iterative reweighted algorithms for matrix rank minimization the problem of minimizing the rank of a matrix subject to affine constraints has applications in several areas including machine learning and is known to be np hard a tractable relaxation for this problem is nuclear norm or trace norm minimization which is guaranteed to find the minimum rank matrix under suitable assumptions in this paper we propose a family of iterative reweighted least squares algorithms irls p with 0 p 1 as a computationally efficient way to improve over the performance of nuclear norm minimization the algorithms can be viewed as locally minimizing certain smooth approximations to the rank function when p 1 we give theoretical guarantees similar to those for nuclear norm minimization that is recovery of low rank matrices under certain assumptions on the operator defining the constraints for p 1 irls p shows better empirical performance in terms of recovering low rank matrices than nuclear norm minimization we provide an efficient implementation for irls p and also present a related family of algorithms sirls p these algorithms exhibit competitive run times and improved recovery when compared to existing algorithms for random instances of the matrix completion problem as well as on the movielens movie recommendation data set

regularity of renormalized solutions in the boltzmann equation with long range interactions it is well established that renormalized solutions of the boltzmann equation enjoy some kind of regularity or at least compactness in the velocity variable when the angular collision kernel is nonintegrable however obtaining explicit estimates in convenient and natural functional settings proves rather difficult in this work we derive a velocity smoothness estimate from the a priori control of the renormalized dissipation as a direct consequence of our result we show that in the presence of long range interactions any renormalized solution f t x v to the boltzmann equation satisfies locally for every and for some s 0 depending on p we also provide an application of this new estimate to the hydrodynamic limit of the boltzmann equation without cutoff 2012 wiley periodicals inc

fluid dynamic limits of kinetic equations i formal derivations the connection between kinetic theory and the macroscopic equations of fluid dynamics is described in particular our results concerning the incompressible navier stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of hilbert or chapman enskog the moment formalism shows that the limit leading to the incompressible navier stokes equations like that leading to the compressible euler equations is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible navier stokes equations some indications of the validity of these limits are given more specifically the connection between the diperna lions renormalized solution of the classical boltzmann equation and the leray solution of the navier stokes equations is discussed

perturbation of the svd in the presence of small singular values abstract this paper gives svd perturbation bounds and expansions that are of use when an m n m n matrix a has small singular values the first part of the paper gives subspace bounds that are closely related to those of wedin but are stated so as to isolate the effect of any small singular values to the left singular subspace in the second part first and second order approximations are given for perturbed singular values the subspace bounds are used to show that all approximations retain accuracy when applied to small singular values the paper concludes by deriving a subspace bound for multiplicative perturbations and using that bound to give a simple approximation to a singular value perturbed by a multiplicative perturbation

heterogeneous tensor decomposition for clustering via manifold optimization tensor clustering is an important tool that exploits intrinsically rich structures in real world multiarray or tensor datasets often in dealing with those datasets standard practice is to use subspace clustering that is based on vectorizing multiarray data however vectorization of tensorial data does not exploit complete structure information in this paper we propose a subspace clustering algorithm without adopting any vectorization process our approach is based on a novel heterogeneous tucker decomposition model taking into account cluster membership information we propose a new clustering algorithm that alternates between different modes of the proposed heterogeneous tensor model all but the last mode have closed form updates updating the last mode reduces to optimizing over the multinomial manifold for which we investigate second order riemannian geometry and propose a trust region algorithm numerical experiments show that our proposed algorithm compete effectively with state of the art clustering algorithms that are based on tensor factorization

the navier stokes limit of the boltzmann equation for bounded collision kernels the present work establishes a navier stokes limit for the boltzmann equation considered over the infinite spatial domain r3 appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations whose limit points in the w l1 topology are governed by leray solutions of the limiting navier stokes equations this completes the arguments in bardos golse levermore commun pure appl math 46 5 667 753 1993 for the steady case and in lions masmoudi arch ration mech anal 158 3 173 193 2001 for the time dependent case

