General Subspaces Research Articles

Interactive Subspace Cluster Analysis Guided by Semantic Attribute Associations.

Multivariate datasets with many variables are increasingly common in many application areas. Most methods approach multivariate data from a singular perspective. Subspace analysis techniques, on the other hand. provide the user a set of subspaces which can be used to view the data from multiple perspectives. However, many subspace analysis methods produce a huge amount of subspaces, a number of which are usually redundant. The enormity of the number of subspaces can be overwhelming to analysts, making it difficult for them to find informative patterns in the data. In this article, we propose a new paradigm that constructs semantically consistent subspaces. These subspaces can then be expanded into more general subspaces by ways of conventional techniques. Our framework uses the labels/meta-data of a dataset to learn the semantic meanings and associations of the attributes. We employ a neural network to learn a semantic word embedding of the attributes and then divide this attribute space into semantically consistent subspaces. The user is provided with a visual analytics interface that guides the analysis process. We show via various examples that these semantic subspaces can help organize the data and guide the user in finding interesting patterns in the dataset.

Stability of deficiency indices of Hermitian subspaces under relatively bounded perturbations

This paper is concerned with stability of deficiency indices of Hermitian subspaces (i.e. linear relations) under relatively bounded perturbations in Hilbert spaces. Several results about invariance of deficiency indices of Hermitian subspaces under relatively bounded perturbations are established. As a consequence, invariance of self-adjointness of Hermitian subspaces under relatively bounded perturbations is obtained. In addition, it is shown that the deficiency indices may shrink in the special case that the relative bound is equal to 1. The results obtained in the present paper generalize the corresponding results for symmetric operators to more general Hermitian subspaces, some of which relax or improve certain conditions of the related results in existing literatures.

The deflated conjugate gradient method: Convergence, perturbation and accuracy

Deflation techniques for Krylov subspace methods have seen a lot of attention in recent years. They provide means to improve the convergence speed of these methods by enriching the Krylov subspace with a deflation subspace. The most common approach for the construction of deflation subspaces is to use (approximate) eigenvectors, but also more general subspaces are applicable.In this paper we discuss two results concerning the accuracy requirements within the deflated CG method. First we show that the effective condition number which bounds the convergence rate of the deflated conjugate gradient method depends asymptotically linearly on the size of the perturbations in the deflation subspace. Second, we discuss the accuracy required in calculating the deflating projection. This is crucial concerning the overall convergence of the method, and also allows to save some computational work.To show these results, we use the fact that as a projection approach deflation has many similarities to multigrid methods. In particular, recent results relate the spectra of the deflated matrix to the spectra of the error propagator of twogrid methods. In the spirit of these results we show that the effective condition number can be bounded by the constant of a weak approximation property.

Example Based Repetitive Structure Synthesis

AbstractWe present an example based geometry synthesis approach for generating general repetitive structures. Our model is based on a meshless representation, unifying and extending previous synthesis methods. Structures in the example and output are converted into a functional representation, where the functions are defined by point locations and attributes. We then formulate synthesis as a minimization problem where patches from the output function are matched to those of the example. As compared to existing repetitive structure synthesis methods, the new algorithm offers several advantages. It handles general discrete and continuous structures, and their mixtures in the same framework. The smooth formulation leads to employing robust optimization procedures in the algorithm. Equipped with an accurate patch similarity measure and dedicated sampling control, the algorithm preserves local structures accurately, regardless of the initial distribution of output points. It can also progressively synthesize output structures in given subspaces, allowing users to interactively control and guide the synthesis in real‐time. We present various results for continuous/discrete structures and their mixtures, residing on curves, submanifolds, volumes, and general subspaces, some of which are generated interactively.

A Note on the Structure of Affine Subspaces of <i>L</i><sup>2</sup>(R<i><sup>d</sup></i>)

This paper investigates the structure of general affine subspaces of L2(Rd) . For a d × d expansive matrix A, it shows that every affine subspace can be decomposed as an orthogonal sum of spaces each of which is generated by dilating some shift invariant space in this affine subspace, and every non-zero and non-reducing affine subspace is the orthogonal direct sum of a reducing subspace and a purely non-reducing subspace, and every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces when |detA| = 2.

Construction of error-tolerance pooling designs in symplectic spaces

The paper provides the construction of error-correcting pooling designs with the incidence matrix of two types of subspaces of symplectic spaces over finite fields. As a generalization of Guo et al.'s matrix, we use the general subspaces of type $$(m,s)$$ to substitute special subspaces of type $$(2s,s)$$ . If $$\nu $$ is big enough, we can get the higher degree of error-tolerant property.

