identifiability in blind deconvolution with subspace or sparsity constraints blind deconvolution bd the resolution of a signal and a filter given their convolution arises in many applications without further constraints bd is ill posed in practice subspace or sparsity constraints have been imposed to reduce the search space and have shown some empirical success however the existing theoretical analysis on uniqueness in bd is rather limited in an effort to address the still open question we derive sufficient conditions under which two vectors can be uniquely identified from their circular convolution subject to subspace or sparsity constraints these sufficient conditions provide the first algebraic sample complexities for bd we first derive a sufficient condition that applies to almost all bases or frames for bd of vectors in mathbb c n with two subspace constraints of dimensions m 1 and m 2 the required sample complexity is n geq m 1 m 2 then we impose a sub band structure on one basis and derive a sufficient condition that involves a relaxed sample complexity n geq m 1 m 2 1 which we show to be optimal we present the extensions of these results to bd with sparsity constraints or mixed constraints with the sparsity level replacing the subspace dimension the cost for the unknown support in this case is an extra factor of 2 in the sample complexity

further relaxations of the semidefinite programming approach to sensor network localization recently a semidefinite programming sdp relaxation approach has been proposed to solve the sensor network localization problem although it achieves high accuracy in estimating the sensor locations the speed of the sdp approach is not satisfactory for practical applications in this paper we propose methods to further relax the sdp relaxation more precisely to relax the single semidefinite matrix cone into a set of small size semidefinite submatrix cones which we call a sub sdp ssdp approach we present two such relaxations although they are weaker than the original sdp relaxation they retain the key theoretical property and numerical experiments show that they are both efficient and accurate the speed of the ssdp is even faster than that of other approaches based on weaker relaxations the ssdp approach may also pave a way to efficiently solving general sdp problems without sacrificing the solution quality

equations differentielles sur l espace de wiener et formules de cameron martin non lineaires proprietes algebriques des approximations sur r n une formule de perturbation des flots et convergence des approximations formule de changement de variable sur x et equation differentielle

graph theoretic methods for matrix completion problems abstract a pattern is a list of positions in an n n real matrix a matrix completion problem for the class of matrices asks whether every partial matrix whose specified entries are exactly the positions of the pattern can be completed to a matrix we survey the current state of research on matrix completion problems for many subclasses of p 0 matrices including positive definite matrices m matrices inverse m matrices p matrices and matrices defined by various sign symmetry and positivity conditions on p 0 and p matrices graph theoretic techniques used to study completion problems are discussed several new results are also presented including the solution to the m 0 matrix completion problem and the sign symmetric p 0 matrix completion problem

exploiting sparsity in sdp relaxation for sensor network localization a sensor network localization problem can be formulated as a quadratic optimization problem qop for qops semidefinite programming sdp relaxation by lasserre with relaxation order 1 for general polynomial optimization problems pops is known to be equivalent to the sparse sdp relaxation by waki et al with relaxation order 1 except for the size and sparsity of the resulting sdp relaxation problems we show that the sparse sdp relaxation applied to the qop is at least as strong as the biswas ye sdp relaxation for the sensor network localization problem a sparse variant of the biswas ye sdp relaxation which is equivalent to the original biswas ye sdp relaxation is also derived we compare numerically the sparse sdp relaxation applied to the qop the biswas ye sdp relaxation its sparse variant and the edge based sdp relaxation by wang et al to confirm the effectiveness of the proposed techniques for exploiting the sparsity in sdp relaxation for sensor network localization problems the sparse sdp relaxation applied to the qop is much faster than the biswas ye sdp relaxation and the sparse variant of the biswas ye sdp relaxation outperforms all other sdp relaxations in speed

blind deconvolution using convex programming we consider the problem of recovering two unknown vectors w and x of length l from their circular convolution we make the structural assumption that the two vectors are members of known subspaces one with dimension n and the other with dimension k although the observed convolution is nonlinear in both w and x it is linear in the rank 1 matrix formed by their outer product wx this observation allows us to recast the deconvolution problem as low rank matrix recovery problem from linear measurements whose natural convex relaxation is a nuclear norm minimization program we prove the effectiveness of this relaxation by showing that for generic signals the program can deconvolve w and x exactly when the maximum of n and k is almost on the order of l that is we show that if x is drawn from a random subspace of dimension n and w is a vector in a subspace of dimension k whose basis vectors are spread out in the frequency domain then nuclear norm minimization recovers wx without error we discuss this result in the context of blind channel estimation in communications if we have a message of length n which we code using a random l x n coding matrix and the encoded message travels through an unknown linear time invariant channel of maximum length k then the receiver can recover both the channel response and the message when l n k to within constant and log factors

convergence behavior of interior point algorithms we show that most interior point algorithms for linear programming generate a solution sequence in which every limit point satisfies the strict complementarity condition these algorithms include all path following algorithms and some potential reduction algorithms the result also holds for the monotone complementarity problem if a strict complementarity solution exists in general the limit point is a solution that maximizes the number of its nonzero components among all solutions

