a generalized kernel approach to dissimilarity based classification usually objects to be classified are represented by features in this paper we discuss an alternative object representation based on dissimilarity values if such distances separate the classes well the nearest neighbor method offers a good solution however dissimilarities used in practice are usually far from ideal and the performance of the nearest neighbor rule suffers from its sensitivity to noisy examples we show that other more global classification techniques are preferable to the nearest neighbor rule in such cases for classification purposes two different ways of using generalized dissimilarity kernels are considered in the first one distances are isometrically embedded in a pseudo euclidean space and the classification task is performed there in the second approach classifiers are built directly on distance kernels both approaches are described theoretically and then compared using experiments with different dissimilarity measures and datasets including degraded data simulating the problem of missing values

convex optimization learning of faithful euclidean distance representations in nonlinear dimensionality reduction classical multidimensional scaling only works well when the noisy distances observed in a high dimensional space can be faithfully represented by euclidean distances in a low dimensional space advanced models such as maximum variance unfolding mvu and minimum volume embedding mve use semi definite programming sdp to reconstruct such faithful representations while those sdp models are capable of producing high quality configuration numerically they suffer two major drawbacks one is that there exist no theoretically guaranteed bounds on the quality of the configuration the other is that they are slow in computation when the data points are beyond moderate size in this paper we propose a convex optimization model of euclidean distance matrices we establish a non asymptotic error bound for the random graph model with sub gaussian noise and prove that our model produces a matrix estimator of high accuracy when the order of the uniform sample size is roughly the degree of freedom of a low rank matrix up to a logarithmic factor our results partially explain why mvu and mve often work well moreover the convex optimization model can be efficiently solved by a recently proposed 3 block alternating direction method of multipliers numerical experiments show that the model can produce configurations of high quality on large data points that the sdp approach would struggle to cope with

a simpler approach to matrix completion this paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix these results improve on prior work by candes and recht 2009 candes and tao 2009 and keshavan et al 2009 the reconstruction is accomplished by minimizing the nuclear norm or sum of the singular values of the hidden matrix subject to agreement with the provided entries if the underlying matrix satisfies a certain incoherence condition then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition the proof of this assertion is short self contained and uses very elementary analysis the novel techniques herein are based on recent work in quantum information theory

characterization of the subdifferential of some matrix norms abstract a characterization is given of the subdifferential of matrix norms from two classes orthogonally invariant norms and operator or subordinate norms specific results are derived for some special cases

on some difference equations with eventually periodic solutions abstract we study the boundedness and the periodic character of the solutions of the equation formula where x 1 x 0 z and 1 1

subharmonic solutions of systems of difference equations with periodic perturbations part i existence we consider the existence of subharmonic solutions of systems of difference equations with periodic perturbations the theory of coincidence degree coupled with some detailed a priori estimates is applied to show that the perturbed system admits subharmonic solutions near a hyperbolic periodic orbit of the unperturbed system if the perturbation is sufficiently small

a schur complement based semi proximal admm for convex quadratic conic programming and extensions this paper is devoted to the design of an efficient and convergent semi proximal alternating direction method of multipliers admm for finding a solution of low to medium accuracy to convex quadratic conic programming and related problems for this class of problems the convergent two block semi proximal admm can be employed to solve their primal form in a straightforward way however it is known that it is more efficient to apply the directly extended multi block semi proximal admm though its convergence is not guaranteed to the dual form of these problems naturally one may ask the following question can one construct a convergent multi block semi proximal admm that is more efficient than the directly extended semi proximal admm indeed for linear conic programming with 4 block constraints this has been shown to be achievable in a recent paper by sun et al 2014 arxiv 1404 5378 inspired by the aforementioned work and with the convex quadratic conic programming in mind we propose a schur complement based convergent semi proximal admm for solving convex programming problems with a coupling linear equality constraint whose objective function is the sum of two proper closed convex functions plus an arbitrary number of convex quadratic or linear functions our convergent semi proximal admm is particularly suitable for solving convex quadratic semidefinite programming qsdp with constraints consisting of linear equalities a positive semidefinite cone and a simple convex polyhedral set the efficiency of our proposed algorithm is demonstrated by numerical experiments on various examples including qsdp

