Abstract: We consider the problem of outlier rejection in single subspace learning. Classical approaches work directly with a low-dimensional representation of the subspace. Our approach works with a dual representation of the subspace and hence aims to find its orthogonal complement. We pose this problem as an l1-minimization problem on the sphere and show that, under certain conditions on the distribution of the data, any global minimizer of this non-convex problem gives a vector orthogonal to the subspace. Moreover, we show that such a vector can still be found by relaxing the non-convex problem with a sequence of linear programs. Experiments on synthetic and real data show that the proposed approach, which we call Dual Principal Component Pursuit (DPCP), outperforms state-of-the art methods, especially in the case of high-dimensional subspaces.

Abstract: The Markov chain Monte Carlo (MCMC) method is the computational workhorse for Bayesian inverse problems. However, MCMC struggles in high-dimensional parameter spaces, since its iterates must sequentially explore the high-dimensional space. This struggle is compounded in physical applications when the nonlinear forward model is computationally expensive. One approach to accelerate MCMC is to reduce the dimension of the state space. Active subspaces are part of an emerging set of tools for subspace-based dimension reduction. An active subspace in a given inverse problem indicates a separation between a low-dimensional subspace that is informed by the data and its orthogonal complement that is constrained by the prior. With this information, one can run the sequential MCMC on the active variables while sampling independently according to the prior on the inactive variables. However, this approach to increase efficiency may introduce bias. We provide a bound on the Hellinger distance between the true posterior and its active subspace- exploiting approximation. And we demonstrate the active subspace-accelerated MCMC on two computational examples: (i) a two-dimensional parameter space with a quadratic forward model and one-dimensional active subspace and (ii) a 100-dimensional parameter space with a PDE-based forward model and a two-dimensional active subspace.

Abstract: We consider the problem of learning a linear subspace from data corrupted by outliers. Classical approaches are typically designed for the case in which the subspace dimension is small relative to the ambient dimension. Our approach works with a dual representation of the subspace and hence aims to find its orthogonal complement; as such, it is particularly suitable for subspaces whose dimension is close to the ambient dimension (subspaces of high relative dimension). We pose the problem of computing normal vectors to the inlier subspace as a non-convex $\ell_1$ minimization problem on the sphere, which we call Dual Principal Component Pursuit (DPCP) problem. We provide theoretical guarantees under which every global solution to DPCP is a vector in the orthogonal complement of the inlier subspace. Moreover, we relax the non-convex DPCP problem to a recursion of linear programs whose solutions are shown to converge in a finite number of steps to a vector orthogonal to the subspace. In particular, when the inlier subspace is a hyperplane, the solutions to the recursion of linear programs converge to the global minimum of the non-convex DPCP problem in a finite number of steps. We also propose algorithms based on alternating minimization and iteratively re-weighted least squares, which are suitable for dealing with large-scale data. Experiments on synthetic data show that the proposed methods are able to handle more outliers and higher relative dimensions than current state-of-the-art methods, while experiments in the context of the three-view geometry problem in computer vision suggest that the proposed methods can be a useful or even superior alternative to traditional RANSAC-based approaches for computer vision and other applications.

Abstract: Motivated by results of linear algebra over fields, rings and tropical semirings, we present a systematic way to understand the behaviour of matrices with entries in an arbitrary semiring. We focus on three closely related problems concerning the row and column spaces of matrices. This allows us to isolate and extract common properties that hold for different reasons over different semirings, yet also lets us identify which features of linear algebra are specific to particular types of semiring. For instance, the row and column spaces of a matrix over a field are isomorphic to each others' duals, as well as to each other, but over a tropical semiring only the first of these properties holds in general (this in itself is a surprising fact). Instead of being isomorphic, the row space and column space of a tropical matrix are anti-isomorphic in a certain order-theoretic and algebraic sense.The first problem is to describe the kernels of the row and column spaces of a given matrix. These equivalence relations generalise the orthogonal complement of a set of vectors, and the nature of their equivalence classes is entirely dependent upon the kind of semiring in question. The second, Hahn-Banach type, problem is to decide which linear functionals on row and column spaces of matrices have a linear extension. If they all do, the underlying semiring is called exact, and in this case the row and column spaces of any matrix are isomorphic to each others' duals. The final problem is to explain the connection between the row space and column space of each matrix. Our notion of a conjugation on a semiring accounts for the different possibilities in a unified manner, as it guarantees the existence of bijections between row and column spaces and lets us focus on the peculiarities of those bijections.Our main original contribution is the systematic approach described above, but along the way we establish several new results about exactness of semirings. We give sufficient conditions for a subsemiring of an exact semiring to inherit exactness, and we apply these conditions to show that exactness transfers to finite group semirings. We also show that every Boolean ring is exact. This result is interesting because it allows us to construct a ring which is exact (also known as FP-injective) but not self-injective. Finally, we consider exactness for residuated lattices, showing that every involutive residuated lattice is exact. We end by showing that the residuated lattice of subsets of a finite monoid is exact if and only if the monoid is a group.

