Map Problem Research Articles

MAP Estimation With Bernoulli Randomness, and Its Application to Text Analysis and Recommender Systems

MAP estimation plays an important role in many probabilistic models. However, in many cases, the MAP problem is non-convex and intractable. In this work, we propose a novel algorithm, called BOPE, which uses Bernoulli randomness for Online Maximum a Posteriori Estimation. We show that BOPE has a fast convergence rate. In particular, BOPE implicitly employs a prior which plays as regularization. Such a prior is different from the one of the MAP problem and will be vanishing as BOPE does more iterations. This property of BOPE is significant and enables to reduce severe overfitting for probabilistic models in ill-posed cases, including short text, sparse data, and noisy data. We validate the practical efficiency of BOPE in two contexts: text analysis and recommender systems. Both contexts show the superior of BOPE over the baselines.

Extremum Seeking for Nonlinear Uncertain Systems: A Small-Gain Synthesis

This paper studies the extremum seeking problem for static maps with the inputs of the maps generated by a nonlinear uncertain system. A new small-gain approach is developed which uses an extremum seeking strategy to generate a reference signal, and employs a control law for reference-tracking of the nonlinear uncertain systems. The notions of input-to-state stability (ISS) and input-to-output stability (IOS) are used to characterize the interconnection between the extremum seeking strategy and the reference-tracking controller, and the nonlinear small-gain theorem is employed to guarantee the stability of the closed-loop extremum seeking system. With the proposed approach, the extremum seeking problem for a complex nonlinear system is solvable as long as one can design a proper reference-tracking controller for the system. Examples are given to show the feasibility of the proposed approach, and a numerical simulation is employed to show the effectiveness of the proposed design.

On the problem of discontinuity at fixed point

In this paper we obtain two known solutions of the problem of continuity of contractive mappings at fixed point under alternative set of conditions; these known solutions followed the first solution of Rhoades problem by Pant in 1999. We show that these two solutions characterize completeness and we also compare these with some recent solutions of the Rhoades problem.

Parallel extragradient algorithms for a family of pseudomonotone equilibrium problems and fixed point problems of nonself-nonexpansive mappings in Hilbert space

Parallel extragradient algorithms for a family of pseudomonotone equilibrium problems and fixed point problems of nonself-nonexpansive mappings in Hilbert space

Generalized extragradient iterative methods for solving split feasibility and fixed point problems in Hilbert spaces

In the paper, we analyze the extragradient method with regularization for finding a common element of the solution set of the split feasibility and fixed point problems of pseudo-contractive mappings. Moreover we propose the extragradient with regularization iterative method due to a generalized Ishikawa-type and Mann-type iterative methods. The weak convergence theorems of the sequences generated by the proposed iterative methods are obtained under the certain assumptions on pseudo-contractive mappings in real Hilbert spaces. Finally, we give the numerical example to demonstrate the effectiveness of our theoretical results and compare its behavior with the iterative methods of Ceng et al. (Fixed Point Theory Appl 192, 2015).

Sequences of contractions on cone metric spaces over Banach algebras and applications to nonlinear systems of equations and systems of differential equations

It is well known that fixed point problems of contractive-type mappings defined on cone metric spaces over Banach algebras are not equivalent to those in usual metric spaces (see [3] and [10]). In this framework, the novelty of the present paper represents the development of some fixed point results regarding sequences of contractions in the setting of cone metric spaces over Banach algebras. Furthermore, some examples are given in order to strengthen our new concepts. Also, based on the powerful notion of a cone metric space over a Banach algebra, we present important applications to systems of differential equations and coupled functional equations, respectively, that are linked to the concept of sequences of contractions.