tangent processes on wiener space abstract this paper deals with the study of the malliavin calculus of euclidean motions on wiener space i e transformations induced by general measure preserving transformations called rotations and h valued shifts and the associated flows on abstract wiener spaces

robust edge based semidefinite programming relaxation of sensor network localization recently wang zheng boyd and ye siam j optim 19 655 673 2008 proposed a further relaxation of the semidefinite programming sdp relaxation of the sensor network localization problem named edge based sdp esdp in simulation the esdp is solved much faster by interior point method than sdp relaxation and the solutions found are comparable or better in approximation accuracy we study some key properties of the esdp relaxation showing that when distances are exact zero individual trace is not only sufficient but also necessary for a sensor to be correctly positioned by an interior solution we also show via an example that when distances are inexact zero individual trace is insufficient for a sensor to be accurately positioned by an interior solution we then propose a noise aware robust version of esdp relaxation for which small individual trace is necessary and sufficient for a sensor to be accurately positioned by a certain analytic center solution assuming the noise level is sufficiently small for this analytic center solution the position error for each sensor is shown to be in the order of the square root of its trace lastly we propose a log barrier penalty coordinate gradient descent method to find such an analytic center solution in simulation this method is much faster than interior point method for solving esdp and the solutions found are comparable in approximation accuracy moreover the method can distribute its computation over the sensors via local communication making it practical for positioning and tracking in real time

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

fast and guaranteed blind multichannel deconvolution under a bilinear system model we consider the multichannel blind deconvolution problem where we observe the output of multiple channels that are all excited with the same unknown input from these observations we wish to estimate the impulse responses of each of the channels we show that this problem is well posed if the channels follow a bilinear model where the ensemble of channel responses is modeled as lying in a low dimensional subspace but with each channel modulated by an independent gain under this model we show how the channel estimates can be found by minimizing a quadratic function over a non convex set we analyze two methods for solving this non convex program and provide performance guarantees for each the first is a method of alternating eigenvectors that breaks the program down into a series of eigenvalue problems the second is a truncated power iteration which can roughly be interpreted as a method for finding the largest eigenvector of a symmetric matrix with the additional constraint that it adheres to our bilinear model as with most non convex optimization algorithms the performance of both of these algorithms is highly dependent on having a good starting point we show how such a starting point can be constructed from the channel measurements our performance guarantees are non asymptotic and provide a sufficient condition on the number of samples observed per channel in order to guarantee channel estimates of certain accuracy our analysis uses a model with a generic subspace that is drawn at random and we show the performance bounds hold with high probability mathematically the key estimates are derived by quantifying how well the eigenvectors of certain random matrices approximate the eigenvectors of their mean we also present a series of numerical results demonstrating that the empirical performance is consistent with the presented theory

iterative greedy algorithm for solving the fir paraunitary approximation problem in this paper a method for approximating a multi input multi output mimo transfer function by a causal finite impulse response fir paraunitary pu system in a weighted least squares sense is presented using a complete parameterization of fir pu systems in terms of householder like building blocks an iterative algorithm is proposed that is greedy in the sense that the observed mean squared error at each iteration is guaranteed to not increase for certain design problems in which there is a phase type ambiguity in the desired response which is formally defined in the paper a phase feedback modification is proposed in which the phase of the fir approximant is fed back to the desired response with this modification in effect it is shown that the resulting iterative algorithm not only still remains greedy but also offers a better magnitude type fit to the desired response simulation results show the usefulness and versatility of the proposed algorithm with respect to the design of principal component filter bank pcfb like filter banks and the fir pu interpolation problem concerning the pcfb design problem it is shown that as the mcmillan degree of the fir pu approximant increases the resulting filter bank behaves more and more like the infinite order pcfb consistent with intuition in particular this pcfb like behavior is shown in terms of filter response shape multiresolution coding gain noise reduction with zeroth order wiener filtering in the subbands and power minimization for discrete multitone dmt type transmultiplexers

on the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues we derive some basic results on the geometry of semidefinite programming sdp and eigenvalue optimization i e the minimization of the sum of the k largest eigenvalues of a smooth matrix valued function we provide upper bounds on the rank of extreme matrices in sdps and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue optimization in the spectrum of an optimal matrix the kth and k 1 st largest eigenvalues tend to be equal and frequently have multiplicity greater than two this clustering is intuitively plausible and has been observed as early as 1975 when the matrix valued function is affine we prove that clustering must occur at extreme points of the set of optimal solutions if the number of variables is sufficiently large we also give a lower bound on the multiplicity of the critical eigenvalue these results generalize to the case of a general matrix valued function under appropriate conditions