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results

nuclear norm system identification with missing inputs and outputs we present a system identification method for problems with partially missing inputs and outputs the method is based on a subspace formulation and uses the nuclear norm heuristic for structured low rank matrix approximation with the missing input and output values as the optimization variables we also present a fast implementation of the alternating direction method of multipliers admm to solve regularized or non regularized nuclear norm optimization problems with hankel structure this makes it possible to solve quite large system identification problems experimental results show that the nuclear norm optimization approach to subspace identification is comparable to the standard subspace methods when no inputs and outputs are missing and that the performance degrades gracefully as the percentage of missing inputs and outputs increases

the fourier spectrum of critical percolation consider the indicator function f of a 2 dimensional percolation crossing event in this paper the fourier transform of f is studied and sharp bounds are obtained for its lower tail in several situations various applications of these bounds are derived in particular we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension frac 31 36 almost surely and the corresponding dimension in the half plane is frac 5 9 it is also proved that critical bond percolation on the square grid has exceptional times almost surely also the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined

exceptional planes of percolation consider the standard continuous percolation in 4 and choose the parameters so that the induced percolation on a fixed two dimensional linear subspace is critical although two dimensional critical percolation dies we show that there are exceptional two dimensional linear subspaces in which percolation occurs

stokes fourier and acoustic limits for the boltzmann equation convergence proofs we establish a stokes fourier limit for the boltzmann equation considered over any periodic spatial domain of dimension 2 or more appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations that globally in time converge weakly to a unique limit governed by a solution of stokes fourier motion and heat equations provided that the fluid moments of their initial fluctuations converge to appropriate l 2 initial data of the stokesfourier equations both the motion and heat equations are recovered in the limit by controlling the fluxes and the local conservation defects of the diperna lions solutions with dissipation rate estimates the scaling of the fluctuations with respect to knudsen number is essentially optimal the assumptions on the collision kernel are little more than those required for the diperna lions theory and that the viscosity and heat conduction are finite for the acoustic limit these techniques also remove restrictions to bounded collision kernels and improve the scaling of the fluctuations both weak limits become strong when the initial fluctuations converge entropically to appropriate l 2 initial data c 2002 john wiley sons inc

from the boltzmann equations to the equations of incompressible fluid mechanics i we consider here the problem of deriving rigorously from boltzmann s equation globally in time and for general initial conditions fluid mechanics equations such as the navier stokes or euler equations

low rank matrix recovery via modified schatten p norm minimization with convergence guarantees in recent years low rank matrix recovery problems have attracted much attention in computer vision and machine learning the corresponding rank minimization problems are both combinational and np hard in general which are mainly solved by both nuclear norm and schatten p 0 norm based optimization algorithms however inspired by weighted nuclear norm and schatten p norm as the relaxations of rank function the main merits of this work firstly provide a modified schatten p norm in the affine matrix rank minimization problem denoted as the modified schatten p norm minimization ms p nm secondly its surrogate function is constructed and the equivalence relationship with the ms p nm is further achieved thirdly the iterative singular value thresholding algorithm isvta is devised to optimize it and its accelerated version i e aisvta is also obtained to reduce the number of iterations through the well known nesterov s acceleration strategy most importantly the convergence guarantees and their relationship with objective function stationary point and variable sequence generated by the proposed algorithms are established under some specific assumptions e g kurdyka ojasiewicz k property finally numerical experiments demonstrate the effectiveness of the proposed algorithms in the matrix completion problem for image inpainting and recommender systems it should be noted that the accelerated algorithm has a much faster convergence speed and a very close recovery precision when comparing with the proposed non accelerated one

smooth parafac decomposition for tensor completion in recent years low rank based tensor completion which is a higher order extension of matrix completion has received considerable attention however the low rank assumption is not sufficient for the recovery of visual data such as color and 3d images when the ratio of missing data is extremely high in this paper we consider smoothness constraints as well as low rank approximations and propose an efficient algorithm for performing tensor completion that is particularly powerful regarding visual data the proposed method admits significant advantages owing to the integration of smooth parafac decomposition for incomplete tensors and the efficient selection of models in order to minimize the tensor rank thus our proposed method is termed as smooth parafac tensor completion spc in order to impose the smoothness constraints we employ two strategies total variation spc tv and quadratic variation spc qv and invoke the corresponding algorithms for model learning extensive experimental evaluations on both synthetic and real world visual data illustrate the significant improvements of our method in terms of both prediction performance and efficiency compared with many state of the art tensor completion methods

sets avoided by brownian motion a fixed two dimensional projection of a three dimensional brownian motion is almost surely neighborhood recurrent is this simultaneously true of all the two dimensional projections with probability 1 equivalently three dimensional brownian motion hits any infinite cylinder with probability 1 does it hit all cylinders this papers shows that the answer is no brownian motion in three dimensions avoids random cylinders and in fact avoids bodies of revolution that grow almost as fast as cones

