subspace averaging and order determination for source enumeration in this paper we address the problem of subspace averaging with special emphasis placed on the question of estimating the dimension of the average the results suggest that the enumeration of sources in a multi sensor array which is a problem of estimating the dimension of the array manifold and as a consequence the number of radiating sources may be cast as a problem of averaging subspaces this point of view stands in contrast to conventional approaches which cast the problem as one of identifiying covariance models in a factor model we present a robust formulation of the proposed order fitting rule based on majorization minimization algorithms a key element of the proposed method is to construct a bootstrap procedure based on a newly proposed discrete distribution on the manifold of projection matrices for stochastically generating subspaces from a function of experimentally determined eigenvalues in this way the proposed subspace averaging sa technique determines the order based on the eigenvalues of an average projection matrix rather than on the likelihood of a covariance model penalized by functions of the model order by means of simulation examples we show that the proposed sa criterion is especially effective in high dimensional scenarios with low sample support

markovianizing cost of tripartite quantum states we introduce and analyze a task that we call markovianization in which a tripartite quantum state is transformed to a quantum markov chain by a randomizing operation on one of the three subsystems we consider cases where the initial state is the tensor product of n copies of a tripartite state rho abc and is transformed to a quantum markov chain conditioned by b n with a small error using a random unitary operation on a n in an asymptotic limit of infinite copies and vanishingly small error we analyze the markovianizing cost that is the minimum cost of randomness per copy required for markovianization for tripartite pure states we derive a single letter formula for the markovianizing costs counterintuitively the markovianizing cost is not a continuous function of states and can be arbitrarily large even if the state is close to a quantum markov chain our results have an application in analyzing the cost of resources for simulating a bipartite unitary gate by local operations and classical communication

a bayesian approach to geometric subspace estimation this paper presents a geometric approach to estimating subspaces as elements of the complex grassmann manifold with each subspace represented by its unique complex projection matrix variation between the subspaces is modeled by rotating their projection matrices via the action of unitary matrices elements of the unitary group u n subspace estimation or tracking then corresponds to inferences on u n taking a bayesian approach a posterior density is derived on u n and certain expectations under this posterior are empirically generated for the choice of the hilbert schmidt norm on u n to define estimation errors an optimal mmse estimator is derived it is shown that this estimator achieves a lower bound on the expected squared errors associated with all possible estimators the estimator and the bound are computed using metropolis adjusted langevin s diffusion algorithm for sampling from the posterior for use in subspace tracking a prior model on subspace rotation that utilizes newtonian dynamics is suggested

hilbert schmidt lower bounds for estimators on matrix lie groups for atr deformable template representations of observed imagery model the variability of target pose via the actions of the matrix lie groups on rigid templates in this paper we study the construction of minimum mean squared error estimators on the special orthogonal group so n for pose estimation due to the nonflat geometry of so n the standard bayesian formulation of optimal estimators and their characteristics requires modifications by utilizing hilbert schmidt metric defined on gl n a larger group containing so n a mean squared criterion is defined on so n the hilbert schmidt estimate hse is defined to be a minimum mean squared error estimator restricted to so n the expected error associated with the hse is shown to be a lower bound called the hilbert schmidt bound hsb on the error incurred by any other estimator analysis and algorithms are presented for evaluating the hse and the hsb in cases of both ground based and airborne targets

a robust comparison of biological shapes summary localized differences in the form of two related animal skeletons are more effectively determined when resistant fitting techniques are used rather than least squares the repeated median resistant fitting algorithm is introduced the methods are tested by comparing hominid skulls with those of other primates and the male ostracode with the female in each case the resistant fit is superior to the least squares fit in that the difierences are more readily identified and agl ee more closely with the structural differences perceived on biological grounds le llltroductioll studies of allometric growth and of morphological change often run into difficulties in identifying and measuring the location and extent of relative shape deformations a reference system is needed for measuring these deformations and in the absence of an inherent fixed reference system one must be estimated the problem of estimating shape change in animal morphology provides an elegant setting for the comparison of least squares with recently developed robust and resistant statistical techniques being robust these new methods are protected against departures from the assumptions of analysis such as independent identically and normally distributed errors being resistant they are also protected against the potentially strong influences of atypical or incorrect data values two shapes can rarely be superimposed perfectly different fitting criteria will generally yield different results by allowing regions with large deformations to have a large impact on the fit least squares methods can minimize true shape differences and thereby obscure them a resistant technique however limits the influence of large deformations and the resulting fit is close in similar regions and not close in relatively deformed regions in this way resistant techniques can help to identify similarities and diflerences in form more effectively than least squares methods when sneath 1967 used least squares as a basis for some of the first quantitative estimates of relative deformation and so extended the fundamental descriptive work of thompson 1917 useful robust methods were not generally available sneath did in fact consider using the available robust methods but concluded that alternative methods that are less affected by wild values are not attractive in practice since then however the development of new theory and methods as reported by huber 1972 1981 hampel

quantization on the grassmann manifold the problem of quantization in an euclidean space with unitary constraints can be formulated as an unconstrained problem on a grassmann manifold such constraints arise in areas such as wireless communication with multiple antennas at the transmitter and at the receiver due to the constraints the distortion rate analysis developed for euclidean spaces cannot be applied directly this paper extends gersho s asymptotic large rate small distortion distortion bounds to the case when the source is distributed on the complex grassmann manifold the special structure of the grassmann manifold and the distortion measures defined on it differentiate this problem from the traditional vector quantization in euclidean spaces