unicite de solutions d equations differentielles sur l espace de wiener soit x l espace de wiener et a un champ de vecteurs sur x on definit la notion de solution de l equation differentielle du dt 1 u t u 0 x on demontre un theoreme d unicite

a class of vector fields on path spaces abstract in this paper we show that the vector fieldx hon a based path spacewo m over a riemannian manifoldmdefined by parallel translating a curvehin the initial tangent spacetomvia an affine connection induces a solution flow which preserves the wiener measure on the based path spacewo m provided the affine connection is adjoint skew symmetric in the case when is a metric connection then is adjoint skew symmetric if and only if is torsion skew symmetric

anticipating flows on the wiener space generated by vector fields of low regularity abstract sobolev spaces for banach space valued wiener functionals are constructed using the operator norm for the h derivatives the corresponding sobolev norms are shown to be strictly weaker than the usual malliavin sobolev norms existing results on flows on an abstract wiener space generated by a vector field are generalized to the case where the vector field is a sum of a not necessarily bounded skew adjoint operator and a non linear part whose regularity is measured by the new sobolev norms this is equivalent to a non linear perturbation of a semi group of rotations

graph rigidity via euclidean distance matrices abstract let g v e be an incomplete graph with node set v edge set e and nonnegative weights ij s on the edges let each edge v i v j be viewed as a rigid bar of length ij which can rotate freely around its end nodes a realization of a graph g is an assignment of coordinates in some euclidean space to each node of g in this paper we consider the problem of determining whether or not a given realization of a graph g is rigid we show that each realization of g can be epresented as a point in a compact convex set r m and that a generic realization of g is rigid if and only if its corresponding point is a vertex of i e an extreme point with full dimensional normal cone

flows associated to tangent processes on the wiener space we prove under certain regularity assumptions on the coefficients that tangent processes namely semimartingales d a dx b d where a is an antisymmetric matrix generate flows on the classical wiener space main applications of the result can be found in the study of the geometry of path spaces

sobolev spaces of banach valued functions associated with a markov process we discuss sobolev spaces of banach valued functions they are extensions of sobolev spaces of scalar functions we use a gamma transform of a semigroup associated with a markov process a typical example is the ornstein uhlenbeck process on the wiener space

global continuation for distance geometry problems distance geometry problems arise in the determination of protein structure we consider the case where only a subset of the distances between atoms is given and formulate this distance geometry problem as a global minimization problem with special structure we show that global smoothing techniques and a continuation approach for global optimization can be used to determine global solutions of this problem reliably and efficiently the global continuation approach determines a global solution with less computational effort than is required by a standard multistart algorithm moreover the continuation approach usually finds the global solution from any given starting point while the multistart algorithm tends to fail

the molecule problem exploiting structure in global optimization the molecule problem is that of determining the relative locations of a set of objects in euclidean space relying only upon a sparse set of pairwise distance measurements this np hard problem has applications in the determination of molecular conformation the molecule problem can be naturally expressed as a continuous global optimization problem but it also has a rich combinatorial structure this paper investigates how that structure can be exploited to simplify the optimization problem in particular we present a novel divide and conquer algorithm in which a large global optimization problem is replaced by a sequence of smaller ones since the cost of the optimization can grow exponentially with problem size this approach holds the promise of a substantial improvement in performance our algorithmic development relies upon some recently published results in graph theory we describe an implementation of this algorithm and report some results of its performance on a sample molecule

explicit sensor network localization using semidefinite representations and facial reductions the sensor network localization snl problem in embedding dimension r consists of locating the positions of wireless sensors given only the distances between sensors that are within radio range and the positions of a subset of the sensors called anchors current solution techniques relax this problem to a weighted nearest positive semidefinite programming sdp completion problem by using the linear mapping between euclidean distance matrices edm and semidefinite matrices the resulting sdp is solved using primal dual interior point solvers yielding an expensive and inexact solution this relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the sdp cone implying that the slater constraint qualification fails cliques in the graph of the snl problem give rise to this degeneracy in the sdp relaxation in this paper we take advantage of the absence of the slater constraint qualification and derive a technique for the snl problem with exact data that explicitly solves the corresponding rank restricted sdp problem no sdp solvers are used for randomly generated instances we are able to efficiently solve many huge instances of this np hard problem to high accuracy by finding a representation of the minimal face of the sdp cone that contains the sdp matrix representation of the edm the main work of our algorithm consists in repeatedly finding the intersection of subspaces that represent the faces of the sdp cone that correspond to cliques of the snl problem

cooperative wireless sensor network positioning via implicit convex feasibility we propose a distributed positioning algorithm to estimate the unknown positions of a number of target nodes given distance measurements between target nodes and between target nodes and a number of reference nodes at known positions based on a geometric interpretation we formulate the positioning problem as an implicit convex feasibility problem in which some of the sets depend on the unknown target positions and apply a parallel projection onto convex sets approach to estimate the unknown target node positions the proposed technique is suitable for parallel implementation in which every target node in parallel can update its position and share the estimate of its location with other targets we mathematically prove convergence of the proposed algorithm simulation results reveal enhanced performance for the proposed approach compared to available techniques based on projections especially for sparse networks

