Problem In Metric Spaces Research Articles

Approximation Scheme for Weighted Metric Clustering via Sherali-Adams

Motivated by applications to classification problems on metric data, we study Weighted Metric Clustering problem: given a metric d over n points and a k x k symmetric matrix A with non-negative entries, the goal is to find a k-partition of these points into clusters C1,...,Ck, while minimizing the sum of A[i,j] * d(u,v) over all pairs of clusters Ci and Cj and all pairs of points u from Ci and v from Cj. Specific choices of A lead to Weighted Metric Clustering capturing well-studied graph partitioning problems in metric spaces, such as Min-Uncut, Min-k-Sum, Min-k-Cut, and more. Our main result is that Weighted Metric Clustering admits a polynomial-time approximation scheme (PTAS). Our algorithm handles all the above problems using the Sherali-Adams linear programming relaxation. This subsumes several prior works, unifies many of the techniques for various metric clustering objectives, and yields a PTAS for several new problems, including metric clustering on manifolds and a new family of hierarchical clustering objectives. Our experiments on the hierarchical clustering objective show that it better captures the ground-truth structural information compared to the popular Dasgupta's objective.

Approximating All-Points Furthest Pairs and Maximum Spanning Trees in Metric Spaces

Consider the problem of finding a point furthest from [Formula: see text] for each point [Formula: see text] in a metric space [Formula: see text], where [Formula: see text]. We prove this problem to have a deterministic [Formula: see text]-time [Formula: see text]-approximation algorithm. As a corollary, the maximum spanning tree problem in metric spaces has a deterministic [Formula: see text]-time [Formula: see text]-approximation algorithm. We also give a Monte Carlo [Formula: see text]-time algorithm outputting, for each [Formula: see text], a point [Formula: see text] satisfying [Formula: see text], where [Formula: see text]. As a corollary, we have a Monte Carlo [Formula: see text]-time algorithm for finding a spanning tree of weight at least [Formula: see text] in [Formula: see text].

On a threshold descent method for quasi-equilibria

We describe a special class of quasi-equilibrium problems in metric spaces and propose a novel simple threshold descent method for solving these problems. Due to the framework, the convergence of the method cannot be established with the usual convexity or generalized convexity assumptions. Under mild conditions, the iterative procedure gives solutions of the quasi-equilibrium problem. We apply this method to scalar and vector generalized quasi-equilibrium problems and to some classes of relative optimization problems.

The Branch Set of Minimal Disks in Metric Spaces

AbstractWe study the structure of the branch set of solutions to Plateau’s problem in metric spaces satisfying a quadratic isoperimetric inequality. In our 1st result, we give examples of spaces with isoperimetric constant arbitrarily close to the Euclidean isoperimetric constant $(4\pi )^{-1}$ for which solutions have large branch set. This complements recent results of Lytchak–Wenger and Stadler stating, respectively, that any space with Euclidean isoperimetric constant is a CAT($0$) space and solutions to Plateau’s problem in a CAT($0$) space have only isolated branch points. We also show that any planar cell-like set can appear as the branch set of a solution to Plateau’s problem. These results answer two questions posed by Lytchak and Wenger. Moreover, we investigate several related questions about energy-minimizing parametrizations of metric disks: when such a map is quasisymmetric, when its branch set is empty, and when it is unique up to a conformal diffeomorphism.

On the k-means/median cost function

In this work, we study the k-means cost function. Given a dataset X⊆Rd and an integer k, the goal of the Euclidean k-means problem is to find a set of k centers C⊆Rd such that Φ(C,X)≡∑x∈Xminc∈C⁡‖x−c‖2 is minimized. Let Δ(X,k)≡minC⊆Rd⁡Φ(C,X) denote the cost of the optimal k-means solution. For any dataset X, Δ(X,k) decreases as k increases. In this work, we try to understand this behavior more precisely. For any dataset X⊆Rd, integer k≥1, and a precision parameter ε>0, let L(X,k,ε) denote the smallest integer such that Δ(X,L(X,k,ε))≤ε⋅Δ(X,k). We show upper and lower bounds on this quantity. Our techniques generalize for the metric k-median problem in metric spaces with bounded doubling dimension. Finally, we observe that for any dataset X, we can compute a set S of size O(L(X,k,ε/c)) using D2-sampling such that Φ(S,X)≤ε⋅Δ(X,k) for some fixed constant c. Some of the applications include new pseudo-approximation guarantees for k-means++ and bounds for movement-based coreset constructions.

Existence of Solutions of Bifunction-Set Optimization Problems in Metric Spaces

Existence of Solutions of Bifunction-Set Optimization Problems in Metric Spaces

Near-linear Time Approximation Schemes for Clustering in Doubling Metrics

We consider the classic Facility Location, k -Median, and k -Means problems in metric spaces of doubling dimension d . We give nearly linear-time approximation schemes for each problem. The complexity of our algorithms is Õ(2 (1/ε) O(d2) n) , making a significant improvement over the state-of-the-art algorithms that run in time n (d/ε) O(d) . Moreover, we show how to extend the techniques used to get the first efficient approximation schemes for the problems of prize-collecting k -Median and k -Means and efficient bicriteria approximation schemes for k -Median with outliers, k -Means with outliers and k -Center.

A general class of relative optimization problems

We consider relative or subjective optimization problems where the goal function and feasible set are dependent on the current state of the system under consideration. In general, they are formulated as quasi-equilibrium problems, hence finding their solutions may be rather difficult. We describe a rather general class of relative optimization problems in metric spaces, which in addition depend on the starting state. We also utilize quasi-equilibrium type formulations of these problems and show that they admit rather simple descent solution methods. This approach gives suitable trajectories tending to a relatively optimal state. We describe several examples of applications of these problems. Preliminary results of computational experiments confirmed efficiency of the proposed method.