principal geodesic analysis for the study of nonlinear statistics of shape a primary goal of statistical shape analysis is to describe the variability of a population of geometric objects a standard technique for computing such descriptions is principal component analysis however principal component analysis is limited in that it only works for data lying in a euclidean vector space while this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points it does not handle more complex representations of shape we have been developing representations of geometry based on the medial axis description or m rep while the medial representation provides a rich language for variability in terms of bending twisting and widening the medial parameters are not elements of a euclidean vector space they are in fact elements of a nonlinear riemannian symmetric space in this paper we develop the method of principal geodesic analysis a generalization of principal component analysis to the manifold setting we demonstrate its use in describing the variability of medially defined anatomical objects results of applying this framework on a population of hippocampi in a schizophrenia study are presented

general formulas for capacity of classical quantum channels the capacity of a classical quantum channel or in other words the classical capacity of a quantum channel is considered in the most general setting where no structural assumptions such as the stationary memoryless property are made on a channel a capacity formula as well as a characterization of the strong converse property is given just in parallel with the corresponding classical results of verdu han 1994 which are based on the so called information spectrum method the general results are applied to the stationary memoryless case with or without cost constraint on inputs whereby a deep relation between the channel coding theory and the hypothesis testing for two quantum states is elucidated

intrinsic analysis of statistical estimation the parametric statistical models with suitable regularity conditions have a natural riemannian manifold structure given by the information metric since the parameters are merely labels for the probability measures an inferential statement should be formulated through intrinsic objects invariant under reparametrizations in this context the estimators will be random objects valued on the manifold corresponding to the statistical model in spite of these considerations classical measures of an estimator s performance like the bias and the mean square error are clearly dependent on the statistical model parametrizations in this paper the authors work with extended notions of mean value and moments of random objects which take values on a hausdorff and connected manifold equipped with an affine connection in particular the riemannian manifold case is considered this extension is applied to the bias and the mean square error study in statistical point estimation theory under this approach an intrinsic version of the cramer rao lower bound is obtained a lower bound which depends on the intrinsic bias and the curvature of the statistical model for the mean square of the rao distance the invariant measure analogous to the mean square error further the behavior of the mean square of the rao distance of an estimator when conditioning with respect to a sufficient statistic is considered obtaining intrinsic versions of the rao blackwell and lehmann scheffe theorems asymptotic properties complete the study

quantum rate distortion coding with auxiliary resources we extend quantum rate distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression the first setting we consider is that of quantum rate distortion coding with the help of a classical side channel our result here is that the regularized entanglement of formation characterizes the quantum rate distortion function extending earlier work of devetak and berger we also combine this bound with the entanglement assisted bound from our prior work to obtain the best known bounds on the quantum rate distortion function for an isotropic qubit source the second setting we consider is that of quantum rate distortion coding with quantum side information qsi available to the receiver in order to prove results in this setting we first state and prove a quantum reverse shannon theorem with qsi for tensor power states which extends the known tensor power quantum reverse shannon theorem the achievability part of this theorem relies on the quantum state redistribution protocol while the converse relies on the fact that the protocol can cause only a negligible disturbance to the joint state of the reference and the receiver s qsi this quantum reverse shannon theorem with qsi naturally leads to quantum rate distortion theorems with qsi with or without entanglement assistance

large sample theory of intrinsic and extrinsic sample means on manifolds ii this article develops nonparametric inference procedures for estimation and testing problems for means on manifolds a central limit theorem for frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on riemannian manifolds central limit theorems are also obtained for extrinsic sample means w r t an arbitrary embedding of a differentiable manifold in a euclidean space bootstrap methods particularly suitable for these problems are presented applications are given to distributions on the sphere s d directional spaces real projective space rp n 1 axial spaces complex projective space cp k 2 planar shape spaces w r t veronese whitney embeddings and a three dimensional shape space 4 3

covariance subspace and intrinsic crame spl acute r rao bounds crame spl acute r rao bounds on estimation accuracy are established for estimation problems on arbitrary manifolds in which no set of intrinsic coordinates exists the frequently encountered examples of estimating either an unknown subspace or a covariance matrix are examined in detail the set of subspaces called the grassmann manifold and the set of covariance positive definite hermitian matrices have no fixed coordinate system associated with them and do not possess a vector space structure both of which are required for deriving classical crame spl acute r rao bounds intrinsic versions of the crame spl acute r rao bound on manifolds utilizing an arbitrary affine connection with arbitrary geodesics are derived for both biased and unbiased estimators in the example of covariance matrix estimation closed form expressions for both the intrinsic and flat bounds are derived and compared with the root mean square error rmse of the sample covariance matrix scm estimator for varying sample support k the accuracy bound on unbiased covariance matrix estimators is shown to be about 10 log 10 n k sup 1 2 db where n is the matrix order remarkably it is shown that from an intrinsic perspective the scm is a biased and inefficient estimator and that the bias term reveals the dependency of estimation accuracy on sample support observed in theory and practice the rmse of the standard method of estimating subspaces using the singular value decomposition svd is compared with the intrinsic subspace crame spl acute r rao bound derived in closed form by varying both the signal to noise ratio snr of the unknown p dimensional subspace and the sample support in the simplest case the crame spl acute r rao bound on subspace estimation accuracy is shown to be about p n p sup 1 2 1 2 snr sup 1 2 rad for p dimensional subspaces it is seen that the svd based method yields accuracies very close to the crame spl acute r rao bound establishing that the principal invariant subspace of a random sample provides an excellent estimator of an unknown subspace the analysis approach developed is directly applicable to many other estimation problems on manifolds encountered in signal processing and elsewhere such as estimating rotation matrices in computer vision and estimating subspace basis vectors in blind source separation