on equivalence of semidefinite relaxations for quadratic matrix programming we analyze two popular semidefinite programming relaxations for quadratically constrained quadratic programs with matrix variables these relaxations are based on vector lifting and on matrix lifting they are of different size and expense we prove under mild assumptions that these two relaxations provide equivalent bounds thus our results provide a theoretical guideline for how to choose a less expensive semidefinite programming relaxation and still obtain a strong bound the main technique used to show the equivalence and that allows for the simplified constraints is the recognition of a class of nonchordal sparse patterns that admit a smaller representation of the positive semidefinite constraint

on bar frameworks stress matrices and semidefinite programming a bar framework g p in r dimensional euclidean space is a graph g on the vertices 1 2 n where each vertex i is located at point p i in mathbb r r given a framework g p in mathbb r r a problem of great interest is that of determining whether or not there exists another framework g q not obtained from g p by a rigid motion such that q i q j 2 p i p j 2 for each edge i j of g this problem is known as either the global rigidity problem or the universal rigidity problem depending on whether such a framework g q is restricted to be in the same r dimensional space or not the stress matrix s of a bar framework g p plays a key role in these and other related problems in this paper semidefinite programming sdp theory is used to address in a unified manner several problems concerning universal rigidity new results are presented as well as new proofs of previously known theorems in particular we use the notion of sdp non degeneracy to obtain a sufficient condition for universal rigidity and we show that this condition yields the previously known sufficient condition for generic universal rigidity we present new results concerning positive semidefinite stress matrices and we use a semidefinite version of farkas lemma to characterize bar frameworks that admit a nonzero positive semidefinite stress matrix s

quasi sure analysis of stochastic flows and banach space valued smooth functionals on the wiener space abstract we prove meyer s inequalities for functionals on the wiener space that take values on a banach space belonging to the burkbolder u m d class as an application we analyse the quasi sure regularity in time of the stochastic flow of diffeomorphisms generated by a stochastic differential equation with smooth coefficients

interior point trajectories in semidefinite programming in this paper we study interior point trajectories in semidefinite programming sdp including the central path of an sdp this work was inspired by the seminal work of megiddo on linear programming trajectories progress in math programming interior point algorithms and related methods n megiddo ed springer verlag berlin 1989 pp 131 158 under an assumption of primal and dual strict feasibility we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of respectively the primal and the dual problems we consider a class of trajectories that are similar to the central path but can be constructed to pass through any given interior feasible or infeasible point and study their convergence finally we study the derivatives of these trajectories and their convergence

on rigidity and realizability of weighted graphs abstract recently a new characterization of rigid graphs was introduced using euclidean distance matrices edms a y alfakih linear algebra appl 310 2000 149 in this paper we address the computational aspects of this characterization also we present a characterization of graphs which are realizable in r r for some 1 r n 2 but not realizable in r n 1 where n is the number of nodes

variational principles for stochastic fluid dynamics this paper derives stochastic partial differential equations spdes for fluid dynamics from a stochastic variational principle svp the paper proceeds by taking variations in the svp to derive stochastic stratonovich fluid equations writing their ito representation and then investigating the properties of these stochastic fluid models in comparison with each other and with the corresponding deterministic fluid models the circulation properties of the stochastic stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models as with deterministic ideal flows motion along the stochastic stratonovich paths also preserves the helicity of the vortex field lines in incompressible stochastic flows however these stratonovich properties are not apparent in the equivalent ito representation because they are disguised by the quadratic covariation drift term arising in the stratonovich to ito transformation this term is a geometric generalization of the quadratic covariation drift term already found for scalar densities in stratonovich s famous 1966 paper the paper also derives motion equations for two examples of stochastic geophysical fluid dynamics namely the euler boussinesq and quasi geostropic approximations

euclidean distance geometry and applications euclidean distance geometry is the study of euclidean geometry based on the concept of distance this is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in euclidean space realizing those given distances we survey the theory of euclidean distance geometry and its most important applications with special emphasis on molecular conformation problems

quasi invariance of the wiener measure on the path space over a complete riemannian manifold we prove a generalization of the cameron martin theorem for a geometrically and stochastically complete riemannian manifold namely the wiener measure on the path space over such a manifold is quasiinvariant under the flow generated by a cameron martin vector field

exploiting sparsity in semidefinite programming via matrix completion i general framework a critical disadvantage of primal dual interior point methods compared to dual interior point methods for large scale semidefinite programs sdps has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse based on some fundamental results about positive semidefinite matrix completion this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage our method is used in two ways one is a conversion of a sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior point method for sdps employing a standard block diagonal matrix data structure the other way is an incorporation of our method into primal dual interior point methods which we can apply directly to a given sdp in part ii of this article we will investigate an implementation of such a primal dual interior point method based on positive definite matrix completion and report some numerical results