The first attempts to create archaeological maps of Samara and Simbirsk governorates at the turn of the XIX-XX centuries

The paper discusses the experience of Russians and their attempt to create the first archaeological maps of Samara and Simbirsk Provinces, organized by V.N.Polivanov from Simbirsk and V.A.Miller and F.T.Yakovlev from Samara at the end of the XIXXX centuries. The author introduces an archaeological map used by V.A.Miller and M.A.Miller into science. Due to the unpublished archival materials the author identifies the sources of information used to compile the archaeological map of the Simbirsk Province. For example, the use of a questionnaire describes the first steps in the centralized collection of information on ancient monuments in the Russian province in the 19thcentury. The questionnaires of the Central Statistical Committee of 1873 and the Moscow Archaeological Society of 1888 are noted for solving the problem of creating archaeological maps of individual provinces. For example, the questionnaires of the Simbirsk provincial scientific archival commission of 1896 solved the map problem of the Simbirsk province. The time gap is considered between the creation of provincial scholars of archival commissions in Samara and Simbirsk, organizations that were not only in-charge of the preservation and processing of archival documents, but were involved in archaeological research and conducted activities to study antiquities. It is summarized that the Samara and Simbirsk provinces were at different levels of involvement in the all-Russian processes, but, one way or another, prerequisites were laid in both provinces for drawing up plans for further archaeological study of the two regions.

On the Aleksandrov problem for mappings preserving fuzzy n-distance in fuzzy n-normed spaces

On the Aleksandrov problem for mappings preserving fuzzy n-distance in fuzzy n-normed spaces

Ky Fan minimax inequalities for set-valued mappings on dense sets

We give a natural Ky Fan minimax inequality version of set-valued maps, and we deal with a type of vector equilibrium problem for set-valued mappings on a special dense set not on the whole domain. We use these results as applications to study the solutions of a generalized set-valued vector variational inequality.

Fixed point problems of nonexpansive mappings for nononvex set in Hilbert spaces

In this paper, we introduce a new concept of W-nonexpansive mappings and obtain fixed point theorems for nonexpansive mappings for non-convex set. Our results resolve fixed pointed problem that nonexpansive mappings be not on closed convex set, and it extends fixed point theorems for nonexpansive mappings.

An iterative method for solving proximal split feasibility problems and fixed point problems

The purpose of this research is to introduce a regularized algorithm based on the viscosity method for solving the proximal split feasibility problem and the fixed point problem in Hilbert spaces. A strong convergence result of our proposed algorithm for finding a common solution of the proximal split feasibility problem and the fixed point problem for nonexpansive mappings is established. We also apply our main result to the split feasibility problem, and the fixed point problem of nonexpansive semigroups, respectively. Finally, we give numerical examples for supporting our main result.

On VT-harmonic maps

VT-harmonic maps generalize the standard harmonic maps, with respect to the structure of both domain and target. These can be manifolds with natural connections other than the Levi-Civita connection of Riemannian geometry, like Hermitian, affine or Weyl manifolds. The standard harmonic map semilinear elliptic system is augmented by a term coming from a vector field V on the domain and another term arising from a 2-tensor T on the target. In fact, this geometric structure then also includes other geometrically defined maps, for instance magnetic harmonic maps. In this paper, we treat VT-harmonic maps and their parabolic analogues with PDE tools. We establish a Jäger–Kaul type maximum principle for these maps. Using this maximum principle, we prove an existence theorem for the Dirichlet problem for VT-harmonic maps. As applications, we obtain results on Weyl/affine/Hermitian harmonic maps between Weyl/affine/Hermitian manifolds, as well as on magnetic harmonic maps from two-dimensional domains. We also derive gradient estimates and obtain existence results for such maps from noncompact complete manifolds.

Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space

We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings, including the Vuorinen injectivity problem. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms M with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such bounded locally homeomorphic quasiregular mappings are defined in the unit 3-ball B3 ⊂ ℝ3 as mappings equivariant with the standard conformal action of uniform hyperbolic lattices Γ ⊂ Isom H3 in the unit 3-ball and with its discrete representation G = ρ(Γ) ⊂ Isom H4. Here, G is the fundamental group of our non-trivial hyperbolic 4-cobordism M = (H4 ∪ Ω(G))/G, and the kernel of the homomorphism ρ: Γ → G is a free group F3 on three generators.

Convergence Theorem of Two Sequences for Solving the Modified Generalized System of Variational Inequalities and Numerical Analysis

The purpose of this paper is to introduce an iterative algorithm of two sequences which depend on each other by using the intermixed method. Then, we prove a strong convergence theorem for solving fixed-point problems of nonlinear mappings and we treat two variational inequality problems which form an approximate modified generalized system of variational inequalities (MGSV). By using our main theorem, we obtain the additional results involving the split feasibility problem and the constrained convex minimization problem. In support of our main result, a numerical example is also presented.