Abstract: We prove a Goldberg?Sachs theorem in dimension three. To be precise, given a three-dimensional Lorentzian manifold satisfying the topological massive gravity equations, we provide necessary and sufficient conditions on the tracefree Ricci tensor for the existence of a null line distribution whose orthogonal complement is integrable and totally geodetic. This includes, in particular, Kundt spacetimes that are solutions of the topological massive gravity equations.

Abstract: Linnik proved in the late 1950's the equidistribution of integer points on large spheres under a congruence condition. The congruence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec) using completely different techniques. We conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement. We use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition.

Abstract: We prove a Goldberg-Sachs theorem in dimension three. To be precise, given a three-dimensional Lorentzian manifold satisfying the topological massive gravity equations, we provide necessary and sufficient conditions on the tracefree Ricci tensor for the existence of a null line distribution whose orthogonal complement is integrable and totally geodetic. This includes, in particular, Kundt spacetimes that are solutions of the topological massive gravity equations.

Abstract: The invention discloses an underdetermined blind separation source signal recovery method based on an SCMP (Subspace Complementary Matching Pursuit) algorithm, and mainly solves the problems of high complexity and low recovery precision in an existing underdetermined blind separation technology. The method comprises the following steps: 1, performing QR decomposition on transpose AT of a mixed matrix to obtain matrix space SA and corresponding orthogonal complement space described in the specification; 2, working out the component s1 of a source signal in the space SA by a received signal x(t); 3, searching the positions of N-M zero elements in the source signal s by an approximate l[0] norm process according to complementary properties of the source signal in the space SA and the space described in the specification; 4, working out other M elements by the complementary properties according to the searched positions of the N-M zero elements to obtain a recovery signal. The method is high in search speed and recovery precision and can be used in the fields of communication, signal processing and the like.

Abstract: The rational fraction polynomial algorithm is the entry-level model of the high-order, frequency-domain modal parameter estimation methods However, it has some well-known issues with numerical ill-conditioning for a high model order and a wide frequency range Among the alternatives that have been proposed over the years to address this shortcoming is a change of basis functions from power polynomials to orthogonal polynomials While this approach does cure the numerical ill-conditioning issues, this algorithm has not yet achieved mainstream acceptance, with the reasons for this reluctance typically cited being additional complication or increased computation time This paper introduces an improved implementation of the orthogonal polynomial algorithm that uses the orthogonal complement, coupled with QR decomposition, to greatly reduce the time of the accumulation phase The neat trick performed by the orthogonal complement is to get all of the overdetermination possible without having to do all of the work

Abstract: The invention discloses a submatrix-level OP wave beam forming method based on covariance matrix normalization. The method can be used to effectively inhibit interference, make a main lobe of a self-adaptive directional diagram conformal, reduce side lobes the self-adaptive directional diagram and obtain higher output SINR and convergence speed. The method comprises that submatrix-level reception signals are normalized, and a corresponding normalized sampling covariance matrix is calculated; the amount of interference signal sources is estimated by utilizing the MDL criterion, and further an interference subspace is obtained; and finally, the static weight vector is projected to an orthogonal complement space of the interference subspace to obtain a self-adaptive weight vector.