a cameron martin type quasi invariance theorem for brownian motion on a compact riemannian manifold abstract let m be a compact manifold without boundary o be a fixed base point in m g be a riemannian metric on m and be a g compatible covariant derivative on tm the tangent space to m assume the torsion t of satisfies the skew symmetry condition g t x y y 0 for all vector fields x and y on m for example take to be the levi civita covariant derivative on m g also let denote the wiener measure on w0 m c 0 1 m 0 o and let h s denote stochastic parallel translation relative to along the path w0 m up to time s 0 1 given a c1 function h 0 1 t0m it is shown that the differential equation t h t h with initial condition 0 id w m w m has a solution r maps w m w m the measurable maps from w m to w m this function is a flow on w m i e for all t r t t furthermore t has the quasi invariance property the law t of t with respect to the wiener measure is equivalent to for all t r this result is used to prove an integration by parts formula for the h derivative hf defined by h f d dt 0 f t where f is a c2 cylinder function on w m

solving large scale sparse semidefinite programs for combinatorial optimization we present a dual scaling interior point algorithm and show how it exploits the structure and sparsity of some large scale problems we solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints we report the first computational results of interior point algorithms for approximating maximum cut semidefinite programs with dimension up to 3 000

on a registration based approach to sensor network localization we consider a registration based approach for localizing sensor networks from range measurements this is based on the assumption that one can find overlapping cliques spanning the network that is for each sensor one can identify geometric neighbors for which all inter sensor ranges are known such cliques can be efficiently localized using multidimensional scaling however since each clique is localized in some local coordinate system we are required to register them in a global coordinate system in other words our approach is based on transforming the localization problem into a problem of registration in this context the main contributions are as follows first we describe an efficient method for partitioning the network into overlapping cliques second we study the problem of registering the localized cliques and formulate a necessary rigidity condition for uniquely recovering the global sensor coordinates in particular we present a method for efficiently testing rigidity and a proposal for augmenting the partitioned network to enforce rigidity a recently proposed semidefinite relaxation of global registration is used for registering the cliques we present simulation results on random and structured sensor networks to demonstrate that the proposed method compares favourably with state of the art methods in terms of run time accuracy and scalability

flows associated to adapted vector fields on the wiener space the existence and uniqueness of a flow associated to an adapted vector field on the wiener space with d a d b d are proved by a modified picard s iteration method mainly under the conditions of exponential integrability concerning b as well as the first order malliavin gradients of a and b a newton leibnitz type inequality for a kind of malliavin differentiable functionals is also proved which is the key point to prove the above result and has an independent interest

fir perfect signal reconstruction from multiple convolutions minimum deconvolver orders given p co prime finite impulse response fir filters h sub i spl isin r m sub h it well known that there exist q sub i spl isin r m sub q such that spl sigma sub i h sub i q sub i spl delta importantly this enables signal recovery from its convolutions with p spl ges 2 co prime fir filters by fir filtering we show that such q sub i exist almost surely if and only if m sub q spl ges m sub h 1 p 1 where x is the smallest integer greater than or equal to x the results also provide conditions for full rank of certain key matrices arising in the blind multichannel deconvolution problem

normalized incremental subgradient algorithm and its application the problem of minimizing the sum of a number of component functions is of great importance in the real world in this paper a new incremental optimization algorithm named normalized incremental subgradient nis algorithm is proposed for a class of such problems where the component functions have common local minima the nis algorithm is performed incrementally just as the general incremental subgradient is algorithm and thus can be implemented in a distributed way in the nis algorithm the update of each subiteration is based on a search direction obtained by individually normalizing each component of subgradients of component functions resulting in much better convergence performance as compared to the is algorithm and other traditional optimization methods e g gauss newton method the convergence of the nis algorithm with both diminishing stepsizes and constant stepsizes is proved and analyzed theoretically two important applications are presented one is to solve a class of convex feasibility problems in a distributed way and the other is distributed maximum likelihood estimation numerical examples arising from two important topics in the area of wireless sensor networks source localization and node localization demonstrate the effectiveness and efficiency of the nis algorithm

stochastic analysis on the path space of a riemannian manifold i markovian stochastic calculus abstract on the brownian flow of a compact riemannian manifold an intrinsic stochastic calculus is defined this calculus is markovian the stochastic calculus of variation on the bundle of orthonormal frames appears only as an intermediate tool the bismut connection plays a paramount role in the final construction