acoustic limit for the boltzmann equation in optimal scaling based on a recent l 2 l 1 framework we establish the acoustic limit of the boltzmann equation for general collision kernels the scaling of the fluctuations with respect to knudsen number is optimal our approach is based on a new analysis of the compressible euler limit of the boltzmann equation as well as refined estimates of euler and acoustic solutions

convergence results for projected line search methods on varieties of low rank matrices via łojasiewicz inequality the aim of this paper is to derive convergence results for projected line search methods on the real algebraic variety mathcal m le k of real m times n matrices of rank at most k such methods extend riemannian optimization methods which are successfully used on the smooth manifold mathcal m k of rank k matrices to its closure by taking steps along gradient related directions in the tangent cone and afterwards projecting back to mathcal m le k considering such a method circumvents the difficulties which arise from the nonclosedness and the unbounded curvature of mathcal m k the pointwise convergence is obtained for real analytic functions on the basis of a ojasiewicz inequality for the projection of the antigradient to the tangent cone if the derived limit point lies on the smooth part of mathcal m le k i e in mathcal m k this boils down to more or less known results but with the benefit that asymptotic convergence rate estimates for specific step sizes

low rank matrix completion by riemannian optimization the matrix completion problem consists of finding or approximating a low rank matrix based on a few samples of this matrix we propose a new algorithm for matrix completion that minimizes the least square distance on the sampling set over the riemannian manifold of fixed rank matrices the algorithm is an adaptation of classical nonlinear conjugate gradients developed within the framework of retraction based optimization on manifolds we describe all the necessary objects from differential geometry necessary to perform optimization over this low rank matrix manifold seen as a submanifold embedded in the space of matrices in particular we describe how metric projection can be used as retraction and how vector transport lets us obtain the conjugate search directions finally we prove convergence of a regularized version of our algorithm under the assumption that the restricted isometry property holds for incoherent matrices throughout the iterations the numerical experiments indicate that our approach

phase retrieval via wirtinger flow theory and algorithms we study the problem of recovering the phase from magnitude measurements specifically we wish to reconstruct a complex valued signal boldsymbol x in mathbb c n about which we have phaseless samples of the form y r left langle boldsymbol a r boldsymbol x rangle right 2 r 1 ldots m knowledge of the phase of these samples would yield a linear system this paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm in a nutshell this algorithm starts with a careful initialization obtained by means of a spectral method and then refines this initial estimate by iteratively applying novel update rules which have low computational complexity much like in a gradient descent scheme the main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements indeed the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources in theory a variation on this scheme leads to a near linear time algorithm for a physically realizable model based on coded diffraction patterns we illustrate the effectiveness of our methods with various experiments on image data underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

fluid dynamic limits of kinetic equations ii convergence proofs for the boltzmann equation using relative entropy estimates about an absolute maxwellian it is shown that any properly scaled sequence of diperna lions renormalized solutions of some classical boltzmann equations has fluctuations that converge to an infinitesimal maxwellian with fluid variables that satisfy the incompressibility and boussinesq relations moreover if the initial fluctuations entropically converge to an infinitesimal maxwellian then the limiting fluid variables satisfy a version of the leray energy inequality if the sequence satisfies a local momentum conservation assumption the momentum densities globaly converge to a solution of the stokes equation a similar discrete time version of this result holds for the navier stokes limit with an additional mild weak compactness assumption the continuous time navier stokes limit is also discussed 01993 john wiley sons inc

the acoustic limit for the boltzmann equation the acoustic equations are the linearization of the compressible euler equations about a spatially homogeneous fluid state we first derive them directly from the boltzmann equation as the formal limit of moment equations for an appropriately scaled family of boltzmann solutions we then establish this limit for the boltzmann equation considered over a periodic spatial domain for bounded collision kernels appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations that converge entropically and hence strongly in l 1 to a unique limit governed by a solution of the acoustic equations for all time provided that its initial fluctuations converge entropically to an appropriate limit associated to any given l 2 initial data of the acoustic equations the associated local conservation laws are recovered in the limit

convergence of solutions to the boltzmann equation in the incompressible euler limit we consider here the problem of deriving rigorously for well prepared initial data and without any additional assumption dissipative or smooth solutions of the incompressible euler equations from renormalized solutions of the boltzmann equation this completes the partial results obtained by golse b perthame and l desvillettes eds series in applied mathematics4 2000 gauthier villars paris and lions masmoudi arch rational mech anal 158 2001 195 211

