edge exchangeable models for interaction networks abstractmany modern network datasets arise from processes of interactions in a population such as phone calls email exchanges co authorships and professional collaborations in such interaction networks the edges comprise the fundamental statistical units making a framework for edge labeled networks more appropriate for statistical analysis in this context we initiate the study of edge exchangeable network models and explore its basic statistical properties several theoretical and practical features make edge exchangeable models better suited to many applications in network analysis than more common vertex centric approaches in particular edge exchangeable models allow for sparse structure and power law degree distributions both of which are widely observed empirical properties that cannot be handled naturally by more conventional approaches our discussion culminates in the hollywood model which we identify here as the canonical family of edge exchangeable distributions the hollywood model i

the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

minimum resistance shapes in linear continuum mechanics a necessary optimality condition for the problem of the minimum resistance shape for a rigid three dimensional inclusion displaced in an unbounded isotropic elastic medium subject to a constraint o

an algebraic riccati equation approach to h optimization abstract this paper considers the general so called four block h optimal control problem with the assumption that system states are available for feedback it is shown that infimization of the h norm of the closed loop transfer function over all linear constant i e nondynamic stabilizing state feedback laws can be completely characterized via an algebraic riccati equation it is further shown that the optimal norm is not improved by allowing feedback to be dynamic thus the general state feedback h optimal control problem can be solved by iteratively solving one are and the controller can be chosen to be static gain

existence of an optimal solution of a shape control problem for the stationary navier stokes equations this paper is concerned with an optimal shape control problem for the stationary navier stokes system a two dimensional channel flow of an incompressible viscous fluid is examined to determine the shape of a bump on a part of the boundary that minimizes the viscous drag after giving a precise formulation of the extremal problem in a function analytic setting it is shown that optimal solutions exist

a revisit to inverse optimality of linear systems in this article we revisit the problem of inverse optimality for linear systems by applying certain explicit formulae for coprime matrix fraction descriptions cmfd of linear systems we propose a necessary and sufficient condition for a given stable state feedback law to be optimal for some quadratic performance index compared to existing results in the literature the proposed condition is simpler to check and interpret moreover it reduces the redundancy in the solutions of the associated algebraic riccati equation are as a direct application of the proposed results we consider the problem of inverse optimality of observer based state feedback to be specific for the case where the state is not fully known we consider the inverse optimality problem of an observer based state feedback for the closed loop system augmented by an observer more precisely it is shown that the observer based state feedback is inverse optimal for the closed loop system with some general forms of cost functions on

optimal graphon estimation in cut distance consider the twin problems of estimating the connection probability matrix of an inhomogeneous random graph and the graphon of a w random graph we establish the minimax estimation rates with respect to the cut metric for classes of block constant matrices and step function graphons surprisingly our results imply that from the minimax point of view the raw data that is the adjacency matrix of the observed graph is already optimal and more involved procedures cannot improve the convergence rates for this metric this phenomenon contrasts with optimal rates of convergence with respect to other classical distances for graphons such as the l 1 or l 2 metrics

co clustering of nonsmooth graphons performance bounds are given for exploratory co clustering blockmodeling of bipartite graph data where we assume the rows and columns of the data matrix are samples from an arbitrary population this is equivalent to assuming that the data is generated from a nonsmooth graphon it is shown that co clusters found by any method can be extended to the row and column populations or equivalently that the estimated blockmodel approximates a blocked version of the generative graphon with estimation error bounded by o p n 1 2 analogous performance bounds are also given for degree corrected blockmodels and random dot product graphs with error rates depending on the dimensionality of the latent variable space

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

on semidefinite relaxations for the block model the stochastic block model sbm is a popular tool for community detection in networks but fitting it by maximum likelihood mle involves a computationally infeasible optimization problem we propose a new semidefinite programming sdp solution to the problem of fitting the sbm derived as a relaxation of the mle we put ours and previously proposed sdps in a unified framework as relaxations of the mle over various subclasses of the sbm which also reveals a connection to the well known problem of sparse pca our main relaxation which we call sdp 1 is tighter than other recently proposed sdp relaxations and thus previously established theoretical guarantees carry over however we show that sdp 1 exactly recovers true communities over a wider class of sbms than those covered by current results in particular the assumption of strong assortativity of the sbm implicit in consistency conditions for previously proposed sdps can be relaxed to weak assortativity for our approach thus significantly broadening the class of sbms covered by the consistency results we also show that strong assortativity is indeed a necessary condition for exact recovery for previously proposed sdp approaches and not an artifact of the proofs our analysis of sdps is based on primal dual witness constructions which provides some insight into the nature of the solutions of various sdps in particular we show how to combine features from sdp 1 and already available sdps to achieve the most flexibility in terms of both assortativity and block size constraints as our relaxation has the tendency to produce communities of similar sizes this tendency makes it the ideal tool for fitting network histograms a method gaining popularity in the graphon estimation literature as we illustrate on an example of a social networks of dolphins we also provide empirical evidence that sdps outperform spectral methods for fitting sbms with a large number of blocks