a diffusion approach to network localization the localization of nodes on a network is a challenging research topic it arises in a variety of applications such as communications and sensor network analysis we propose a computational approach to recovering the positions of network nodes given partial and corrupted distance measurements and the positions of a small subset of anchor nodes first we show how to derive geometrically adaptive diffusion bases defined over the entire network given only partial distance measurements second we propose to utilize several diffusion bases simultaneously to derive multiscale diffusion frames last we utilize the diffusion frames to formulate a l1 regression based extension of the anchor points coordinates to the entire network we experimentally show that under a wide range of conditions our method compares favorably with state of the art approaches

the clark ocone formula for vector valued wiener functionals abstract the classical representation of random variables as the ito integral of nonanticipative integrands is extended to include banach space valued random variables on an abstract wiener space equipped with a filtration induced by a resolution of the identity on the cameron martin space the ito integral is replaced in this case by an extension of the divergence to random operators and the operators involved in the representation are adapted with respect to this filtration in a suitably defined sense a complete characterization of measure preserving transformations in wiener space is presented as an application of this generalized clark ocone formula

cooperative simultaneous localization and mapping by exploiting multipath propagation an affordable and reliable indoor positioning is a highly needed service moreover maps of the indoor environment are vital to many applications in this paper a method for joint localization and mapping using multipath delay estimates is developed required high resolution estimates of multipath delays may be obtained using radio frequency or acoustic measurements among a set of nodes in a network in this paper the problem is modeled in two dimensional space with arbitrary node configuration and assuming a convex polygonal room shape joint localization and mapping is formulated as an optimization problem it is subdivided and relaxed into two convex subproblems which can be solved in an alternating manner a method for data association and a low complexity mapping algorithm stemming from hough transform are proposed both the estimation performance and identifiability of the indoor localization problem are improved moreover a basic map of the propagation environment is produced

on convergence rates of linearized proximal algorithms for convex composite optimization with applications in the present paper we investigate a linearized proximal algorithm lpa for solving a convex composite optimization problem each iteration of the lpa is a proximal minimization of the convex composite function with the inner function being linearized at the current iterate the lpa has the attractive computational advantage that the solution of each subproblem is a singleton which avoids the difficulty as in the gauss newton method gnm of finding a solution with minimum norm among the set of minima of its subproblem while still maintaining the same local convergence rate as that of the gnm under the assumptions of local weak sharp minima of order p p in 1 2 and a quasi regularity condition we establish a local superlinear convergence rate for the lpa we also propose a globalization strategy for the lpa based on a backtracking line search and an inexact version of the lpa we further apply the lpa to solve a possibly nonconvex feasibility problem as well as a sensor network localiza

renormalized differential geometry on path space structural equation curvature abstract the theory of integration in infinite dimensions is in some sense the backbone of probability theory on this backbone the stochastic calculus of variations has given rise to the flesh of differential calculus its first step is the construction at each point of the probability space of a cameron martin like tangent space in which the desired differential calculus can be developed this construction proceeds along the lines of first order differential geometry in this paper we address the following questions what could be the meaning of curvature of the probability space how and why how can curvatures be defined and computed why could a second order differential geometry be relevant to stochastic analysis we try to answer these questions for the probability space associated to the brownian motion of a compact riemannian manifold why a basicenergy identity for anticipative stochastic integralswill be obtained as a byproduct of our computation of curvature how there are essentially four bottlenecks in the development of differential geometry on wiener riemann manifolds i the difficulty of finding an atlas of local charts such that the changes of charts preserve the class of the wiener like measures together with their associated cameron martin like tangent spaces ii the difficulty of finding cylindrical approximations preserving the natural geometrical objects iii the difficulty of renormalizing the divergent series to which the summation operations of finite dimensional differential geometry give rise in the non intrinsic context of local charts iv the nonavailability of the computational procedures analogous to the local coordinates systems of the classical differential geometry in the context of path space the ito filtration provides a much richer structure than that available in the framework of an abstract wiener riemann manifold our work is a systematic attempt to replace the machinery of local charts with amethodology of moving frames in our context stochastic parallel transport provides a canonical moving frame on the path space the concept of a cylindrical approximation has to be reshaped in our new situation into some geometric limit theorems establishing that the riemannian geometric objects of the cylindrical approximations induce by a limiting procedure geometric objects on the path space those limit theorems are reminiscent of the classical theorems which say that a stratonovich sde is the limit of an appropriate sequence of ode the canonical coordinate system provided by the moving frame will make it possible to proceed to the needed renormalizations byintrinsicstochastic integrals in this context the anticipative stochastic integral theory of nualart and pardoux will play a decisive role finally the moving frame will provide aneffective algorithm of computation for this differential geometry in infinite dimensions in our study we encounter a new type of renormalization thehypoelliptic renormalization which corresponds to the fact that the bracket of smooth vector fields taking their values in the cameron martin space can get out of this cameron martin space this hypoelliptic problem induces the nonrenormalizability of some geometrical objects it leads also to a concept oftangent processesto probability spaces extending that based on the cameron martin theorem for tangent processes a formula of integration by parts still holds furthermore the tangent processes form a lie algebra under the bracket on the other hand tangent processes cannot be stochastically integrated this operation is well defined only for cameron martin type vector fields