matrix completion from noisy entries given a matrix m of low rank we consider the problem of reconstructing it from noisy observations of a small random subset of its entries the problem arises in a variety of applications from collaborative filtering the netflix problem to structure from motion and positioning we study a low complexity algorithm introduced by keshavan montanari and oh 2010 based on a combination of spectral techniques and manifold optimization that we call here optspace we prove performance guarantees that are order optimal in a number of circumstances

a rank corrected procedure for matrix completion with fixed basis coefficients for the problems of low rank matrix completion the efficiency of the widely used nuclear norm technique may be challenged under many circumstances especially when certain basis coefficients are fixed for example the low rank correlation matrix completion in various fields such as the financial market and the low rank density matrix completion from the quantum state tomography to seek a solution of high recovery quality beyond the reach of the nuclear norm in this paper we propose a rank corrected procedure using a nuclear semi norm to generate a new estimator for this new estimator we establish a non asymptotic recovery error bound more importantly we quantify the reduction of the recovery error bound for this rank corrected procedure compared with the one obtained for the nuclear norm penalized least squares estimator this reduction can be substantial around 50 we also provide necessary and sufficient conditions for rank consistency in the sense of bach j mach learn res 9 1019 1048 2008 very interestingly these conditions are highly related to the concept of constraint nondegeneracy in matrix optimization as a byproduct our results provide a theoretical foundation for the majorized penalty method of gao and sun a majorized penalty approach for calibrating rank constrained correlation matrix problems http www math nus edu sg matsundf majorpen may5 pdf 2010 and gao 2010 for structured low rank matrix optimization problems extensive numerical experiments demonstrate that our proposed rank corrected procedure can simultaneously achieve a high recovery accuracy and capture the low rank structure

on subharmonic solutions of systems of difference equations with periodic perturbations part ii multiplicity and stability we consider the multiplicity and stability of subharmonic solutions of discrete dynamic systems with periodic perturbations whose existence was established in the first part of this series under the hypotheses that the perturbation is locally lipschitz continuous we determine the precise number of subharmonic solutions and we also show that these subharmonic solutions are stable unstable provide the periodic orbit of the unperturbed system is stable unstable

best constants for the hardy littlewood maximal operator on finite graphs we study the behavior of averages for functions defined on finite graphs g in terms of the hardy littlewood maximal operator mg we explore the relationship between the geometry of a graph and its maximal operator and prove that mg completely determines g even though embedding properties for the graphs do not imply pointwise inequalities for the maximal operators optimal bounds for the p quasi norm of a general graph g in the range 0 p 1 are given and it is shown that the complete graph kn and the star graph sn are the extremal graphs attaining respectively the lower and upper estimates finally we study weak type estimates and some connections with the dilation and overlapping indices of a graph

two classes of piecewise linear difference equations with eventual periodicity three in this paper we will study two classes of difference equations which are piecewise linear and of similar forms we will show that all nontrivial solutions of one equation are eventually periodic with prime period three we will show this result for one case of the second equation

hardy s inequalities for monotone functions on partly ordered measure spaces the theory of weighted inequalities for the hardy operator acting on monotone functions in r was first introduced in 2 extensions of these results to higher dimension have been considered only in very specific cases in particular in the diagonal case only for p 1 see 5 the main difficulty in this context is that the level sets of the monotone functions are not totally ordered contrary to the one dimensional case where one considers intervals of the form 0 a a 0 it is also worth to point out that with no monotonicity restriction the boundedness of the hardy operator is only known in dimension n 2 see 15 12 and also 3 for an extension in the case of product weights in this work we completely characterize the weighted hardy s inequalities for all values of p 0 namely the boundedness of the operator

restricted strong convexity and weighted matrix completion optimal bounds with noise we consider the matrix completion problem under a form of row column weighted entrywise sampling including the case of uniformentrywise sampling as a special case we analyze the associated random observation operator and prove that with high probability it satisfies a form of restricted strong convexity with respect to weighted frobenius norm using this property we obtain as corollaries a number of error bounds on matrix completion in the weighted frobenius norm under noisy sampling and for both exact and near low rank matrices our results are based on measures of the spikiness and low rankness of matrices that are less restrictive than the incoherence conditions imposed in previous work our technique involves an m estimator that includes controls on both the rank and spikiness of the solution and we establish non asymptotic error bounds in weighted frobenius norm for recovering matrices lying with lq balls of bounded spikiness using information theoretic methods we show that no algorithm can achieve better estimates up to a logarithmic factor over these same sets showing that our conditions on matrices and associated rates are essentially optimal