reconstructing undirected graphs from eigenspaces in this paper we aim at recovering an undirected weighted graph of n vertices from the knowledge of a perturbed version of the eigenspaces of its adjacency matrix w for instance this situation arises for stationary signals on graphs or for markov chains observed at random times our approach is based on minimizing a cost function given by the frobenius norm of the commutator mathsf a mathsf b mathsf b mathsf a between symmetric matrices mathsf a and mathsf b in the erd h o s r enyi model with no self loops we show that identifiability i e the ability to reconstruct w from the knowledge of its eigenspaces follows a sharp phase transition on the expected number of edges with threshold function n log n 2 given an estimation of the eigenspaces based on a n sample we provide support selection procedures from theoretical and practical point of views in particular when deleting an edge from the active support our study unveils that our test statistic is the order of mathcal o 1 n when we overestimate the true support and lower bounded by a positive constant when the estimated support is smaller than the true support this feature leads to a powerful practical support estimation procedure simulated and real life numerical experiments assert our new methodology

a second order shape optimization approach for image segmentation the problem of segmentation of a given image using the active contour technique is considered an abstract calculus to find appropriate speed functions for active contour models in image segmentation or related problems based on variational principles is presented the speed method from shape sensitivity analysis is used to derive speed functions which correspond to gradient or newton type directions for the underlying optimization problem the newton type speed function is found by solving an elliptic problem on the current active contour in every time step numerical experiments comparing the classical gradient method with newton s method are presented

path following methods for a class of constrained minimization problems in function space path following methods for primal dual active set strategies requiring a regularization parameter are introduced existence of a primal dual path and its differentiability properties are analyzed monotonicity and convexity of the primal dual path value function are investigated both feasible and infeasible approximations are considered numerical path following strategies are developed and their efficiency is demonstrated by means of examples

sparse exchangeable graphs and their limits via graphon processes in a recent paper caron and fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in r here we show that by generalizing the classical definition of graphons as functions over probability spaces to functions over finite measure spaces we can model a large family of exchangeable graphs including the caron fox graphs and the traditional exchangeable dense graphs as special cases explicitly modelling the underlying space of features by a finite measure space s s and the connection probabilities by an integrable function w s s 0 1 we construct a random family gt t 0 of growing graphs such that the vertices of gt are given by a poisson point process on s with intensity t with two points x y of the point process connected with probability w x y we call such a random family a graphon process we prove that a graphon process has convergent subgraph frequencies with possibly infinite limits and that in the natural extension of the cut metric to our setting the sequence converges to the generating graphon we also show that the underlying graphon is identifiable only as an equivalence class over graphons with cut distance zero more generally we study metric convergence for arbitrary not necessarily random sequences of graphs and show that a sequence of graphs has a convergent subsequence if and only if it has a subsequence satisfying a property we call uniform regularity of tails finally we prove that every graphon is equivalent to a graphon on r equipped with lebesgue measure

matrix estimation by universal singular value thresholding consider the problem of estimating the entries of a large matrix when the observed entries are noisy versions of a small random fraction of the original entries this problem has received widespread attention in recent times especially after the pioneering works of emmanuel candes and collaborators this paper introduces a simple estimation procedure called universal singular value thresholding usvt that works for any matrix that has a little bit of structure surprisingly this simple estimator achieves the minimax error rate up to a constant factor the method is applied to solve problems related to low rank matrix estimation blockmodels distance matrix completion latent space models positive definite matrix completion graphon estimation and generalized bradley terry models for pairwise comparison

on hermitian solutions of the symmetric algebraic riccati equation the structure of the set of hermitian solutions of the matrix quadratic equation xdx xa a x c 0 is studied under the conditions that c c d is positive semidefinite and a d is stabilizable new features e g nonexistence of the minimal solution appear in contrast with the known case when a d is controllable