on certain linear mappings between inner product and squared distance matrices abstract we obtain the spectral decomposition of four linear mappings the first is a mapping of the linear hull of all centered inner product matrices onto the linear hull of all the induced squared distance matrices it is based on the natural generalization of the cosine law of elementary euclidean geometry the other three mappings studied are 1 the adjoint and 1 extensions and applications particularly to multidimensional scaling are discussed in some detail

an upper bound on the second order asymptotic expansion for the quantum communication cost of state redistribution state redistribution is the protocol in which given an arbitrary tripartite quantum state with two of the subsystems initially being with alice and one being with bob the goal is for alice to send one of her subsystems to bob possibly with the help of prior shared entanglement we derive an upper bound on the second order asymptotic expansion for the quantum communication cost of achieving state redistribution with a given finite accuracy in proving our result we also obtain an upper bound on the quantum communication cost of this protocol in the one shot setting by using the protocol of coherent state merging as a primitive

trading classical communication quantum communication and entanglement in quantum shannon theory in this paper we give tradeoffs between classical communication quantum communication and entanglement for processing information in the shannon theoretic setting we first prove a unit resource capacity theorem that applies to the scenario where only the above three noiseless resources are available for consumption or generation the optimal strategy mixes the three fundamental protocols of teleportation superdense coding and entanglement distribution we then provide an achievable rate region and a matching multiletter converse for the direct static capacity theorem this theorem applies to the scenario where a large number of copies of a noisy bipartite state are available in addition to consumption or generation of the above three noiseless resources our coding strategy involves a protocol that we name the classically assisted state redistribution protocol and the three fundamental protocols we finally provide an achievable rate region and a matching multiletter converse for the direct dynamic capacity theorem this theorem applies to the scenario where a large number of uses of a noisy quantum channel are available in addition to the consumption or generation of the three noiseless resources our coding strategy combines the classically enhanced father protocol with the three fundamental unit protocols

tangent bundle maps on the grassmann manifold application to empirical arithmetic averaging the present paper elaborates on tangent bundle maps on the grassmann manifold with application to subspace arithmetic averaging in particular the present contribution elaborates on the work about retraction lifting maps devised for the stiefel manifold in the recently published paper t kaneko s fiori and t tanaka empirical arithmetic averaging over the compact stiefel manifold ieee trans signal process vol 61 no 4 pp 883 894 february 2013 and discusses the extension of such maps to the grassmann manifold tangent bundle maps are devised on the basis of the thin qr matrix decomposition the polar matrix decomposition and the exponential map also tangent bundle pseudo maps based on the matrix cayley transform are devised theoretical and numerical comparisons about the devised tangent bundle maps are performed in order to get an insight into their relative merits and demerits with special emphasis to their computational burden the averaging algorithm based on the thin qr decomposition maps stands out as it exhibits the best trade off between numerical precision and computational burden such algorithm is further compared with two grassmann averaging algorithms drawn from the scientific literature on an handwritten digits recognition data set the thin qr tangent bundle maps based algorithm exhibits again numerical features that make it preferable over such algorithms

faithful squashed entanglement squashed entanglement is a measure for the entanglement of bipartite quantum states in this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states this implies that squashed entanglement is faithful that is it is strictly positive if and only if the state is entangled we derive the lower bound on squashed entanglement from a lower bound on the quantum conditional mutual information which is used to define squashed entanglement the quantum conditional mutual information corresponds to the amount by which strong subadditivity of von neumann entropy fails to be saturated our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality the proof is based on two recent results from quantum information theory the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing the distance to the set of separable states is measured in terms of the locc norm an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states each shared by two parties using any protocol formed by local quantum operations and classical communication locc between the parties a similar result for the frobenius or euclidean norm follows as an immediate consequence the result has two applications in complexity theory the first application is a quasipolynomial time algorithm solving the weak membership problem for the set of separable states in locc or euclidean norm the second application concerns quantum merlin arthur games here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to locc operations thereby providing a new characterisation of the complexity class qma

a resource framework for quantum shannon theory quantum shannon theory is loosely defined as a collection of coding theorems such as classical and quantum source compression noisy channel coding theorems entanglement distillation etc which characterize asymptotic properties of quantum and classical channels and states in this paper we advocate a unified approach to an important class of problems in quantum shannon theory consisting of those that are bipartite unidirectional and memoryless

