describes a common pattern of deviation from the marginal density of the pattern ensemble in different combinations they can describe nonstationary distributions adapting the model to image or audio data yields a nonlinear distributed code for higher order statistical regularities that reflect more abstract invariant properties of the signal

primal methods are better than dual methods for solving overdetermined linear systems in the l sense it is generally believed that methods based upon solving the dual of the linear programming formulation of the linear l infty problem are superior to methods based upon the primal formulation a primal approach is presented along with evidence that for random problems primal methods are to be preferred it is then shown that the method generally considered to be the best algorithm for discrete linear approximation in the l infty norm actually owes much of its efficiency to effective choice of a starting point this phenomenon is explained with the aid of results from classical minimax theory which are then used to determine a suitable initial point for a primal method with this enhancement it would appear that the primal approach is also preferable for problems arising from function approximations

optimizing quantum process tomography with unitary 2 designs we show that weighted unitary 2 designs define optimal measurements on the system ancilla output state for ancilla assisted process tomography of unital quantum channels examples include complete sets of mutually unbiased unitary operator bases each of these specifies a minimal series of optimal orthogonal measurements general quantum channels are also considered

control system design via infinite linear programming in this paper we deal with the problem of designing non overshooting feedback control systems when the input is a step we also consider the more general problem of designing controllers to track a step optimally with some predetermined amount of allowable overshoot these problems are cast as infinite linear programming problems we show that both these problems always have a solution under some standard assumptions it is then shown that the counterpart of this problem that is the problem of designing non undershooting feedback control systems need not have a solution in general but it is proved that one can always design a feedback controller to achieve as little percentage undershoot as desired finally it is shown that the problem of designing controllers to simultaneously satisfy bounds on the overshoot and undershoot always has a solution if some undershoot is allowed for the duals to the infinite linear programming problems are formulated and the absense of a duality gap is established fo

training products of experts by minimizing contrastive divergence it is possible to combine multiple latent variable models of the same data by multiplying their probability distributions together and then renormalizing this way of combining individual expert models makes it hard to generate samples from the combined model but easy to infer the values of the latent variables of each expert because the combination rule ensures that the latent variables of different experts are conditionally independent when given the data a product of experts poe is therefore an interesting candidate for a perceptual system in which rapid inference is vital and generation is unnecessary training a poe by maximizing the likelihood of the data is difficult because it is hard even to approximate the derivatives of the renormalization term in the combination rule fortunately a poe can be trained using a different objective function called contrastive divergence whose derivatives with regard to the parameters can be approximated accurately and efficiently examples are presented of contrastive divergence learning using several types of expert on several types of data

a linear non gaussian acyclic model for causal discovery in recent years several methods have been proposed for the discovery of causal structure from non experimental data such methods make various assumptions on the data generating process to facilitate its identification from purely observational data continuing this line of research we show how to discover the complete causal structure of continuous valued data under the assumptions that a the data generating process is linear b there are no unobserved confounders and c disturbance variables have non gaussian distributions of non zero variances the solution relies on the use of the statistical method known as independent component analysis and does not require any pre specified time ordering of the variables we provide a complete matlab package for performing this lingam analysis short for linear non gaussian acyclic model and demonstrate the effectiveness of the method using artificially generated data and real world data

blind separation of mixture of independent sources through a quasi maximum likelihood approach we propose two methods for separating mixture of independent sources without any precise knowledge of their probability distribution they are obtained by considering a maximum likelihood ml solution corresponding to some given distributions of the sources and relaxing this assumption afterward the first method is specially adapted to temporally independent non gaussian sources and is based on the use of nonlinear separating functions the second method is specially adapted to correlated sources with distinct spectra and is based on the use of linear separating filters a theoretical analysis of the performance of the methods has been made a simple procedure for optimally choosing the separating functions is proposed further in the second method a simple implementation based on the simultaneous diagonalization of two symmetric matrices is provided finally some numerical and simulation results are given illustrating the performance of the method and the good agreement between the experiments and the theory

beyond independent components trees and clusters we present a generalization of independent component analysis ica where instead of looking for a linear transform that makes the data components independent we look for a transform that makes the data components well fit by a tree structured graphical model this tree dependent component analysis tca provides a tractable and flexible approach to weakening the assumption of independence in ica in particular tca allows the underlying graph to have multiple connected components and thus the method is able to find clusters of components such that components are dependent within a cluster and independent between clusters finally we make use of a notion of graphical models for time series due to brillinger 1996 to extend these ideas to the temporal setting in particular we are able to fit models that incorporate tree structured dependencies among multiple time series

a technique for choosing zero locations for minimal overshoot an approach is presented to the specification of optimal overshoot controllers when the controller order is fixed and the closed loop system poles are at a fixed location the essential effect of the technique is to pick the best possible locations of the free zeros of the system in terms of minimizing the overshoot the solution is obtained by solving an affine minimax optimization problem an example illustrates how this technique can be used as a design aid when it is necessary to trade off conflicting design requirements such as overshoot and settling time specifications it is shown that the method is not restricted to the deadbeat case but can also be used for any prescribed set of closed loop poles

simultaneous estimation of nongaussian components and their correlation structure the statistical dependencies that independent component analysis ica cannot remove often provide rich information beyond the linear independent components it would thus be very useful to estimate the dependency structure from data while such models have been proposed they have usually concentrated on higher order correlations such as energy square correlations yet linear correlations are a fundamental and informative form of dependency in many real data sets linear correlations are usually completely removed by ica and related methods so they can only be analyzed by developing new methods that explicitly allow for linearly correlated components in this article we propose a probabilistic model of linear nongaussian components that are allowed to have both linear and energy correlations the precision matrix of the linear components is assumed to be randomly generated by a higher order process and explicitly parameterized by a parameter matrix the estimation of the parameter matrix is shown to be particularly simple because using score matching hyvarinen 2005 the objective function is a quadratic form using simulations with artificial data we demonstrate that the proposed method improves the identifiability of nongaussian components by simultaneously learning their correlation structure applications on simulated complex cells with natural image input as well as spectrograms of natural audio data show that the method finds new kinds of dependencies between the components

model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data we consider the problem of estimating the parameters of a gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse our approach is to solve a maximum likelihood problem with an added l1 norm penalty term the problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes we present two new algorithms for solving problems with at least a thousand nodes in the gaussian case our first algorithm uses block coordinate descent and can be interpreted as recursive l1 norm penalized regression our second algorithm based on nesterov s first order method yields a complexity estimate with a better dependence on problem size than existing interior point methods using a log determinant relaxation of the log partition function wainwright and jordan 2006 we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case we test our algorithms on synthetic data as well as on gene expression and senate voting records data

random quantum circuits are approximate 2 designs given a universal gate set on two qubits it is well known that applying random gates from the set to random pairs of qubits will eventually yield an approximately haar distributed unitary however this requires exponential time we show that random circuits of only polynomial length will approximate the first and second moments of the haar distribution thus forming approximate 1 and 2 designs previous constructions required longer circuits and worked only for specific gate sets as a corollary of our main result we also improve previous bounds on the convergence rate of random walks on the clifford group

topographic independent component analysis in ordinary independent component analysis the components are assumed to be completely independent and they do not necessarily have any meaningful order relationships in practice however the estimated independent components are often not at all independent we propose that this residual dependence structure could be used to define a topographic order for the components in particular a distance between two components could be defined using their higher order correlations and this distance could be used to create a topographic representation thus we obtain a linear decomposition into approximately independent components where the dependence of two components is approximated by the proximity of the components in the topographic representation