Abstract: In this paper, first we give a characterization of spacelike inclined curves according to the type-2 Bishop frame in Minkowski 3-space, and then define rectifying curves of spacelike curves according to the type-2 Bishop frame in Minkowski 3-space as their position vectors always lie in the orthogonal complement of their vector field . Moreover we characterize Bertrand curves in the same space via the new frame. In particular, we study Mannheim partner curves according to type-2 Bishop frame in and express such curves in terms of their curvature functions.

Abstract: It is known that the minimal cone for the constraint system of a conic linear optimization problem is a key component in obtaining strong duality without any constraint qualification. In the particular case of semidefinite optimization, an explicit expression for the dual cone of the minimal cone allows for a dual program of polynomial size that satisfies strong duality. This is achieved due to the fact that we can express the orthogonal complement of a face of the cone of positive semidefinite matrices completely in terms of a system of semidefinite inequalities. In this paper, we extend this result to cones that are either faces of the cone of positive semidefinite matrices or the dual cones of faces of the cone of positive semidefinite matrices. The newly proved result was used in Zhang (4OR 9:403–416, 2011). However, a proof was not given in Zhang (4OR 9:403–416, 2011).

Abstract: Subspace clustering is the problem of clustering data that lie close to a union of linear subspaces. In the abstract form of the problem, where no noise or other corruptions are present, the data are assumed to lie in general position inside the algebraic variety of a union of subspaces, and the objective is to decompose the variety into its constituent subspaces. Prior algebraic-geometric approaches to this problem require the subspaces to be of equal dimension, or the number of subspaces to be known. Subspaces of arbitrary dimensions can still be recovered in closed form, in terms of all homogeneous polynomials of degree $m$ that vanish on their union, when an upper bound m on the number of the subspaces is given. In this paper, we propose an alternative, provably correct, algorithm for addressing a union of at most $m$ arbitrary-dimensional subspaces, based on the idea of descending filtrations of subspace arrangements. Our algorithm uses the gradient of a vanishing polynomial at a point in the variety to find a hyperplane containing the subspace S passing through that point. By intersecting the variety with this hyperplane, we obtain a subvariety that contains S, and recursively applying the procedure until no non-trivial vanishing polynomial exists, our algorithm eventually identifies S. By repeating this procedure for other points, our algorithm eventually identifies all the subspaces by returning a basis for their orthogonal complement. Finally, we develop a variant of the abstract algorithm, suitable for computations with noisy data. We show by experiments on synthetic and real data that the proposed algorithm outperforms state-of-the-art methods on several occasions, thus demonstrating the merit of the idea of filtrations.

Abstract: Roughly speaking, a regular subspace of a Dirichlet form is a subspace, which is also a regular Dirichlet form, on the same state space. In particular, the domain of regular subspace is a closed subspace of the Hilbert space induced by the domain and �-inner product of original Dirichlet form. We shall investigate the or- thogonal complement of regular subspace of 1-dimensional Brownian motion in this paper. Our main results indicate that this orthogonal complement has a very close connection with the �-harmonic equation under Neumann boundary condition.

Abstract: We provide a direct, intersection theoretic, argument that the Jordan models of an operator of class C_{0}, of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where `inequality' is replaced by `divisibility'. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. The result is true even for operators acting on nonseparable Hilbert spaces. For such operators the usual Horn inequalities are supplemented so as to apply to all the Jordan blocks in the model.

Abstract: Recently, Li et al. (Int. J. Theor. Phys. 46, 2599, 2007) has constructed the quantum superimposing multiple anti-cloning machine, moreover established the sufficient and necessary condition of this machine exists. In the proofs given by Li et al. (Int. J. Theor. Phys. 46, 2599, 2007), claimed that the following key fact to hold : Fact For an arbitrary unknown state |ψ〉 belongs to n-dimensional Hilbert space, there exists an antiunitary operator K such that K|ψ〉=|ψ⊥〉 here the state |ψ⊥〉 is an orthogonal complement state of |ψ〉, that is, it satisfies the following two conditions 〈ψ|K|ψ〉=〈ψ|ψ⊥〉=0 and 〈ψ|ψ〉=〈ψ⊥|ψ⊥〉=1 In this Comment, we would like to point out that (a). In 1-dimensional Hilbert space, for an arbitrary unknown state |ψ〉, the antiunitary operator K and the orthogonal complement state both do not exist in general. (b). In 3-dimensional Hilbert space, for an arbitrary unknown state |ψ〉, the antiunitary operator K do not exist in general, there are uncountably many orthogonal complement states that can be constructed through the skew-symmetric operator, but is not unitary one. Which shows that above Fact given by Li et al. [1] is incorrect in general for both 1 and 3-dimensional Hilbert space