renyi squashed entanglement discord and relative entropy differences the squashed entanglement quantifies the amount of entanglement in a bipartite quantum state and it satisfies all of the axioms desired for an entanglement measure the quantum discord is a measure of quantum correlations that are different from those due to entanglement what these two measures have in common is that they are both based upon the conditional quantum mutual information in berta et al 2015 j math phys 56 022205 we recently proposed renyi generalizations of the conditional quantum mutual information of a tripartite state on abc with c being the conditioning system which were shown to satisfy some properties that hold for the original quantity such as non negativity duality and monotonicity with respect to local operations on the system b with it being left open to show that the renyi quantity is monotone with respect to local operations on system a here we define a renyi squashed entanglement and a renyi quantum discord based on a renyi conditional quantum mutual information and investigate these quantities in detail taking as a conjecture that the renyi conditional quantum mutual information is monotone with respect to local operations on both systems a and b we prove that the renyi squashed entanglement and the renyi quantum discord satisfy many of the properties of the respective original von neumann entropy based quantities in our prior work berta et al 2015 phys rev a 91 022333 we also detailed a procedure to obtain renyi generalizations of any quantum information measure that is equal to a linear combination of von neumann entropies with coefficients chosen from the set 1 0 1 here we extend this procedure to include differences of relative entropies using the extended procedure and a conjectured monotonicity of the renyi generalizations in the renyi parameter we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy joint convexity of the relative entropy and the holevo bound

strong converse theorems using renyi entropies we use a renyi entropy method to prove strong converse theorems for certain information theoretic tasks which involve local operations and quantum or classical communication between two parties these include state redistribution coherent state merging quantum state splitting measurement compression with quantum side information randomness extraction against quantum side information and data compression with quantum side information the method we employ in proving these results extends ideas developed by sharma preprint arxiv 1404 5940 quant ph 2014 which he used to give a new proof of the strong converse theorem for state merging for state redistribution we prove the strong converse property for the boundary of the entire achievable rate region in the e q plane where e and q denote the entanglement cost and quantum communication cost respectively in the case of measurement compression with quantum side information we prove a strong converse theorem for the classical communication cost which is a new result extending the previously known weak converse for the remaining tasks we provide new proofs for strong converse theorems previously established using smooth entropies for each task we obtain the strong converse theorem from explicit bounds on the figure of merit of the task in terms of a renyi generalization of the optimal rate hence we identify candidates for the strong converse exponents for each task discussed in this paper to prove our results we establish various new entropic inequalities which might be of independent interest these involve conditional entropies and mutual information derived from the sandwiched renyi divergence in particular we obtain novel bounds relating these quantities as well as the renyi conditional mutual information to the fidelity of two quantum states

robust multidimensional matched subspace classifiers based on weighted least squares we propose and design two classes of robust subspace classifiers for classification of multidimensional signals our classifiers are based on robust m estimators and the least median of squares principle and we show that they may be unified as iterated reweighted oblique subspace classifiers the performance of the proposed classifiers are demonstrated by two examples noncoherent detection of space time frequency shift keying signals and shape classification of partially occluded two dimensional 2 d objects in both cases the proposed robust subspace classifiers outperform the conventional subspace classifiers

the fidelity of recovery is multiplicative fawzi and renner recently established a lower bound on the conditional quantum mutual information cqmi of tripartite quantum states abc in terms of the fidelity of recovery for i e the maximal fidelity of the state abc with a state reconstructed from its marginal bc by acting only on the c system the for measures quantum correlations by the local recoverability of global states and has many properties similar to the cqmi here we generalize the for and show that the resulting measure is multiplicative by utilizing semi definite programming duality this allows us to simplify an operational proof by brandao et al of the above mentioned lower bound that is based on quantum state redistribution in particular in contrast to the previous approaches our proof does not rely on de finetti reductions

shape procrustes tangent projections and bilateral symmetry summary in shape analysis with concentrated data the procrustes tangent projection of shapes plays a fundamental role since this projection can be used to convert shape analysis to standard multivariate analysis in view of its importance a systematic and rigorous treatment of the procrustes tangent projection is developed here to facilitate its use in applications one important application involves an analytical description of bilateral symmetry of objects together with some related distributional results an explicit expression for the tangent projection is also derived

bounds on entanglement distillation and secret key agreement for quantum broadcast channels the squashed entanglement of a quantum channel is an additive function of quantum channels which finds application as an upper bound on the rate at which secret key and entanglement can be generated when using a quantum channel a large number of times in addition to unlimited classical communication this quantity has led to an upper bound of log 1 eta 1 eta on the capacity of a pure loss bosonic channel for such a task where eta is the average fraction of photons that make it from the input to the output of the channel the purpose of this paper is to extend these results beyond the single sender single receiver setting to the more general case of a single sender and multiple receivers a quantum broadcast channel we employ multipartite generalizations of the squashed entanglement to constrain the rates at which secret key and entanglement can be generated between any subset of the users of such a channel along the way developing several new properties of these measures we apply our results to the case of a pure loss broadcast channel with one sender and two receivers

spherical principal curves this paper presents a new approach for dimension reduction of data observed on spherical surfaces several dimension reduction techniques have been developed in recent years for non euclidean data analysis as a pioneer work hauberg 2016 attempted to implement principal curves on riemannian manifolds however this approach uses approximations to process data on riemannian manifolds resulting in distorted results this study proposes a new approach to project data onto a continuous curve to construct principal curves on spherical surfaces our approach lies in the same line of hastie and stuetzle 1989 that proposed principal curves for data on euclidean space we further investigate the stationarity of the proposed principal curves that satisfy the self consistency on spherical surfaces the results on the real data analysis and simulation examples show promising empirical characteristics of the proposed approach