analysis of shape optimization for magnetic microswimmers we analyze an infinite dimensional geometrically constrained shape optimization problem for magnetically driven microswimmers locomotors in three dimensional 3 d stokes flow and give a well posed descent scheme for computing optimal shapes the problem is inspired by recent experimental work in this area we show the existence of a minimizer of the optimization problem using analytical tools for elastic rods that respect the excluded volume constraint we derive a variational gradient descent method for computing optimal locomotor shapes using the tools of shape differential calculus the descent direction is obtained by solving a saddle point system which we prove is well posed we also introduce a finite element approximation of the gradient descent method and prove its stability we present numerical results illustrating our method and the effect that finite aspect ratio and external cargo can have on the optimal shape the 3 d stokes equations are solved by a boundary integral method

a variational shape optimization approach for image segmentation with a mumford shah functional we introduce a novel computational method for a mumford shah functional which decomposes a given image into smooth regions separated by closed curves casting this as a shape optimization problem we develop a gradient descent approach at the continuous level that yields nonlinear pde flows we propose time discretizations that linearize the problem and space discretization by continuous piecewise linear finite elements the method incorporates topological changes such as splitting and merging for detection of multiple objects space time adaptivity and a coarse to fine approach to process large images efficiently we present several simulations that illustrate the performance of the method and investigate the model sensitivity to various parameters

general matrix pencil techniques for the solution of algebraic riccati equations a unified approach we present a unified theory of matrix pencil techniques for solving both continuous and discrete time algebraic riccati equations ares under fairly general conditions on the coefficient matrices the theory applies to a large class of ares and riccati like equations arising from the singular h sup spl infin and h sup 2 control problems singular linear quadratic control the 4 block nehari problem or from singular j spectral factorizations the underlying concept is the so called proper deflating subspace of a possibly singular matrix pencil in terms of which necessary and sufficient conditions for the solvability of riccati equations are given it is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm

analytic feedback control and the algebraic riccati equation the central result of this correspondence is lemma 1 1 which states that the stabilizing solution p to the algebraic riccati equation in a b c depends analytically on a b c in the remainder of the correspondence various control and system theoretic ramifications of the analyticity lemma are considered an important consequence of lemma 1 1 is that any smooth state feedback control law for a finite dimensional plant may be implemented using dynamic input output compensators whose transfer functions depend smoothly on the plant transfer function

robust h infty control of an uncertain system via a stable output feedback controller this technical note presents a new approach to the robust control of an uncertain system via a stable output feedback controller the uncertain systems under consideration contain structured uncertainty described by integral quadratic constraints the controller is designed to achieve absolute stabilization with a specified level of disturbance attenuation the main result involves solving a state feedback version of the problem by solving an algebraic riccati equation dependent on a set of scaling parameters then two further algebraic riccati equations are solved which depend on a further set of scaling parameters

novel insights on the stabilising solution to the continuous time algebraic riccati equation in the present paper we present a closed form solution as a function of the closed loop poles for the continuous time algebraic riccati equations cares related to single input single output systems with non repeated poles the proposed solution trades the standard numerical algorithm approach for one based on a spectral factorisation argument offering potential insight into any control technique based on a care and its solution as an example we present the equivalence of two fairly recent control over network results furthermore we apply the proposed result to the formula for the optimal regulator gain matrix k or equivalently the luenberger s observer gain l and present an example finally we conclude by discussing the possible extension of the proposed closed form solution to the repeated eigenvalues case and to the case when the care is related to multiple input multiple output systems

oracle inequalities for network models and sparse graphon estimation inhomogeneous random graph models encompass many network models such as stochastic block models and latent position models we consider the problem of statistical estimation of the matrix of connection probabilities based on the observations of the adjacency matrix of the network taking the stochastic block model as an approximation we construct estimators of network connection probabilities the ordinary block constant least squares estimator and its restricted version we show that they satisfy oracle inequalities with respect to the block constant oracle as a consequence we derive optimal rates of estimation of the probability matrix our results cover the important setting of sparse networks another consequence consists in establishing upper bounds on the minimax risks for graphon estimation in the l 2 norm when the probability matrix is sampled according to a graphon model these bounds include an additional term accounting for the agnostic error induced by the variability of the latent unobserved variables of the graphon model in this setting the optimal rates are influenced not only by the bias and variance components as in usual nonparametric problems but also include the third component which is the agnostic error the results shed light on the differences between estimation under the empirical loss the probability matrix estimation and under the integrated loss the graphon estimation

technical communique closed form solution for a class of continuous time algebraic riccati equations in the present paper we obtain a closed form solution for a class of continuous time algebraic riccati equations ares with vanishing state weight the are in such a class solves a minimum energy control problem the obtained closed form solution is used to prove a link between two independent fundamental limitation results in control over networks