General alternative regularization method for solving split equality common fixed point problem for quasi-pseudocontractive mappings in Hilbert spaces

We propose a general alternative regularization algorithm for solving the split equality fixed point problem for the class of quasi-pseudocontractive mappings in Hilbert spaces. We also illustrate the performance of our algorithm with numerical example and compare the result with some other algorithms in the literature in this direction. We found out that our algorithm requires a lesser number of iterations and CPU time for its convergence than some of the existing algorithms. Our results extend and generalize some existing results in the literature in this direction.

CR-analogue of the Siu-∂\overline{∂}-formula and applications to the rigidity problem for pseudo-Hermitian harmonic maps

We give several versions of Siu’s ∂ ∂ ¯ \partial \overline {\partial } -formula for maps from a strictly pseudoconvex pseudo-Hermitian manifold ( M 2 m + 1 , θ ) (M^{2m+1}, \theta ) into a Kähler manifold ( N n , g ) (N^n, g) . We also define and study the notion of pseudo-Hermitian harmonicity for maps from M M into N N . In particular, we prove a CR version of the Siu Rigidity Theorem for pseudo-Hermitian harmonic maps from a pseudo-Hermitian manifold with vanishing Webster torsion into a Kähler manifold having strongly negative curvature.

Viscosity Methods and Split Common Fixed Point Problems for Demicontractive Mappings

We, first, propose a new method for solving split common fixed point problems for demicontractive mappings in Hilbert spaces, and then establish the strong convergence of such an algorithm, which extends the Halpern type algorithm studied by Wang and Xu to a viscosity iteration. Above all, the step sizes in this algorithm are chosen without a priori knowledge of the operator norms.

Seeking affinity structure: Strategies for improving m-best graph matching

State-of-the-art methods for finding the m-best solutions to graph matching (QAP) rely on exclusion strategies. The k-th best solution is found by excluding all better ones from the search space. This provides diversity, a natural requirement for transforming a MAP problem into a m-best one. Since diversity enforces mode hopping, it is usually combined with a mode-approximation strategy such as marginalisation. However, these methods are generic insofar they do not incorporate the detailed structure of the problem at hand, i.e. the properties of the global affinity matrix which characterise the search space. Without this knowledge, it is thus hard to devise a practical criterion for choosing the next variable to clamp. In this paper, we propose several strategies to select the next variable to clamp, spanning the whole range between depth-first and breadth-first search, and we contribute with a unifying view for characterising the search space on the fly. Our strategies are: a) Number of factors in which the variables participate, b) centrality measures associated with the affinity matrix, and c) discrete pooling. Our experiments show that max number of factors and centrality provide a trade-off between efficiency and accuracy, whereas discrete pooling leads to an improvement of the state-of-the-art

A hybrid iterative algorithm for solving monotone variational inclusion and hierarchical fixed point problems

This paper deals with a strong convergence theorem for a hybrid iterative algorithm without extrapolating step to approximate a common solution of monotone variational inclusion and hierarchical fixed point problems for nonexpansive mappings. Some consequences of the strong convergence theorem are also derived. An application to mixed equilibrium problem is also discussed. Finally, we give a numerical example to justify the main result. The method and results presented in this paper generalize and unify previously known corresponding results of this area.

A Halpern-Type Iteration Method for Bregman Nonspreading Mapping and Monotone Operators in Reflexive Banach Spaces

In this paper, we introduce an iterative method for approximating a common solution of monotone inclusion problem and fixed point of Bregman nonspreading mappings in a reflexive Banach space. Using the Bregman distance function, we study the composition of the resolvent of a maximal monotone operator and the antiresolvent of a Bregman inverse strongly monotone operator and introduce a Halpern-type iteration for approximating a common zero of a maximal monotone operator and a Bregman inverse strongly monotone operator which is also a fixed point of a Bregman nonspreading mapping. We further state and prove a strong convergence result using the iterative algorithm introduced. This result extends many works on finding a common solution of the monotone inclusion problem and fixed-point problem for nonlinear mappings in a real Hilbert space to a reflexive Banach space.