Order isomorphisms on function spaces

The classical theorems of Banach and Stone, Gelfand and Kolmogorov, and Kaplansky show that a compact Hausdorff space $X$ is uniquely determined by the linear isometric structure, the algebraic structure, and the lattice structure, respectively, of the space $C(X)$. In this paper, it is shown that for rather general subspaces $A(X)$ and $A(Y)$ of $C(X)$ and $C(Y)$ respectively, any linear bijection $T: A(X) \to A(Y)$ such that $f \geq 0$ if and only if $Tf \geq 0$ gives rise to a homeomorphism $h: X \to Y$ with which $T$ can be represented as a weighted composition operator. The three classical results mentioned above can be derived as corollaries. Generalizations to noncompact spaces and other function spaces such as spaces of uniformly continuous functions, Lipschitz functions and differentiable functions are presented.

Graph-like continua, augmenting arcs, and Menger’s theorem

We show that an adaptation of the augmenting path method for graphs proves Menger’s Theorem for wide classes of topological spaces. For example, it holds for locally compact, locally connected, metric spaces, as already known. The method lends itself particularly well to another class of spaces, namely the locally arcwise connected, hereditarily locally connected, metric spaces. Finally, it applies to every space where every point can be separated from every closed set not containing it by a finite set, in particular to every subspace of the Freudenthal compactification of a locally finite, connected graph. While closed subsets of such a space behave nicely in that they are compact and locally connected (and therefore locally arcwise connected), the general subspaces do not: They may be connected without being arcwise connected. Nevertheless, they satisfy Menger’s Theorem.

Theory of Potential with Respect to Consistent Kernels; Theorem on Completeness and Sequences of Potentials

The concept of consistent kernels introduced by Fuglede in 1960 is widely used in extremal problems of the theory of potential on classes of positive measures. In the present paper, we show that this concept is also efficient for the investigation of extremal problems on fairly broad classes of signed measures. In particular, for an arbitrary consistent kernel in a locally compact space, we prove a theorem on the strong completeness of fairly general subspaces E of all measures with finite energy. (Note that, according to the well-known Cartan counterexample, the entire space E is strongly incomplete even in the classical case of the Newton kernel in ℝn Using this theorem, we obtain new results for the Gauss variational problem, namely, in the non-compact case, we give a description of vague and (or) strong limiting measures of minimizing sequences and obtain sufficient solvability conditions.

General reflexivity and pattern subspaces

This paper extends the theory of reflexivity triples by extending notions and results in the theory of dual operator algebras to this more general setting. In addition, a complete and simple characterization is given for the reflexivity of two-dimensional pattern spaces of finite matrices.

Additive rank-one preserving mappings on triangular matrix algebras

We classify surjective additive maps on the space of block upper triangular matrices that preserve matrices of rank one as well as linear maps preserving matrices of rank one on fairly general subspaces of matrices.

Real rational curves in Grassmannians

Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some given general figures [5]. For the problem of plane conics tangent to five general (real) conics, the surprising answer is that all 3264 may be real [13]. Similarly, given any problem of enumerating p-planes incident on some given general subspaces, there are real subspaces such that each of the (finitely many) incident p-planes is real [17]. We show that the problem of enumerating parameterized rational curves in a Grassmannian satisfying simple (codimension 1) conditions may have all of its solutions real. This problem of enumerating rational curves in a Grassmannian arose in at least two distinct areas of mathematics. The number of such curves was predicted by the formula of Vafa and Intriligator [20, 8] from mathematical physics. It is also the number of complex dynamic compensators which stabilize a given linear system, and the enumeration was solved in this context [12, 11]. The question of real solutions also arose in systems theory [3]. This application will be discussed in Section 4.

Tensor products in p-adic Hodge theory

There is a classical relation between the p-adic absolute value of the eigenvalues of Frobenius on crystalline cohomology and Hodge numbers, for a variety in characteristic p: “the Newton polygon lies on or above the Hodge polygon” [14], [1]. For a variety in characteristic p with a lift to characteristic 0, Fontaine conjectured and Faltings proved a more precise statement: there is an inequality which relates the slope of Frobenius on any Frobenius-invariant subspace of the crystalline cohomology to the Hodge filtration, restricted to that subspace [7], [4]. A vector space over a p-adic field together with a σ-linear endomorphism and a filtration which satisfies this inequality is called a weakly admissible filtered isocrystal (see section 1 for the precise definition). The category of such objects is one possible p-adic analogue of the category of Hodge structures: in particular, it is an abelian category. We give a new proof of Faltings’s theorem that the tensor product of weakly admissible filtered isocrystals over a p-adic field is weakly admissible [5]. By a similar argument, we also prove a characterization of weakly admissible filtered isocrystals with G-structure in terms of geometric invariant theory, which was conjectured by Rapoport and Zink [19]. Before Faltings, Laffaille [12] had proved the tensor product theorem in the case of filtered isocrystals over an unramified extension of Qp. Faltings’s proof works by reducing this problem of σ-linear algebra to a different problem of pure linear algebra, the problem of showing that the tensor product of two vector spaces, each equipped with a finite “semistable” set of filtrations, is semistable. The latter problem is solved by constructing suitable integral lattices (in [5]) or hermitian metrics (in [20]) on vector spaces with a semistable set of filtrations, just as one can prove that the tensor product of semistable bundles on an algebraic curve is semistable using Narasimhan-Seshadri’s hermitian metrics ([6], [16]). In this paper, we can avoid the reduction from filtered isocrystals to filtered vector spaces. The point is that Ramanan and Ramanathan’s algebraic proof [17] that the tensor product of semistable vector bundles is semistable can be modified to apply directly to filtered isocrystals. We have an inequality to prove for a class of linear subspaces S of a tensor product V ⊗W . The inequality is obvious for sufficiently general subspaces S and also if S is a very special subspace, say if S is a decomposable subspace S1 ⊗ S2 ⊂ V ⊗W . But it is not clear how to prove the inequality we want if S is somewhere in the middle. The solution, following Ramanan and Ramanathan, is to use geometric invariant theory to give a sharp dichotomy between “general” subspaces and “special” subspaces of V ⊗W , in such a way that we get useful information in either case.