random tensor theory extending random matrix theory to mixtures of random product states we consider a problem in random matrix theory that is inspired by quantum information theory determining the largest eigenvalue of a sum of p random product states in mathbb c d otimes k where k and p d k are fixed while d when k 1 the marcenko pastur law determines up to small corrections not only the largest eigenvalue 1 sqrt p d k 2 but the smallest eigenvalue min 0 1 sqrt p d k 2 and the spectral density in between we use the method of moments to show that for k 1 the largest eigenvalue is still approximately 1 sqrt p d k 2 and the spectral density approaches that of the marcenko pastur law generalizing the random matrix theory result to the random tensor case our bound on the largest eigenvalue has implications both for sampling from a particular heavy tailed distribution and for a recently proposed quantum data hiding and correlation locking scheme due to leung and winter

capacity theorems for quantum multiple access channels classical quantum and quantum quantum capacity regions in this paper we consider quantum channels with two senders and one receiver for an arbitrary such channel we give multiletter characterizations of two different two dimensional capacity regions the first region comprises the rates at which it is possible for one sender to send classical information while the other sends quantum information the second region consists of the rates at which each sender can send quantum information for each region we give an example of a channel for which the corresponding region has a single letter description one of our examples relies on a new result proved here perhaps of independent interest stating that the coherent information over any degradable channel is concave in the input density operator we conclude with connections to other work and a discussion on generalizations where each user simultaneously sends classical and quantum information

the geometry of shape spaces kendall introduced shape spaces which classified n tuples of points in r k and explained their role for statistics the aim of this paper is to study such shape spaces and in particular to explore the corresponding spaces for points on spheres the ultimate purpose of the work is to be able to apply statistical techniques to observations where only the shape of the observations is significant the geometry of the shape space is critical in determining the type of statistical technique which one should use therefore the major part of this paper is devoted to an examination of the geometry and the statistics is only introduced in a simple case when considering random processes on the shape space

a weighted procrustes criterion the procrustes criterion is a common measure for the distance between two matricesx andy and can be interpreted as the sum of squares of the euclidean distances between their respective column vectors often a weighted procrustes criterion using for example a weighted sum of the squared distances between the column vectors is called for this paper describes and analyzes the performance of an algorithm for rotating a matrixx such that the column weighted procrustes distance toy is minimized the problem of rotatingx intoy such that an aggregate measure of tucker s coefficient of congruence is maximized is also discussed

the cost of randomness for converting a tripartite quantum state to be approximately recoverable we introduce and analyze a task in which a tripartite quantum state is transformed to an approximately recoverable state by a randomizing operation on one of the three subsystems we consider cases where the initial state is a tensor product of n copies of a tripartite state rho abc and is transformed by a random unitary operation on a n to another state which is approximately recoverable from its reduced state on a n b n case 1 or b n c n case 2 we analyze the minimum cost of randomness per copy required for the task in an asymptotic limit of infinite copies and vanishingly small error of recovery mainly focusing on the case of pure states we prove that the minimum cost in case 1 is equal to the markovianizing cost of the state for which a single letter formula is known with an additional requirement on the convergence speed of the recovery error we prove that the minimum cost in case 2 is also equal to the markovianizing cost our results have an application for distributed quantum computation

smooth entropy bounds on one shot quantum state redistribution in quantum state redistribution as introduced by luo and devetak and devetak and yard there are four systems of interest the a system held by alice the b system held by bob the c system that is to be transmitted from alice to bob and the r system that holds a purification of the state in the abc registers we give upper and lower bounds on the amount of quantum communication and entanglement required to perform the task of quantum state redistribution in a one shot setting our bounds are in terms of the smooth conditional min and max entropy and the smooth max information the protocol for the upper bound has a clear structure building on the work of oppenheim it decomposes the quantum state redistribution task into two simpler coherent state merging tasks by introducing a coherent relay in the independent and identical i i d asymptotic limit our bounds for the quantum communication cost converge to the quantum conditional mutual information i c r b and our bounds for the total cost converge to the conditional entropy h c b this yields an alternative proof of optimality of these rates for quantum state redistribution in the i i d asymptotic limit in particular we obtain a strong converse for quantum state redistribution which even holds when allowing for feedback

multipartite generalization of quantum discord a generalization of quantum discord to multipartite systems is proposed a key feature of our formulation is its consistency with the conventional definition of discord in bipartite systems it is by construction zero only for systems with classically correlated subsystems and is a non negative quantity giving a measure of the total nonclassical correlations in the multipartite system with respect to a fixed measurement ordering for the tripartite case we show that the discord can be decomposed into contributions resulting from changes induced by nonclassical correlation breaking measurements in the conditional mutual information and tripartite mutual information the former gives a measure of the bipartite nonclassical correlations and is a non negative quantity while the latter is related to the monogamy of the nonclassical correlations

stieltjes integral representation and effective diffusivity bounds for turbulent transport a stieltjes integral representation for the effective diffusivity in turbulent transport is developed this formula is valid for all peclet numbers and yields a rigorous resummation of the divergent perturbation series in peclet number provided that all diagrams are computed exactly another consequence of the integral representation is convergent upper and lower bounds on effective diffusivity for all peclet numbers utilizing a prescribed finite number of terms in their perturbation series