visualizing data as objects by dc difference of convex optimization in this paper we address the problem of visualizing in a bounded region a set of individuals which has attached a dissimilarity measure and a statistical value as convex objects this problem which extends the standard multidimensional scaling analysis is written as a global optimization problem whose objective is the difference of two convex functions dc suitable dc decompositions allow us to use the difference of convex algorithm dca in a very efficient way our algorithmic approach is used to visualize two real world datasets

identifiability and stability in blind deconvolution under minimal assumptions blind deconvolution bd arises in many applications without assumptions on the signal and the filter bd does not admit a unique solution in practice subspace or sparsity assumptions have shown the ability to reduce the search space and yield the unique solution however existing theoretical analysis on uniqueness in bd is rather limited in an earlier paper we provided the first algebraic sample complexities for bd that hold for lebesgue almost all bases or frames we showed that for bd of a pair of vectors in mathbb c n with subspace constraints of dimensions m 1 and m 2 respectively a sample complexity of n geq m 1 m 2 is sufficient this result is suboptimal since the number of degrees of freedom is merely m 1 m 2 1 we provided analogous results with similar suboptimality for bd with sparsity or mixed subspace and sparsity constraints in this paper taking advantage of the recent progress on the information theoretic limits of unique low rank matrix recovery we finally bridge this gap and derive an optimal sample complexity result for bd with generic bases or frames we show that for bd of an arbitrary pair respectively all pairs of vectors in mathbb c n with sparsity constraints of sparsity levels s 1 and s 2 a sample complexity of n s 1 s 2 respectively n 2 s 1 s 2 is sufficient we also present analogous results for bd with subspace constraints or mixed constraints with the subspace dimension replacing the sparsity level last but not least in all the above scenarios if the bases or frames follow a probabilistic distribution specified in this paper the recovery is not only unique but also stable against small perturbations in the measurements under the same sample complexities

sparsity in sums of squares of polynomials representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming sdp relaxation of polynomial optimization problems we discuss effective methods to obtain a simpler representation of a sparse polynomial as a sum of squares of sparse polynomials by eliminating redundancy

spaseloc an adaptive subproblem algorithm for scalable wireless sensor network localization an adaptive rule based algorithm spaseloc is described to solve localization problems for ad hoc wireless sensor networks a large problem is solved as a sequence of very small subproblems each of which is solved by semidefinite programming relaxation of a geometric optimization model the subproblems are generated according to a set of sensor anchor selection rules computational results compared with existing approaches show that the spaseloc algorithm scales well and provides excellent localization accuracy

distributed wireless sensor network localization via sequential greedy optimization algorithm node localization is essential to most applications of wireless sensor networks wsns in this paper we consider both range based node localization and range free node localization with uncertainties in range measurements radio range and anchor positions first a greedy optimization algorithm named sequential greedy optimization sgo algorithm is presented which is more suitable for distributed optimization in networks than the classical nonlinear gauss seidel algorithm then a unified optimization framework is proposed for both range based localization and range free localization and two convex localization formulations are obtained based on semidefinite programming sdp relaxation techniques by applying the sgo algorithm to the edge based sdp relaxation formulation we propose a second order cone programming socp based distributed node localization algorithm two distributed refinement algorithms are also proposed by using the sgo algorithm to nonconvex localization formulations the proposed localization algorithms all can be implemented partially asynchronously in networks finally extensive simulations are conducted to demonstrate the efficiency and accuracy of the proposed distributed localization algorithms

the divergence of banach space valued random variables on wiener space the domain of definition of the divergence operator on an abstract wiener space w h is extended to include w valued and open image in new window valued integrands the main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in s extended domain have divergence zero these results are then applied to the analysis of quasiinvariant flows induced by w valued vector fields and among other results it turns out that these divergence free vector fields are responsible for generating measure preserving flows

the positive definite completion problem relative to a subspace abstract we consider the problem of completing a partial real symmetric matrix so that its quadratic from is positive semi definite on a subspace presented as the null space of a given matrix under the appropriate definition of partial positive definite the classical result regarding chordal patterns is extended to this setting however unlike the classical setting there are differences between the positive semi definite and positive definite completion problems

universal rigidity and edge sparsification for sensor network localization owing to their high accuracy and ease of formulation there has been great interest in applying convex optimization techniques particularly that of semidefinite programming sdp relaxation to tackle the sensor network localization problem in recent years however a drawback of such techniques is that the resulting convex program is often expensive to solve in order to speed up computation various edge sparsification heuristics have been proposed whose aim is to reduce the number of edges in the input graph although these heuristics do reduce the size of the convex program and hence make it faster to solve they are often ad hoc in nature and do not preserve the localization properties of the input as such one often has to face a tradeoff between solution accuracy and computational effort in this paper we propose a novel edge sparsification heuristic that can provably preserve the localization properties of the original input at the heart of our heuristic is a graph decomposition procedure which allows us to identify certain sparse generically universally rigid subgraphs of the input graph our computational results show that the proposed approach can significantly reduce the computational and memory complexities of sdp based algorithms for solving the sensor network localization problem moreover it compares favorably with existing speedup approaches both in terms of accuracy and solution time

