Directionally nondifferentiable metric projection

A closed convex set inR ² is constructed such that the associated metric projection onto that set is not everywhere directionally differentiable.     

Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images

We continue studying the class of weakly convex sets (in the sense of Vial). For points in a sufficiently small neighborhood of a closed weakly convex subset in Hubert space, we prove that the metric projection on this set exists and is unique. In other words, we show that the closed weakly convex sets have a Chebyshev layer. We prove that the metric projection of a point on a weakly convex set satisfies the Lipschitz condition with respect to a point and the Hölder condition with exponent 1/2 with respect to a set. We develop a method for constructing a continuous parametrization of a set-valued mapping with weakly convex images. We obtain an explicit estimate for the modulus of continuity of the parametrizing function.     

半监督距离度量学习内蕴加速投影梯度算法

考虑求解一类半监督距离度量学习问题. 由于样本集(数据库)的规模与复杂性的激增, 在考虑距离度量学习问题时, 必须考虑学习来的距离度量矩阵具有稀疏性的特点. 因此, 在现有的距离度量学习模型中, 增加了学习矩阵的稀疏约束. 为了便于模型求解, 稀疏约束应用了Frobenius 范数约束. 进一步, 通过罚函数方法将Frobenius范数约束罚到目标函数, 使得具有稀疏约束的模型转化成无约束优化问题. 为了求解问题, 提出了正定矩阵群上加速投影梯度算法, 克服了矩阵群上不能直接进行线性组合的困难, 并分析了算法的收敛性. 最后通过UCI数据库的分类问题的例子, 进行了数值实验, 数值实验的结果说明了学习矩阵的稀疏性以及加速投影梯度算法的有效性.     

Characterization of subdual latticial cones in Hilbert spaces by the isotonicity of the metric projection

The subdual latticial cones in Hilbert spaces are characterized by the isotonicity of a generalization of the positive part mapping which can be expressed in terms of the metric projection only. Although Németh characterized the positive cone of Hilbert lattices with the metric projection and ordering only [A.B. Németh, Characterization of a Hilbert vector lattice by the metric projection onto its positive cone, J. Approx. Theory 123 (2) (2003), pp. 295–299.], this has been done for the first time for subdual latticial cones in this article. We also note that the normal generating pointed closed convex cones for which the projection onto the cone is isotone are subdual latticial cones, but there are subdual latticial cones for which the metric projection onto the cone is not isotone [G. Isac, A.B. Németh, Monotonicity of metric projections onto positive cones of ordered Euclidean spaces, Arch. Math. 46 (6) (1986), pp. 568–576; G. Isac, A.B. Néemeth, Every generating isotone projection cone is latticial and correct, J. Math. Anal. Appl. 147 (1) (1990), pp. 53–62; G. Isac, A.B. Németh, Isotone projection cones in Hilbert spaces and the complementarity problem, Boll. Un. Mat. Ital. B 7 (4) (1990), pp. 773–802; G. Isac, A.B. Németh, Projection methods, isotone projection cones, and the complementarity problem, J. Math. Anal. Appl. 153 (1) (1990), pp. 258–275; G. Isac, A.B. Németh, Isotone projection cones in Eucliden spaces, Ann. Sci. Math Québec 16 (1) (1992), pp. 35–52].     

The metric projection onto the soul

We study geometric properties of the metric projection of an open manifold with nonnegative sectional curvature onto a soul . is shown to be up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of , and study the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of .

     

Finding the projection of a point onto the intersection of convex sets via projections onto half-spaces

We present a modification of Dykstra's algorithm which allows us to avoid projections onto general convex sets. Instead, we calculate projections onto either a half-space or onto the intersection of two half-spaces. Convergence of the algorithm is established and special choices of the half-spaces are proposed.The option to project onto half-spaces instead of general convex sets makes the algorithm more practical. The fact that the half-spaces are quite general enables us to apply the algorithm in a variety of cases and to generalize a number of known projection algorithms.The problem of projecting a point onto the intersection of closed convex sets receives considerable attention in many areas of mathematics and physics as well as in other fields of science and engineering such as image reconstruction from projections.In this work we propose a new class of algorithms which allow projection onto certain super half-spaces, i.e., half-spaces which contain the convex sets. Each one of the algorithms that we present gives the user freedom to choose the specific super half-space from a family of such half-spaces. Since projecting a point onto a half-space is an easy task to perform, the new algorithms may be more useful in practical situations in which the construction of the super half-spaces themselves is not too difficult.     

CHARACTERIZATION OF BEST APPROXIMATIONS IN METRIC LINEAR SPACES

Let (X,d) be a real metric linear space, with translation-invariant metric d and G a linear subspace of X. In this paper we use functionals in the Lipschitz dual of X to characterize those elements of G which are best approximations to elements of X. We also give simultaneous characterization of elements of best approximation and also consider elements of ε-approximation.     

Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space

LetE be a real Banach space andL(E) the family of all nonempty compact starshaped subsets ofE. Under the Hausdorff distance,L(E) is a complete metric space. The elements of the complement of a first Baire category subset ofL(E) are called typical elements ofL(E). ForXL(E) we denote by the metrical projection ontoX, i.e. the mapping which associates to eachaE the set of all points inX closest toa. In this note we prove that, ifE is strictly convex and separable with dimE2, then for a typicalXL(E) the map is not single valued at a dense set of points. Moreover, we show that a typical element ofL(E) has kernel consisting of one point and set of directions dense in the unit sphere ofE.     

Convexities and approximative compactness and continuity of metric projection in Banach spaces

In this paper, we investigate the continuities of the metric projection in a nonreflexive Banach space X, which improve the results in [X.N. Fang, J.H. Wang, Convexity and continuity of metric projection, Math. Appl. 14 (1) (2001) 47–51; P.D. Liu, Y.L. Hou, A convergence theorem of martingales in the limit, Northeast. Math. J. 6 (2) (1990) 227–234; H.J. Wang, Some results on the continuity of metric projections, Math. Appl. 8 (1) (1995) 80–84]. Under the assumption that X has some convexities, we discuss the relationship between approximative compactness of a subset A of X and continuity of the metric projection P_A. We also give a representation theorem for the metric projection to a hyperplane in dual space X^∗ and discuss its continuity.     

Closed mappings onto metric spaces

We investigate the classes of spaces that can be mapped onto a metrizable space by a closed mapping with fibers having a given property $\text{P}$. We give some conditions which assure that such classes are closed under the action of perfect or open and compact mappings. Such a treatment includes the investigation of paracompact p-spaces and M-spaces. We also discuss spaces that can be mapped onto a metacompact Moore space.     

Rank-determining sets of metric graphs

A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph Γ is an element of the free abelian group on Γ. The rank of a divisor on a metric graph is a concept appearing in the Riemann-Roch theorem for metric graphs (or tropical curves) due to Gathmann and Kerber, and Mikhalkin and Zharkov. We define a rank-determining set of a metric graph Γ to be a subset A of Γ such that the rank of a divisor D on Γ is always equal to the rank of D restricted on A. We show constructively in this paper that there exist finite rank-determining sets. In addition, we investigate the properties of rank-determining sets in general and formulate a criterion for rank-determining sets. Our analysis is based on an algorithm to derive the v₀-reduced divisor from any effective divisor in the same linear system.     

Supremum metric and relatively compact sets of fuzzy sets

The supremum metric D between fuzzy subsets of a metric space is the supremum of the Hausdorff distances of the corresponding level sets. In this paper some new criteria of compactness with respect to the distance D are given; they concern arbitrary fuzzy sets (see Theorem 7), fuzzy sets having no proper local maximum points (see Theorem 12) and, finally, fuzzy sets with convex sendograph (see Theorem 13). In order to compare results with a previous characterization of compactness of Diamond–Kloeden, the criteria will be expressed by equi-(left/right)-continuity. In the proofs a first author's purely topological criterion of D  -compactness and a variational convergence (called Γ$\Gamma$-convergence) which was introduced by De Giorgi and Franzoni, are fundamental.     

Self-similar sets in complete metric spaces

We develop a theory for Hausdorff dimension and measure of self-similar sets in complete metric spaces. This theory differs significantly from the well-known one for Euclidean spaces. The open set condition no longer implies equality of Hausdorff and similarity dimension of self-similar sets and that has nonzero Hausdorff measure in this dimension. We investigate the relationship between such properties in the general case.

     

Some results on topical functions and upward sets

The purpose of this paper is to introduce and discuss the concept of topical functions on upward sets.We give characterizations of topical functions in terms of upward sets.     

Some results on projection of planar sets

In this paper we define certain types of projections of planar sets and study some properties of such projections.     

Condensations of projective sets onto compacta

For a coanalytic-complete or -complete subspace of a Polish space we prove that there exists a continuous bijection of onto the Hilbert cube . This extends results of Pytkeev. As an application of our main theorem we give an answer to some questions of Arkhangelskii and Christensen.

Under the assumption of Projective Determinacy we also give some generalizations of these results to higher projective classes.

     

The omega limit sets of subsets in a metric space

In this paper, we discuss the properties of limit sets of subsets and attractors in a compact metric space. It is shown that the -limit set (Y) of Y is the limit point of the sequence {(Cl Y) [i, )}_i=1 in 2^X and also a quasi-attractor is the limit point of attractors with respect to the Hausdorff metric. It is shown that if a component of an attractor is not an attractor, then it must be a real quasi-attractor.This paper was completed while the author was visiting the Academy of Mathematics and Systems, Chinese Academy of Sciences. The author was supported by the National Science Foundation of China (Grant No. 10461003).