Abstract: Let G be a finite abelian group. We examine the discrepancy between subspaces of l^2(G) which are diagonalized in the standard basis and subspaces which are diagonalized in the dual Fourier basis. The general principle is that a Fourier subspace whose dimension is small compared to |G| = dim(l^2(G)) tends to be far away from standard subspaces. In particular, the recent positive solution of the Kadison-Singer problem shows that from within any Fourier subspace whose dimension is small compared to |G| there is standard subspace which is essentially indistinguishable from its orthogonal complement.

Abstract: The embodiment of the invention relates to the technical field of wireless communication, specially relates to a signal processing method and equipment, and is used to solve the problems that a current interference reduction scheme existing in the prior art cannot improve the system capacity and the signal-interference and noise ratio and cannot perform inhibition or coordination relative to a remote same-frequency cell The method of the embodiment comprises determining a matrix for representing statistical characteristics of a reference source, performing matrix decomposition on the determined matrix for representing the statistical characteristics of the reference source to obtain an orthogonal complement space matrix, and performing normalized processing on an estimated value of a received signal according to the orthogonal complement space matrix of the reference source

Abstract: When restricted to a subspace, a nonsmooth function can be differentiable. It is known that for a nonsmooth convex function f and a point x, the Euclidean space can be decomposed into two subspaces: U, over which a special Lagrangian can be defined and has nice smooth properties and V, the orthogonal complement subspace of U. In this paper we generalize the definition of UV-decomposition and U-Lagrangian to the context of nonconvex functions, specifically that of a prox-regular function.

Abstract: We consider -linear () spaces of dimension over an algebraically closed field of characteristic 0, whose centre (the analogue of the space of symmetric matrices of a bilinear form) is maximal, as a subalgebra of In a previous work (with S. Ryom-Hansen), we found several properties of the orthogonal group of d-linear spaces whose centre consists of powers of a single lineal map acting cyclically on the underlying vector space, called cyclic spaces. We extend those results to tensor products of cyclic spaces. In particular, we give a description of the orthogonal group of the product of two cyclic spaces.

Abstract: In this paper, we introduce dual Toeplitz operators on the orthogonal complement of the Hardy space of the polydisk and establish their main algebraic properties using an auxiliary transformation of operators. As a byproduct, we exploit this mysterious transformation in the investigation of boundedness and compactness of Hankel products and mixed Toeplitz-Hankel products on the Hardy space of the polydisk.

Abstract: We study large deviations for measurable averaging operators on state spaces of dynamical systems. Our main motivation is the Hecke operators on the modular curve Y_0(p^n) and their generalization to higher rank S-arithmetic quotients. We prove a relatively sharp large deviations result in terms of the norm of the averaging operator restricted to the orthogonal complement of the constant functions in L2. In the self-adjoint case this norm is expressible by the spectral gap. Developing ideas of Linnik and Ellenberg, Michel and Venkatesh, we use this large deviation result to prove an effective equidistribution theorem on a state space. The novelty of our results is that they apply to measures with sub-optimal bounds on the mass of Bowen balls. We present two new applications to our effective equidistribution result. The first one is effective rigidity for the measure of maximal entropy on S-arithmetic quotients with respect to a semisimple action in a non-archimedean place. Measures having large enough metric entropy must also be close on the state space to the Haar measure. This is a partial extension of a recent result of Ruhr to a significantly more general setting. The second one is non-escape of mass for sequences of measures having large entropy with respect to a semisimple element in a non-archimedean place. This generalizes similar known results for real flows. Our methods differ from the methods used by Ruhr and in the previously known non-escape of mass results.