incompressible navier stokes and euler limits of the boltzmann equation we consider solutions of the boltzmann equation in a d dimensional torus d 2 3 for macroscopic times t n 1 t 0 when the space variations are on a macroscopic scale x n 1r n 2 x in the unit torus let u x t be for t t0 a smooth solution of the incompressible navier stokes equations ins for n 2 and of the incompressible euler equation ie for n 2 we prove that has solutions for t t0 which are close to o 2 in a suitable norm to the local maxwellian p 2 t d 2 exp v u x t 2 2t with constant density p and temperature t this is a particular case defined by the choice of initial values of the macroscopic variables of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations for n 3 these equations correspond to variable density ie while for n 2 they involve higher order derivatives of the density

from the boltzmann equation to the incompressible navier stokes equations on the torus a quantitative error estimate we investigate the boltzmann equation depending on the knudsen number in the navier stokes perturbative setting on the torus using hypocoer civity we derive a new proof of existence and exponential decay for solutions close to a global equilibrium with explicit regularity bounds and rates of convergence these results are uniform in the knudsen number and thus allow us to obtain a strong derivation of the incompressible navier stokes equations as the knudsen number tends to 0 moreover our method is also used to deal with other kinetic models finally we show that the study of the hydrodynamical limit is rather different on the torus than the one already proved in the whole space as it requires averaging in time unless the initial layer conditions are satisfied

a riemannian symmetric rank one trust region method the well known symmetric rank one trust region method where the hessian approximation is generated by the symmetric rank one update is generalized to the problem of minimizing a real valued function over a d d dimensional riemannian manifold the generalization relies on basic differential geometric concepts such as tangent spaces riemannian metrics and the riemannian gradient as well as on the more recent notions of first order retraction and vector transport the new method called rtr sr1 is shown to converge globally and d 1 d 1 step q superlinearly to stationary points of the objective function a limited memory version referred to as lrtr sr1 is also introduced in this context novel efficient strategies are presented to construct a vector transport on a submanifold of a euclidean space numerical experiments rayleigh quotient minimization on the sphere and a joint diagonalization problem on the stiefel manifold illustrate the value of the new methods

local time on the exceptional set of dynamical percolation and the incipient infinite cluster in dynamical critical site percolation on the triangular lattice or bond percolation on z2 we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster we show that at a typical time with respect to this measure the percolation configuration has the law of kesten s incipient infinite cluster in the most technical result of this paper we show that on the other hand at the first exceptional time the law of the configuration is different we believe that the two laws are mutually singular but do not show this we also study the collapse of the infinite cluster near typical exceptional times and establish a relation between static and dynamic exponents analogous to kesten s near critical relation

which properties of a random sequence are dynamically sensitive consider a sequence of i i d random variables where each variable is refreshed i e replaced by an independent variable with the same law independently according to a poisson clock at any fixed time t the resulting sequence has the same law as at time 0 but there can be exceptional random times at which certain almost sure properties of the time 0 sequence are violated we prove that there are no such exceptional times for the law of large numbers and the law of the iterated logarithm so these laws are dynamically stable however there are times at which run lengths are exceptionally long that is run tests are dynamically sensitive we obtain a multifractal analysis of exceptional times for run lengths and for prediction in particular starting from an i i d sequence of unbiased random bits the random set of times t where log2 n bits among the first n bits can be predicted from their predecessors has hausdorff dimension 1 a s finally we consider simple random walk in the lattice zd and prove that transience is dynamically stable for d 5 and dynamically sensitive for d 3 4 moreover for d 3 4 the nonempty random set of exceptional times t where the walk is recurrent has hausdorff dimension 4 d 2 a s

harmonic mean iteratively reweighted least squares for low rank matrix recovery we propose a new iteratively reweighted least squares irls algorithm for the recovery of a matrix x cd1 d2 of rank r min d1 d2 from incomplete linear observations solving a sequence of low complexity linear problems the easily implementable algorithm which we call harmonic mean iteratively reweighted least squares hm irls optimizes a non convex schatten p quasi norm penalization to promote low rankness and carries three major strengths in particular for the matrix completion setting first we observe a remarkable global convergence behavior of the algorithm s iterates to the low rank matrix for relevant interesting cases for which any other state of the art optimization approach fails the recovery secondly hm irls exhibits an empirical recovery probability close to 1 even for a number of measurements very close to the theoretical lower bound r d1 d2 r i e already for significantly fewer linear observations than any other tractable approach in the literature thirdly hm irls exhibits a locally superlinear rate of convergence of order 2 p if the linear observations fulfill a suitable null space property while for the first two properties we have so far only strong empirical evidence we prove the third property as our main theoretical result