shape optimization of peristaltic pumping transport is a fundamental aspect of biology and peristaltic pumping is a fundamental mechanism to accomplish this it is also important to many industrial processes we present a variational method for optimizing the wave shape of a peristaltic pump specifically we optimize the wave profile of a two dimensional channel containing a navier stokes fluid with no assumption on the wave profile other than it is a traveling wave e g we do not assume it is the graph of a function hence this is an infinite dimensional optimization problem the optimization criteria consists of minimizing the input fluid power due to the peristaltic wave subject to constraints on the average flux of fluid and area of the channel sensitivities of the cost and constraints are computed variationally via shape differential calculus and we use a sequential quadratic programming sqp method to find a solution of the first order kkt conditions we also use a merit function based line search in order to balance between decreasing the cost and keeping the constraints satisfied when updating the channel shape our numerical implementation uses a finite element method for computing a solution of the navier stokes equations adjoint equations as well as for the sqp method when computing perturbations of the channel shape the walls of the channel are deformed by an explicit front tracking approach in computing functional sensitivities with respect to shape we use l 2 type projections for computing boundary stresses and for geometric quantities such as the tangent field on the channel walls and the curvature we show error estimates for the boundary stress and tangent field approximations as a result we find optimized shapes that are not obvious and have not been previously reported in the peristaltic pumping literature specifically we see highly asymmetric wave shapes that are far from being sine waves many examples are shown for a range of fluxes and reynolds numbers up to re 500 which illustrate the capabilities of our method

limiting performance of optimal linear filters we study the lowest achievable mean square estimation error in two limiting optimal linear filtering problems first when the intensity of the process noise tends to zero the lowest achievable mean square estimation error is a function of the unstable poles of the system second when the intensity of the measurement noise tends to zero the lowest achievable mean square estimation error is a function of the nonminimum phase zeros of the system we link these results with bode integral characterisations of performance limitations in linear filtering

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

droplet footprint control controlling droplet shape via surface tension has numerous technological applications such as droplet lenses and lab on a chip this motivates a partial differential equation constrained shape optimization approach for controlling the shape of droplets on flat substrates by controlling the surface tension of the substrate we use shape differential calculus to derive an l 2 gradient flow approach to compute equilibrium shapes for sessile droplets on substrates we then develop a gradient based optimization method to find the substrate surface tension coefficient yielding an equilibrium droplet shape with a desired footprint i e the liquid solid interface has a desired shape moreover we prove a sensitivity result with respect to the substrate surface tensions for the free boundary problem associated with the footprint numerical results are also presented to showcase the method

the algebraic riccati equation a polynomial approach abstract polynomial models are used to give a unified approach to the problem of classifying the set of all real symmetric solutions of the algebraic riccati equation

universally consistent vertex classification for latent positions graphs in this work we show that using the eigen decomposition of the adjacency matrix we can consistently estimate feature maps for latent position graphs with positive definite link function provided that the latent positions are i i d from some distribution f we then consider the exploitation task of vertex classification where the link

shape optimization of self avoiding curves this paper presents a softened notion of proximity or self avoidance for curves we then derive a sensitivity result based on shape differential calculus for the proximity this is combined with a gradient based optimization approach to compute three dimensional parameterized curves that minimize the sum of an elastic bending energy and a proximity energy that maintains self avoidance by a penalization technique minimizers are computed by a sequential quadratic programming sqp method where the bending energy and proximity energy are approximated by a finite element method we then apply this method to two problems first we simulate adsorbed polymer strands that are constrained to be bound to a surface and be locally inextensible this is a basic model of semi flexible polymers adsorbed onto a surface a current topic in material science several examples of minimizing curve shapes on a variety of surfaces are shown an advantage of the method is that it can be much faster than using molecular dynamics for simulating polymer strands on surfaces second we apply our proximity penalization to the computation of ideal knots we present a heuristic scheme utilizing the sqp method above for minimizing rope length and apply it in the case of the trefoil knot applications of this method could be for generating good initial guesses to a more accurate but expensive knot tightening algorithm a new softened version of proximity or self contact for 1 d curves is proposed sensitivity of the proximity with respect to shape differentiation is given a method for finding minimizing curve shapes that respect self contact is given the method is applied to simulating polymer strands adsorbed onto surfaces a heuristic method is given for approximating ideal knots

disturbance attenuation and h optimization a design method based on the algebraic riccati equation this note presents a method for designing a state feedback control law to reduce the effect of disturbances on the output of a given linear system the problem under consideration involves attenuating the disturbances to a prespecified level the construction of the state feedback control law requires the solution of a certain algebraic riccati equation by applying the procedure with successively smaller values of the prespecifled disturbance level the result also provides an approach to the h infty optimization problem for the state feedback case