eigenshape analysis of a cut grow mapping for triangles and its application to phyllotaxis in plants phyllotaxis the regular arrangement of leaves in plants has been extensively studied geometrically and through theories of mechanism recent observations of the biophysics of cell growth and division at the plant apex during leaf formation lead to a cut and grow algorithm in which a newly formed leaf is first cut off from the meristem and then grows rapidly deforming the adjacent tissue to a new configuration the properties of the sequence of shapes constructed by a cut grow map of triangles are discussed the cut grow map excises a triangular segment of a triangle and dilates the newly formed edge in shape space the cut grow map is equivalent to a reflection followed by inversion each map has a conjugate pair of eigenshapes in the limiting case when the eigenshapes are equal the shapes converge with successive iterations at rate 1 n however an arbitrarily small perturbation from the eigenshape can initialize a divergent sequence of arbitrary length before the shapes begin to converge

successive refinement of information the successive refinement of information consists of first approximating data using a few bits of information then iteratively improving the approximation as more and more information is supplied the goal is to achieve an optimal description at each stage in general an ongoing description which is rate distortion optimal whenever it is interrupted is sought it is shown that in order to achieve optimal successive refinement the necessary and sufficient conditions are that the solutions of the rate distortion problem can be written as a markov chain in particular all finite alphabet signals with hamming distortion satisfy these requirements it is also shown that the same is true for gaussian signals with squared error distortion and for laplacian signals with absolute error distortion a simple counterexample with absolute error distortion and a symmetric source distribution which shows that successive refinement is not always achievable is presented

diffusion advection in cellular flows with large peclet numbers for periodic two dimensional cellular flows in incompressible fluids we provide explicit upper and lower estimates of the effective diffusivity which have the correct scaling behavior for large peclet numbers we demonstrate that all allowed scaling laws can occur the bounds prove that there is no residual diffusion in the infinite peclet number limit for holder continuous flows answering a problem posed by kozlov

distributed compression of correlated classical quantum sources or the price of ignorance we resume the investigation of the problem of independent local compression of correlated quantum sources the classical case of which is covered by the celebrated slepian wolf theorem we focus specifically on classical quantum cq sources for which one edge of the rate region corresponding to the compression of the classical part using the quantum part as side information at the decoder was previously determined by devetak and winter phys rev a 68 042301 2003 whereas the devetak winter protocol attains a rate sum equal to the von neumann entropy of the joint source here we show that the full rate region is much more complex due to the partially quantum nature of the source in particular in the opposite case of compressing the quantum part of the source using the classical part as side information at the decoder typically the rate sum is strictly larger than the von neumann entropy of the total source we determine the full rate region in the generic case showing that apart from the devetak winter point all other points in the achievable region have a rate sum strictly larger than the joint entropy we can interpret the difference as the price paid for the quantum encoder being ignorant of the classical side information in the general case we give an achievable rate region via protocols that are built on the decoupling principle and the protocols of quantum state merging and quantum state redistribution our achievable region is matched almost by a single letter converse which however still involves asymptotic errors and an unbounded auxiliary system

monte carlo extrinsic estimators of manifold valued parameters monte carlo mc methods have become an important tool for inferences in non gaussian and non euclidean settings we study their use in those signal image processing scenarios where the parameter spaces are certain riemannian manifolds finite dimensional lie groups and their quotient sets we investigate the estimation of means and variances of the manifold valued parameters using two popular sampling methods independent and importance sampling using euclidean embeddings we specify a notion of extrinsic means employ monte carlo methods to estimate these means and utilize large sample asymptotics to approximate the estimator covariances experimental results are presented for target pose estimation orthogonal groups and signal subspace estimation grassmann manifolds asymptotic covariances are utilized to construct confidence regions to compare estimators and to determine the sample size for mc sampling

multipartite quantum correlations and local recoverability characterizing genuine multipartite quantum correlations in quantum physical systems has historically been a challenging problem in quantum information theory more recently however the total correlation or multipartite information measure has been helpful in accomplishing this goal especially with the multipartite symmetric quantum msq discord piani et al 2008 phys rev lett 100 090502 doi 10 1103 physrevlett 100 090502 1 and the conditional entanglement of multipartite information cemi yang et al 2008 phys rev lett 101 140501 doi 10 1103 physrevlett 101 140501 2 here we apply a recent and significant improvement of strong subadditivity of quantum entropy fawzi renner 2014 in order to develop these quantities further in particular we prove that the msq discord is nearly equal to zero if and only if the multipartite state for which it is evaluated is approximately locally recoverable after performing measurements on each of its systems furthermore we prove that the cemi is a faithful entanglement measure i e it vanishes if and only if the multipartite state for which it is evaluated is a fully separable state along the way we provide an operational interpretation of the msq discord in terms of the partial state distribution protocol which in turn as a special case gives an interpretation for the original discord quantity finally we prove an inequality that could potentially improve upon the fawzi renner inequality in the multipartite context but it remains an open question to determine whether this is so 1 http dx doi org 10 1103 physrevlett 100 090502 2 http dx doi org 10 1103 physrevlett 101 140501

surface bounded growth modeling applied to human mandibles from a set of longitudinal three dimensional scans of the same anatomical structure the authors have accurately modeled the temporal shape and size changes using a linear shape model on a total of 31 computed tomography scans of the mandible from six patients 14 851 semilandmarks are found automatically using shape features and a new algorithm called geometry constrained diffusion the semilandmarks are mapped into procrustes space principal component analysis extracts a one dimensional subspace which is used to construct a linear growth model the worst case mean modeling error in a cross validation study is 3 7 mm