from boltzmann s equation to the incompressible navier stokes fourier system with long range interactions we establish a rigorous demonstration of the hydrodynamic convergence of the boltzmann equation towards a navier stokes fourier system under the presence of long range interactions this convergence is obtained by letting the knudsen number tend to zero and has been known to hold at least formally for decades it is only more recently that a fully rigorous mathematical derivation of this hydrodynamic limit was discovered however these results failed to encompass almost all physically relevant collision kernels due to a cutoff assumption which requires that the cross sections be integrable indeed as soon as long range intermolecular forces are present non integrable collision kernels have to be considered because of the enormous number of grazing collisions in the gas in this long range setting the boltzmann operator becomes a singular integral operator and the known rigorous proofs of hydrodynamic convergence simply do not carry over to that case in fact the diperna lions renormalized solutions do not even make sense in this situation and the relevant global solutions to the boltzmann equation are the so called renormalized solutions with a defect measure developed by alexandre and villani our work overcomes the new mathematical difficulties coming from the consideration of long range interactions by proving the hydrodynamic convergence of the alexandre villani solutions towards the leray solutions

exceptional times and invariance for dynamical random walks consider a sequence open image in new window of i i d random variables associate to each xi 0 an independent mean one poisson clock every time a clock rings replace that x variable by an independent copy and restart the clock in this way we obtain i i d stationary processes xi t t 0 i 1 2 whose invariant distribution is the law of x1 0

riemannian gradient descent methods for graph regularized matrix completion abstract low rank matrix completion is the problem of recovering the missing entries of a data matrix by using the assumption that the true matrix admits a good low rank approximation much attention has been given recently to exploiting correlations between the column row entities to improve the matrix completion quality in this paper we propose preconditioned gradient descent algorithms for solving the low rank matrix completion problem with graph laplacian based regularizers experiments on synthetic data show that our approach achieves significant speedup compared to an existing method based on alternating minimization experimental results on real world data also show that our methods provide low rank solutions of similar quality in comparable or less time than the state of the art method

hausdorff dimension of the contours of symmetric additive levy processes let x 1 x n denote n independent symmetric levy processes on r d the corresponding additive levy process is defined as the following n parameter random field on r d mathfrak x t x 1 t 1 cdots x n t n quad t in bf r n khoshnevisan and xiao ann probab 30 1 62 100 2002 have found a necessary and sufficient condition for the zero set mathfrak x 1 0 of mathfrak x to be non trivial with positive probability they also provide bounds for the hausdorff dimension of mathfrak x 1 0 which hold with positive probability in the case that mathfrak x 1 0 can be non void here we prove that the hausdorff dimension of mathfrak x 1 0 is a constant almost surely on the event mathfrak x 1 0 neq varnothing moreover we derive a formula for the said constant this portion of our work extends the well known formulas of horowitz israel j math 6 176 182 1968 and hawkes j lond math soc 8 517 525 1974 both of which hold for one parameter levy processes more generally we prove that for every nonrandom borel set f in 0 n the hausdorff dimension of mathfrak x 1 0 cap f is a constant almost surely on the event mathfrak x 1 0 cap f neq varnothing this constant is computed explicitly in many cases

global renormalized solutions and navier stokes limit of the boltzmann equation with incoming boundary condition for long range interaction abstract for the long range interaction we prove the global existence of renormalized solutions to the boltzmann equation with incoming boundary condition furthermore as knudsen number goes to zero the limit to the incompressible navier stokes limit with homogeneous dirichlet boundary condition is justified when the boundary data of the scaled boltzmann equation is close to the maxwellian with order o 3 in the sense of boundary relative entropy

the phase transition of matrix recovery from gaussian measurements matches the minimax mse of matrix denoising let be an unknown m by n matrix in matrix recovery one takes linear measurements of where and each is an m by n matrix a popular approach for matrix recovery is nuclear norm minimization nnm solving the convex optimization problem for all where denotes the nuclear norm namely the sum of singular values empirical work reveals a phase transition curve stated in terms of the undersampling fraction rank fraction and aspect ratio specifically when the measurement matrices ai have independent standard gaussian random entries a curve exists such that if nnm typically succeeds for large m n whereas if it typically fails an apparently quite different problem is matrix denoising in gaussian noise in which an unknown m by n matrix is to be estimated based on direct noisy measurements where the matrix z has independent and identically distributed gaussian entries a popular matrix denoising scheme solves the unconstrained optimization problem when optimally tuned this scheme achieves the asymptotic minimax mean squared error where we report extensive experiments showing that the phase transition in the first problem matrix recovery from gaussian measurements coincides with the minimax risk curve in the second problem matrix denoising in gaussian noise for any rank fraction at each common aspect ratio our experiments considered matrices belonging to two constraint classes real m by n matrices of various ranks and aspect ratios and real symmetric positive semidefinite n by n matrices of various ranks