complete dictionary recovery over the sphere i overview and the geometric picture we consider the problem of recovering a complete i e square and invertible matrix a 0 from y in mathbb r n times p with y a 0 x 0 provided x 0 is sufficiently sparse this recovery problem is central to theoretical understanding of dictionary learning which seeks a sparse representation for a collection of input signals and finds numerous applications in modern signal processing and machine learning we give the first efficient algorithm that provably recovers a 0 when x 0 has o left n right nonzeros per column under suitable probability model for x 0 in contrast prior results based on efficient algorithms either only guarantee recovery when x 0 has o sqrt n zeros per column or require multiple rounds of semidefinite programming relaxation to work when x 0 has o n nonzeros per column our algorithmic pipeline centers around solving a certain nonconvex optimization problem with a spherical constraint in this paper we provide a geometric characterization of the objective landscape in particular we show that the problem is highly structured with high probability 1 there are no spurious local minimizers and 2 around all saddle points the objective has a negative directional curvature this distinctive structure makes the problem amenable to efficient optimization algorithms in a companion paper we design a second order trust region algorithm over the sphere that provably converges to a local minimizer from arbitrary initializations despite the presence of saddle points

simple and fast convex relaxation method for cooperative localization in sensor networks using range measurements we address the sensor network localization problem given noisy range measurements between pairs of nodes we approach the nonconvex maximum likelihood formulation via a known simple convex relaxation we exploit its favorable optimization properties to the full to obtain an approach that is completely distributed has a simple implementation at each node and capitalizes on an optimal gradient method to attain fast convergence we offer a parallel but also an asynchronous flavor both with theoretical convergence guarantees and iteration complexity analysis experimental results establish leading performance our algorithms top the accuracy of a comparable state of the art method by one order of magnitude using one order of magnitude fewer communications

distributed maximum likelihood sensor network localization we propose a class of convex relaxations to solve the sensor network localization problem based on a maximum likelihood ml formulation this class as well as the tightness of the relaxations depends on the noise probability density function pdf of the collected measurements we derive a computational efficient edge based version of this ml convex relaxation class and we design a distributed algorithm that enables the sensor nodes to solve these edge based convex programs locally by communicating only with their close neighbors this algorithm relies on the alternating direction method of multipliers admm it converges to the centralized solution it can run asynchronously and it is computation error resilient finally we compare our proposed distributed scheme with other available methods both analytically and numerically and we argue the added value of admm especially for large scale networks

second order cone programming relaxation of sensor network localization the sensor network localization problem has been much studied recently biswas and ye proposed a semidefinite programming sdp relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed here we study a second order cone programming socp relaxation of this problem motivated by its simpler structure and its potential to be solved faster than sdp we show that the socp relaxation though weaker than the sdp relaxation has nice properties that make it useful as a problem preprocessor in particular sensors that are uniquely positioned among interior solutions of the socp relaxation are accurate up to the square root of the distance error thus these sensors which are easily identified are accurately positioned in our numerical simulation the interior solution found can accurately position up to 80 90p of the sensors we also propose a smoothing coordinate gradient descent method for finding an interior solution that is faster than an interior point method

phaselift exact and stable signal recovery from magnitude measurements via convex programming suppose we wish to recover a signal input amssym font abc cmmib10 def bi 1 hbox abc 1 bi x in bbb c n from m intensity measurements of the form that is from data in which phase information is missing we prove that if the vectors are sampled independently and uniformly at random on the unit sphere then the signal x can be recovered exactly up to a global phase factor by solving a convenient semidefinite program a trace norm minimization problem this holds with large probability provided that m is on the order of and without any assumption about the signal whatsoever this novel result demonstrates that in some instances the combinatorial phase retrieval problem can be solved by convex programming techniques finally we also prove that our methodology is robust vis a vis additive noise 2012 wiley periodicals inc

sensor network localization by augmented dual embedding in this work we propose an anchor based sensor networks localization scheme that utilizes a dual spectral embedding the input noisy distance measurements are first embedded by diffusion embedding and then by isomap this allows to better estimate the intrinsic network geometry and derive improved adaptive bases that are used to estimate the global localization via l 1 regression we then introduce the augmented dual embedding by computationally augmenting the set of measured distances and computing the dual embedding this significantly improves the scheme s robustness and accuracy we also propose a straightforward approach to preprocessing the noisy distances via the triangle inequality the proposed scheme is experimentally shown to outperform contemporary state of the art localization schemes