optimal quantum source coding with quantum side information at the encoder and decoder consider many instances of an arbitrary quadripartite pure state of four quantum systems abcd alice holds the ac part of each state bob holds b while r represents all other parties correlated with abc alice is required to redistribute the c systems to bob while asymptotically preserving the overall purity we prove that this is possible using q qubits of communication and e ebits of shared entanglement between alice and bob provided that q ges 1 2i c d b and q e ges h c b proving the optimality of the luo devetak outer bound the optimal qubit rate provides the first known operational interpretation of quantum conditional mutual information we also show how our protocol leads to a fully operational proof of strong subaddivity and uncover a general organizing principle in analogy to thermodynamics that underlies the optimal rates

morphometric tools for landmark data geometry and biology 1 introduction 2 preliminaries 3 landmarks 4 distance measures 5 shape coordinates 6 principal axes of shape change for triangles 7 features of shape comparison 8 retrospect and prospect

entanglement assisted capacity of a quantum channel and the reverse shannon theorem the entanglement assisted classical capacity of a noisy quantum channel c sub e is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel assuming that the sender and receiver have access to the resource of shared quantum entanglement which may be used up by the communication protocol we show that the capacity c sub e is given by an expression parallel to that for the capacity of a purely classical channel i e the maximum over channel inputs spl rho of the entropy of the channel input plus the entropy of the channel output minus their joint entropy the latter being defined as the entropy of an entangled purification of spl rho after half of it has passed through the channel we calculate entanglement assisted capacities for two interesting quantum channels the qubit amplitude damping channel and the bosonic channel with amplification attenuation and gaussian noise we discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel alone or in the presence of ancillary resources such as prior entanglement in the classical analog of entanglement assisted communication communication over a discrete memoryless channel dmc between parties who share prior random information we show that one parameter is sufficient i e that in the presence of prior shared random information all dmcs of equal capacity can simulate one another with unit asymptotic efficiency

a flag representation for finite collections of subspaces of mixed dimensions abstract given a finite set of subspaces of r n perhaps of differing dimensions we describe a flag of vector spaces i e a nested sequence of vector spaces that best represents the collection based on a natural optimization criterion and we present an algorithm for its computation the utility of this flag representation lies in its ability to represent a collection of subspaces of differing dimensions when the set of subspaces all have the same dimension d the flag mean is related to several commonly used subspace representations for instance the d dimensional subspace in the flag corresponds to the extrinsic manifold mean when the set of subspaces is both well clustered and equidimensional of dimension d then the d dimensional component of the flag provides an approximation to the karcher mean an intermediate matrix used to construct the flag can also be used to recover the canonical components at the heart of multiset canonical correlation analysis two examples utilizing the carnegie mellon university pose illumination and expression database cmu pie serve as visual illustrations of the algorithm

probability convexity and harmonic maps with small image i uniqueness and fine existence this paper uses probability theory to provide an approach to the dirichlet problem for harmonic maps the probabilistic tools used are those of manifold valued brownian motion and martingales probabilistic proofs are given of uniqueness continuous dependence on data and existence of generalized solutions finely harmonic maps for target domains which have convex geometry this class of domain includes all regular geodesic balls links are made with new results in other fields the martingale dirichlet problem riemannian centres of mass

extrinsic mean of brownian distributions on compact lie groups this paper studies brownian distributions on compact lie groups these are defined as the marginal distributions of brownian processes and are intended as a natural extension of the well known normal distributions to compact lie groups it is shown that this definition preserves key properties of normal distributions in particular brownian distributions transform in a nice way under group operations and satisfy an extension of the central limit theorem brownian distributions on a compact lie group g belong to one of two parametric families nl g c and nr g c g g and c a positive definite symmetric matrix in particular the parameter g appears as a location parameter an approach based on the extrinsic mean for estimation of the parameters g and c is studied in detail it is shown that g is the unique extrinsic mean for a brownian distribution nl g c or nr g c resulting estimates are proved to be consistent and asymptotically normal while they may also be used to simultaneously estimate g and c it is seen this requires that g be embedded into a higher dimensional matrix lie group going beyond brownian distributions it is shown the extrinsic mean can be used to recover the location parameter for a wider class of distributions arising more generally from levy processes the compact lie group structure places limitations on the analogy between normal distributions and brownian distributions this is illustrated by the study of multivariate brownian distributions these are introduced as brownian distributions on some product group e g g g this paper describes their covariance structure and considers its transformation under group operations

entanglement assisted communication of classical and quantum information in this paper we consider the problem of transmitting classical and quantum information reliably over an entanglement assisted quantum eaq channel our main result is a capacity theorem that gives a 3 d achievable rate region points in the region are rate triples consisting of the classical communication rate the quantum communication rate and the entanglement consumption rate of a particular coding scheme the crucial protocol in achieving the boundary points of the capacity region is a protocol that we name the classically enhanced father cef protocol the cef protocol is more general than other protocols in the family tree of quantum shannon theoretic protocols in the sense that several previously known quantum protocols are now child protocols of it the cef protocol also shows an improvement over a timesharing strategy for the case of a qubit dephasing channel this result justifies the need for simultaneous coding of classical and quantum information over an eaq channel our capacity theorem is of a multiletter nature requiring a limit over many uses of the channel but it reduces to a single letter characterization for at least three channels the completely depolarizing channel the quantum erasure channel and the qubit dephasing channel

convection of microstructure and related problems we study how two flows that vary spatially in two widely separated scales evolve under the dynamics of euler s equations we use a multiple scale approach which we motivate by studying some simpler model problems