maximal functions on classical lorentz spaces and hardy s inequality with weights for nonincreasing functions a characterization is given of a class of classical lorentz spaces on which the hardy littlewood maximal operator is bounded this is done by determining the weights for which hardy s inequality holds for nonincreasing functions an alternate characterization valid for nondecreasing weights is also derived

the fastest mixing markov process on a graph and a connection to a maximum variance unfolding problem we consider a markov process on a connected graph with edges labeled with transition rates between the adjacent vertices the distribution of the markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue lambda 2 of the laplacian of the weighted graph in this paper we consider the problem of assigning transition rates to the edges so as to maximize lambda 2 subject to a linear constraint on the rates this is the problem of finding the fastest mixing markov process fmmp on the graph we show that the fmmp problem is a convex optimization problem which can in turn be expressed as a semidefinite program and therefore effectively solved numerically we formulate a dual of the fmmp problem and show that it has a natural geometric interpretation as a maximum variance unfolding mvu problem the problem of choosing a set of points to be as far apart as possible measured by their variance while respecting local distance constraints this mvu problem is closely related to a problem recently proposed by weinberger and saul as a method for unfolding high dimensional data that lies on a low dimensional manifold the duality between the fmmp and mvu problems sheds light on both problems and allows us to characterize and in some cases find optimal solutions

the power of convex relaxation near optimal matrix completion this paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries this is known as the matrix completion problem and comes up in a great number of applications including the famous netflix prize and other similar questions in collaborative filtering in general accurate recovery of a matrix from a small number of entries is impossible but the knowledge that the unknown matrix has low rank radically changes this premise making the search for solutions meaningful this paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever the information theoretic limit more importantly the paper shows that under certain incoherence assumptions on the singular vectors of the matrix recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit up to logarithmic factors this convex program simply finds among all matrices consistent with the observed entries that with minimum nuclear norm as an example we show that on the order of nr log n samples are needed to recover a random n x n matrix of rank r by any method and to be sure nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog n

on the boundedness of positive solutions of the reciprocal max type difference equation xn max a1n 1 xn 1 a2n 1 xn 2 atn 1 xn t with periodic parameters we investigate the boundedness of positive solutions of the reciprocal max type difference equation xn max a1n 1 xn 1 a2n 1 xn 2 atn 1 xn t where for each value of i the sequence ain n 0 of positive numbers is periodic with period pi we give both sufficient conditions on the pi s for the boundedness of all solutions and sufficient conditions for all solutions to be unbounded this work essentially complements the work by bidwell and franke who showed that as long as every positive solution of our equation is bounded then every positive solution is eventually periodic thereby leaving open the question as to when solutions are bounded

periodicity of a second order switched difference system over integers abstract in this paper a second order switched difference system which consists of two linear difference equations with a switching rule is proposed to study specifically the periodicity of a particular case is addressed deriving the appropriate rational values for parameter r which possess periodic integer solutions by the transformation method the particular second order difference system is transformed into a first order switched system and we prove that 1 this system possesses periodic integer solutions of prime period two if and only if r 1 2 2 any rational r except for the integers arises periodic integer solutions of prime period three 3 periodic integer solutions of prime period four exist if and only if r 1 2 4 this system possesses no periodic solutions of prime period five we also prove that if r 0 and the system has periodic integer solutions of prime period k 6 then the only possible values of r are reciprocals of integers

positive periodic solutions of nonlinear functional difference equations depending on a parameter in this paper we use the upper and lower solutions method to show that there existsa l such that the nonlinear functional difference equation of the form g n 1 a n g n lh n g n t n n e z has at least one positive t periodic solutions for l e 0 l and does not have any positive t periodic solutions for l l where a n h n and t n are t periodic functions

on the rank minimization problem and its control applications we present several new results on the rank minimization problem over sets which are defined by linear matrix inequalities we proceed to apply some of these results to propose a computational procedure for determining lower and upper bounds for the minimal order dynamic output feedback which stabilizes a given linear time invariant plant

on the convergence of maximum variance unfolding maximum variance unfolding is one of the main methods for nonlinear dimensionality reduction we study its large sample limit providing specific rates of convergence under standard assumptions we find that it is consistent when the underlying submanifold is isometric to a convex subset and we provide some simple examples where it fails to be consistent

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