The Angle Between Complementary Subspaces

Usually the discussion stops right there, and extensions to angles between subspaces of higher dimensions are, more or less tacitly, shoved under the rug. Perhaps this is because most instructors feel that such extensions are difficult to understand, or that further effort in this direction is not worthwhile. Indeed, this makes sense for angles between general subspaces because one would have to introduce concepts like gap or distance between subspaces [7, 12], principal (or canonical) angles [1, 2, 15, 12], the CS decomposition [11, 4, 10, 6, 12], and so on. These topics are better off in a more advanced course. However, angles between complementary subspaces are easier to deal with. The purpose of our article is to draw attention to some simple, though not very well known, expressions for the angle between complementary subspaces which are easily derived from the fundamental theorem of linear algebra [14] and elementary facts about matrix norms and projectors. Angles between complementary subspaces are not just academic. They arise, for instance, in the context of controller robustness [9, 16]. Roughly speaking, the spaces associated with the controller and the plant (a system described by a set of differential equations) are complementary subspaces. The robustness of the controller is defined by the smallest perturbation that renders the system unstable, which means that the associated subspaces are no longer complementary. The system remains stable as long as perturbations are smaller than the distance between the complementary subspaces. One measure of distance is the sine of the angle between the spaces.

Left‐Definite Regular Hamiltonian Systems

AbstractLinear Hamiltonian systems allow us to generalize, as well as consider, self‐adjoint problems of any even order. Such left‐definite problems are interesting, not only because of the generalization, but also because of the new intricacies they expose, some of which have made it possible to go beyond fourth order scale problems.We explore the left definite Sobolev settings for such problems, which are in general subspaces determined by boundary conditions. We show that the Hamiltonian operator remains self‐adjoint, and inherits the same resolvent and spectral resolution from its original L2 space when set in the left‐definite Sobolev space.

Convergence Theory of Nonlinear Newton–Krylov Algorithms

This paper presents some convergence theory for nonlinear Krylov subspace methods. The basic idea of these methods, which have been described by the authors in an earlier paper, is to use variants of Newton’s iteration in conjunction with a Krylov subspace method for solving the Jacobian linear systems. These methods are variants of inexact Newton methods where the approximate Newton direction is taken from a subspace of small dimension. The main focus of this paper is to analyze these methods when they are combined with global strategies such as linesearch techniques and model trust region algorithms. Most of the convergence results are formulated for projection onto general subspaces rather than just Krylov subspaces.

A Choquet theorem for general subspaces of vector-valued functions

Let X be a compact Hausdorff space, let E be a (real or complex) Banach space, and let C(X, E) stand for the Banach space of all continuous E-valued functions defined on X under the supremum norm. If A is an arbitrary linear subspace of C(X, E), then it is shown that each bounded linear functional l on A can be represented by a boundary E*-valued vector measure μ on X that has the same norm as l. This result constitutes an extension to vector-valued functions of the so-called analytic version of Choquet's integral representation theorem.

Necessary Conditions for Universal Interpolation in

Let be the space of Fourier transforms of distributions with compact support, or equivalently, the space of entire functions h satisfying the growth condition(1)where p(z) = |Imz| + log (1 + |z|2) and A, B are constants depending only on h. A sequence with |zk| ↑ ∞ is said to be a universal interpolation sequence for if for all such that(2)for constants A, B independent of k, there exists such that f(zk) = ak. In this note we will consider necessary conditions for universal interpolation in and more general subspaces of the entire functions.

On the Numerical Solutions of Helmholtz’s Equation by the Finite Element Method

We consider the numerical solution of the Helmholtz equation via finite element methods. A two-stage method which gives the same accuracy in the computed gradient as in the computed solution is discussed. This method is applicable to more general finite element subspaces than methods previously considered; it is also of lower computational complexity and the same conditioning as previous methods.