compressive multiplexing of correlated signals we present a general architecture for the acquisition of ensembles of correlated signals the signals are multiplexed onto a single line by mixing each one against a different code and then adding them together and the resulting signal is sampled at a high rate we show that if the m signals each band limited to w 2 hz can be approximated by a superposition of r m underlying signals then the ensemble can be recovered by sampling at a rate within a logarithmic factor of rw as compared with the cumulative nyquist rate of mw this sampling theorem shows that the correlation structure of the signal ensemble can be exploited in the acquisition process even though it is unknown a priori the reconstruction of the ensemble is recast as a low rank matrix recovery problem from linear measurements the architectures we are considering impose a certain type of structure on the linear operators although our results depend on the mixing forms being random this imposed structure results in a very different type of random projection than those analyzed in the low rank recovery literature to date

tree indexed random walks on groups and first passage percolation suppose that i i d random variables are attached to the edges of an infinite tree when the tree is large enough the partial sumss along some of its infinite paths will exhibit behavior atypical for an ordinary random walk this principle has appeared in works on branching random walks first passage percolation and rwre on trees we establish further quantitative versions of this principle which are applicable in these settings in particular different notions of speed for such a tree indexed walk correspond to different dimension notions for trees finally if the labeling variables take values in a group then properties of the group e g polynomial growth or a nontrivial poisson boundary are reflected in the sample path behavior of the resulting tree indexed walk

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

low rank optimization with trace norm penalty the paper addresses the problem of low rank trace norm minimization we propose an algorithm that alternates between fixed rank optimization and rank one updates the fixed rank optimization is characterized by an efficient factorization that makes the trace norm differentiable in the search space and the computation of duality gap numerically tractable the search space is nonlinear but is equipped with a riemannian structure that leads to efficient computations we present a second order trust region algorithm with a guaranteed quadratic rate of convergence overall the proposed optimization scheme converges superlinearly to the global solution while maintaining complexity that is linear in the number of rows and columns of the matrix to compute a set of solutions efficiently for a grid of regularization parameters we propose a predictor corrector approach that outperforms the naive warm restart approach on the fixed rank quotient manifold the performance of the proposed algorithm is illustrated on p

optimization on the hierarchical tucker manifold applications to tensor completion in this work we develop an optimization framework for problems whose solutions are well approximated by hierarchical tucker ht tensors an efficient structured tensor format based on recursive subspace factorizations by exploiting the smooth manifold structure of these tensors we construct standard optimization algorithms such as steepest descent and conjugate gradient for completing tensors from missing entries our algorithmic framework is fast and scalable to large problem sizes as we do not require svds on the ambient tensor space as required by other methods moreover we exploit the structure of the gramian matrices associated with the ht format to regularize our problem reducing overfitting for high subsampling ratios we also find that the organization of the tensor can have a major impact on completion from realistic seismic acquisition geometries these samplings are far from idealized randomized samplings that are usually considered in the literature but are realizable in practical scenarios using these algorithms we successfully interpolate large scale seismic data sets and demonstrate the competitive computational scaling of our algorithms as the problem sizes grow

from the boltzmann equation to the stokes fourier system in a bounded domain we prove that the renormalized solutions of the boltzmann equation considered in a bounded domain with different types of kinetic boundary conditions converge to the stokes fourier system with different types of fluid boundary conditions when the main free path goes to zero this extends the work of f golse and d levermore 9 to the case of a bounded domain 2003 wiley periodicals inc

an extended frank wolfe method with in face directions and its application to low rank matrix completion motivated principally by the low rank matrix completion problem we present an extension of the frank wolfe method that is designed to induce near optimal solutions on low dimensional faces of the feasible region this is accomplished by a new approach to generating in face directions at each iteration as well as through new choice rules for selecting between in face and regular frank wolfe steps our framework for generating in face directions generalizes the notion of away steps introduced by wolfe in particular the in face directions always keep the next iterate within the minimal face containing the current iterate we present computational guarantees for the new method that trade off efficiency in computing near optimal solutions with upper bounds on the dimension of minimal faces of iterates we apply the new method to the matrix completion problem where low dimensional faces correspond to low rank matrices we present computational results that demonstrate the effectiveness of our methodo