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

geometry and multiple direction estimation abstract we consider the problem of the estimation of multiple target directions given the output of a phased array of identical elements the natural mathematical setting for this problem is the grassmannian manifold in order to understand the problems considered here a knowledge of exterior forms lie groups and differential geometry is essential we review some of the essential ideas relevant to the problem including the maurer cartan equations algebraic varieties in projective space and the invariant volume integral for symmetric spaces our purpose is to study probability measures on grassmannians in order to pose and solve the bayes problem on grassmannians

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

the statistical analysis of shape data summary the paper deals with the statistical shape analysis of landmark data bookstein 1986 introduced a plausible model and obtained a useful normal approximation for three landmarks we use exact distribution theory and a normal approximation to consider maximum likelihood estimation the fisher information matrix for the normal approximation is given and we consider the hypothesis of shape change using these results we analyse a practical example based on digitized shapes this example motivated the work and the general direction for future work is indicated

riemann manifold langevin and hamiltonian monte carlo methods the paper proposes metropolis adjusted langevin and hamiltonian monte carlo sampling methods defined on the riemann manifold to resolve the shortcomings of existing monte carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations the methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for metropolis hastings or indeed hamiltonian monte carlo and metropolis adjusted langevin algorithms this allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the markov chain the methodology proposed exploits the riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold providing highly efficient convergence and exploration of the target density the performance of these riemann manifold monte carlo methods is rigorously assessed by performing inference on logistic regression models log gaussian cox point processes stochastic volatility models and bayesian estimation of dynamic systems described by non linear differential equations substantial improvements in the time normalized effective sample size are reported when compared with alternative sampling approaches matlab code that is available from http www ucl ac uk statistics research rmhmc allows replication of all the results reported

analysis of planar shapes using geodesic paths on shape spaces for analyzing shapes of planar closed curves we propose differential geometric representations of curves using their direction functions and curvature functions shapes are represented as elements of infinite dimensional spaces and their pairwise differences are quantified using the lengths of geodesics connecting them on these spaces we use a fourier basis to represent tangents to the shape spaces and then use a gradient based shooting method to solve for the tangent that connects any two shapes via a geodesic using the surrey fish database we demonstrate some applications of this approach 1 interpolation and extrapolations of shape changes 2 clustering of objects according to their shapes 3 statistics on shape spaces and 4 bayesian extraction of shapes in low quality images

enhanced diffusivity and intercell transition layers in 2 d models of passive advection a class of two dimensional isotropic divergence free vector fields is introduced and the effective diffusivity of the corresponding advection diffusion equations is studied these examples are very idealized flows but they can be solved exactly in the limit pe 1 scaling laws d d0 pe are obtained where d0 molecular diffusion pe peclet number with exponents in the range 0 1 and examples of stream functions with logarithmic singularities for which d d0pe the exponent is related by a simple formula to the shape of the stream function along cell boundaries suggesting that similar scaling laws should hold for more general 2 d closed cell flows

channel simulation with quantum side information we study and solve the problem of classical channel simulation with quantum side information at the receiver this is a generalization of both the classical reverse shannon theorem and the classical quantum slepian wolf problem the optimal noiseless communication rate is found to be reduced from the mutual information between the channel input and output by the holevo information between the channel output and the quantum side information our main theorem has two important corollaries the first is a quantum generalization of the wyner ziv problem rate distortion theory with quantum side information the second is an alternative proof of the tradeoff between classical communication and common randomness distilled from a quantum state the fully quantum generalization of the problem considered is quantum state redistribution here the sender and receiver share a mixed quantum state and the sender wants to transfer part of her state to the receiver using entanglement and quantum communication we present outer and inner bounds on the achievable rate pairs

perturbation theory for generalized procrustes analysis summary perturbation theory for procrustes statistics is developed when both configurations are subject to independent normal errors the total sum of squares is shown to partition into sums of squares for translation rotation reflection and scaling plus a residual these results are extended to generalized procrustes analysis this gives a theoretical justification to gower s analysis of variance summary of the results from a generalized procrustes analysis

directions and projective shapes this paper deals with projective shape analysis which is a study of finite configurations of points modulo projective transformations the topic has various applications in machine vision we introduce a convenient projective shape space as well as an appropriate coordinate system for this shape space for generic configurations of k points in m dimensions the resulting projective shape space is identified as a product of k m 1 copies of axial spaces rp m this identification leads to the need for developing multivariate directional and multivariate axial analysis and we propose parametric models as well as nonparametric methods for these areas in particular we investigate the frechet extrinsic mean for the multivariate axial case asymptotic distributions of the appropriate parametric and nonparametric tests are derived we illustrate our methodology with examples from machine vision

a one shot achievability result for quantum state redistribution we study the problem of entanglement assisted quantum state redistribution in the one shot setting and provide a new achievability result on the quantum communication required our bounds are in terms of the max relative entropy and the hypothesis testing relative entropy we use the techniques of convex split and position based decoding to arrive at our result we show that our result is upper bounded by the result obtained in berta et al 2016