moment closure approximations of the boltzmann equation based on varphi divergences this paper is concerned with approximations of the boltzmann equation based on the method of moments we propose a generalization of the setting of the moment closure problem from relative entropy to varphi divergences and a corresponding closure procedure based on minimization of varphi divergences the proposed description encapsulates as special cases grad s classical closure based on expansion in hermite polynomials and levermore s entropy based closure we establish that the generalization to divergence based closures enables the construction of extended thermodynamic theories that avoid essential limitations of the standard moment closure formulations such as inadmissibility of the approximate phase space distribution potential loss of hyperbolicity and singularity of flux functions at local equilibrium the divergence based closure leads to a hierarchy of tractable symmetric hyperbolic systems that retain the fundamental structural properties of the boltzmann equation

sparse density estimation on the multinomial manifold a new sparse kernel density estimator is introduced based on the minimum integrated square error criterion for the finite mixture model since the constraint on the mixing coefficients of the finite mixture model is on the multinomial manifold we use the well known riemannian trust region rtr algorithm for solving this problem the first and second order riemannian geometry of the multinomial manifold are derived and utilized in the rtr algorithm numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with an accuracy competitive with those of existing kernel density estimators

the boltzmann equation with soft potentials near a local maxwellian in this paper we consider the boltzmann equation with soft potentials and prove the stability of a class of non trivial profiles defined as some given local maxwellians the method consists of the analytic techniques for viscous conservation laws properties of burnett functions and the energy method through the micro macro decomposition of the boltzmann equation in particular one of the key observations is a detailed analysis of the burnett functions so that the energy estimates can be obtained in a clear way as an application of the main results in this paper we prove the large time nonlinear asymptotic stability of rarefaction waves to the boltzmann equation with soft potentials

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

compressible navier stokes approximation to the boltzmann equation abstract even though the system of the compressible navier stokes equations is not a limiting system of the boltzmann equation when the knudsen number tends to zero it is the second order approximation by applying the chapman enskog expansion the purpose of this paper is to justify this approximation rigorously in mathematics that is if the difference between the initial data for the compressible navier stokes equations and the boltzmann equation is of the second order of the knudsen number so is the difference between two solutions for all time the analysis is based on a refined energy method for a fluid type system using the techniques for the system of viscous conservation laws

an entropy stable discontinuous galerkin finite element moment method for the boltzmann equation this paper presents a numerical approximation technique for the boltzmann equation based on a moment system approximation in velocity dependence and a discontinuous galerkin finite element approximation in position dependence the closure relation for the moment systems derives from minimization of a suitable divergence this divergence based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the boltzmann equation the approximation in position dependence is based on the discontinuous galerkin finite element method the resulting combined discontinuous galerkin moment method corresponds to a galerkin approximation of the boltzmann equation in renormalized form the new momentclosure formulation engenders a new upwind numerical flux function based on half space integrals of the approximate distribution we establish that the proposed upwind flux ensures entropy dissipation of the discontinuous galerkin finite element approximation numerical results are presented for a one dimensional test case

hilbert expansion from the boltzmann equation to relativistic fluids we study the local in time hydrodynamic limit of the relativistic boltzmann equation using a hilbert expansion more specifically we prove the existence of local solutions to the relativistic boltzmann equation that are nearby the local relativistic maxwellians the maxwellians are constructed from a class of solutions to the relativistic euler equations that includes a large subclass of near constant non vacuum fluid states in particular for small knudsen number these solutions to the relativistic boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic euler equations

boundary layers and incompressible navier stokes fourier limit of the boltzmann equation in bounded domain i we establish the incompressible navier stokes fourier limit for solutions to the boltzmann equation with a general cutoff collision kernel in a bounded domain appropriately scaled families of diperna lions mischler renormalized solutions with maxwell reflection boundary conditions are shown to have fluctuations that converge as the knudsen number goes to 0 every limit point is a weak solution to the navier stokes fourier system with different types of boundary conditions depending on the ratio between the accommodation coefficient and the knudsen number the main new result of the paper is that this convergence is strong in the case of the dirichlet boundary condition indeed we prove that the acoustic waves are damped immediately namely they are damped in a boundary layer in time this damping is due to the presence of viscous and kinetic boundary layers in space as a consequence we also justify the first correction to the infinitesimal maxwellian that one obtains from the chapman enskog expansion with navier stokes scaling this extends the work of golse and saint raymond 20 21 and levermore and masmoudi 28 to the case of a bounded domain the case of a bounded domain was considered by masmoudi and saint raymond 34 for the linear stokes fourier limit and saint raymond 41 for the navier stokes limit for hard potential kernels neither 34 nor 41 studied the damping of the acoustic waves this paper extends the result of 34 41 to the nonlinear case and includes soft potential kernels more importantly for the dirichlet boundary condition this work strengthens the convergence so as to make the boundary layer visible this answers an open problem proposed by ukai 46 2016 wiley periodicals inc

generalized reduced rank latent factor regression for high dimensional tensor fields and neuroimaging genetic applications we propose a generalized reduced rank latent factor regression model grrlf for the analysis of tensor field responses and high dimensional covariates the model is motivated by the need from imaging genetic studies to identify genetic variants that are associated with brain imaging phenotypes often in the form of high dimensional tensor fields grrlf identifies from the structure in the data the effective dimensionality of the data and then jointly performs dimension reduction of the covariates dynamic identification of latent factors and nonparametric estimation of both covariate and latent response fields after accounting for the latent and covariate effects grllf performs a nonparametric test on the remaining factor of interest grrlf provides a better factorization of the signals compared with common solutions and is less susceptible to overfitting because it exploits the effective dimensionality the generality and the flexibility of grrlf also allow various statistical models to be handled in a unified framework and solutions can be efficiently computed within the field of neuroimaging it improves the sensitivity for weak signals and is a promising alternative to existing approaches the operation of the framework is demonstrated with both synthetic datasets and a real world neuroimaging example in which the effects of a set of genes on the structure of the brain at the voxel level were measured and the results compared favorably with those from existing approaches

from the boltzmann equation to an incompressible navier stokes fourier system we establish a navier stokes fourier limit for solutions of the boltzmann equation considered over any periodic spatial domain of dimension two or more we do this for a broad class of collision kernels that relaxes the grad small deflection cutoff condition for hard potentials and includes for the first time the case of soft potentials appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations that are compact every limit point is governed by a weak solution of a navier stokes fourier system for all time

majorized proximal alternating imputation for regularized rank constrained matrix completion abstract low rank approximation in the presence of missing data is ubiquitous in many areas of science and engineering as an np hard problem spectral regularization e g nuclear norm schatten p norm is widely used however solving this problem directly can be computationally expensive due to the unknown rank of variables and full rank singular value decompositions svd bilinear factorization is efficient and effectively benefits from the fast numerical optimization however most of existing methods require explicit knowledge of the rank and often tend to be flatlining or over fitting to balance and take advantage of both we formulate the matrix completion problem under the regularized rank constrained conditions we relax this problem to the surrogates with majorization minimization mm by employing a proximal alternating minimization method to optimize the corresponding surrogates we propose an efficient and accurate algorithm named majorized proximal alternating imputation mpa impute algorithm the proposed method iteratively replaces the missing data at each step with the most recent estimate and only involves simple matrix computation and small scale svd theoretically the proposed method is guaranteed to converge to the stationary point under mild conditions moreover the proposed method can be generalized to a family of non convex regularization problems extensive experiments over synthetic data and real world dataset demonstrate the superior performance of our method

homogenization and hydrodynamic limit for fermi dirac statistics coupled to a poisson equation this paper deals with the diffusion approximation of a boltzmann poisson system modeling fermi dirac statistics in the presence of an extra external oscillating electrostatic potential here we extend the analysis done in 19 to the case of a nonlinear collision operator in addition to the averaging lemma and control from entropy dissipation used in 19 here we use two scale young measures and renormalization techniques to prove the convergence this result rigorously justifies the formal analysis of 3 2015 wiley periodicals inc

on dynamical gaussian random walks motivated by the recent work of benjamini haggstrom peres and steif ann probab 34 2003 1 34 on dynamical random walks we do the following i prove that after a suitable normalization the dynamical gaussian walk converges weakly to the ornstein uhlenbeck process in classical wiener space ii derive sharp tail asymptotics for the probabilities of large deviations of the said dynamical walk and iii characterize by way of an integral test the minimal envelope s for the growth rate of the dynamical gaussian walk this development also implies the tail capacity estimates of mountford for large deviations in classical wiener space the results of this paper give a partial affirmative answer to the problem raised in benjamini haggstrom peres and steif ann probab 34 2003 1 34 question 4 of whether there are precise connections between the ou process in classical wiener space and dynamical random walks