Caristi's fixed point theorem in probabilistic metric spaces

In this work, we define a partial order on probabilistic metric spaces and establish some new Caristi's fixed point theorems and Ekeland's variational principle for the class of (right) continuous and Archimedean t-norms. As an application, a partial answer to Kirk's problem in metric spaces is given.

Fault-Tolerant Covering Problems in Metric Spaces

In this article, we study some fault-tolerant covering problems in metric spaces. In the metric multi-cover problem (MMC), we are given two point sets Y (servers) and X (clients) in an arbitrary metric space $$(X \cup Y, d)$$ , a positive integer k that represents the coverage demand of each client, and a constant $$\alpha \ge 1$$ . Each server can host a single ball of arbitrary radius centered on it. Each client $$x \in X$$ needs to be covered by at least k such balls centered on servers. The objective function that we wish to minimize is the sum of the $$\alpha $$ -th powers of the radii of the balls. We also study some non-trivial generalizations of the MMC, such as (a) the non-uniform MMC, where we allow client-specific demands, and (b) the t-MMC, where we require the number of open servers to be at most some given integer t. We present the first constant approximations for these fault-tolerant covering problems. Our algorithms are based on the following paradigm: for each of the three problems, we present an efficient algorithm that reduces the problem to several instances of the corresponding 1-covering problem, where the coverage demand of each client is 1. The reductions preserve optimality up to a multiplicative constant factor. Applying known constant factor approximation algorithms for 1-covering, we obtain our results for the MMC and these generalizations.

Maximal metric surfaces and the Sobolev-to-Lipschitz property

We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak–Wenger, which satisfies a related maximality condition.

Space of minimal discs and its compactification

We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds. This kind of discs naturally come up in the context of the solution of Plateau’s problem in metric spaces by Lytchak and Wenger as generalizations of minimal surfaces.

Determining Fuzzy Distance between Sets by Application of Fixed Point Technique Using Weak Contractions and Fuzzy Geometric Notions

In the present paper, we solve the problem of determining the fuzzy distance between two subsets of a fuzzy metric space. We address the problem by reducing it to the problem of finding an optimal approximate solution of a fixed point equation. This approach is well studied for the corresponding problem in metric spaces and is known as proximity point problem. We employ fuzzy weak contractions for that purpose. Fuzzy weak contraction is a recently introduced concept intermediate to a fuzzy contraction and a fuzzy non-expansive mapping. Fuzzy versions of some geometric properties essentially belonging to Hilbert spaces are considered in the main theorem. We include an illustrative example and two corollaries, one of which comes from a well-known fixed point theorem. The illustrative example shows that the main theorem properly includes its corollaries. The work is in the domain of fuzzy global optimization by use of fixed point methods.

Proximal Point Algorithm for Quasi-Convex Minimization Problems in Metric Spaces

ABSTRACTIn this paper, the proximal point algorithm for quasi-convex minimization problem in nonpositive curvature metric spaces is studied. We prove Δ-convergence of the generated sequence to a critical point (which is defined in the text) of an objective quasi-convex, proper and lower semicontinuous function with at least a minimum point as well as some strong convergence results to a minimum point with some additional conditions. The results extend the recent results of the proximal point algorithm in Hadamard manifolds and CAT(0) spaces.

Structure of solutions of nonautonomous optimal control problems in metric spaces

Structure of solutions of nonautonomous optimal control problems in metric spaces

Approximate nearest neighbor algorithm based on navigable small world graphs

We propose a novel approach to solving the approximate k-nearest neighbor search problem in metric spaces. The search structure is based on a navigable small world graph with vertices corresponding to the stored elements, edges to links between them, and a variation of greedy algorithm for searching. The navigable small world is created simply by keeping old Delaunay graph approximation links produced at the start of construction. The approach is very universal, defined in terms of arbitrary metric spaces and at the same time it is very simple. The algorithm handles insertions in the same way as queries: by finding approximate neighbors for the inserted element and connecting it to them. Both search and insertion can be done in parallel requiring only local information from the structure. The structure can be made distributed. The accuracy of the probabilistic k-nearest neighbor queries can be adjusted without rebuilding the structure.The performed simulation for data in the Euclidean spaces shows that the structure built using the proposed algorithm has small world navigation properties with log2(n) insertion and search complexity at fixed accuracy, and performs well at high dimensionality. Simulation on a CoPHiR dataset revealed its high efficiency in case of large datasets (more than an order of magnitude less metric computations at fixed recall) compared to permutation indexes. Only 0.03% of the 10 million 208-dimensional vector dataset is needed to be evaluated to achieve 0.999 recall (virtually exact search). For recall 0.93 processing speed 2800queries/s can be achieved on a dual Intel X5675 Xenon server node with Java implementation.

The Monge Problem for Distance Cost in Geodesic Spaces

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and d L is a geodesic Borel distance which makes (X, d L ) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1-dimensional Hausdorff distance induced by d L . It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting d L -cyclical monotonicity is not sufficient for optimality.

On Hölder calmness of solution mappings in parametric equilibrium problems

We consider parametric equilibrium problems in metric spaces. Sufficient conditions for the Holder calmness of solutions are established. We also study the Holder well-posedness for equilibrium problems in metric spaces.

Optimal Transport with Branching Distance Costs and the Obstacle Problem

We address the Monge problem in metric spaces with a geodesic distance: $(X,d)$ is a Polish space and $d_N$ is a geodesic Borel distance which makes $(X,d_N)$ a possibly branching geodesic space. We show that under some assumptions on the transference plan we can reduce the transport problem to transport problems along a family of geodesics. We introduce three assumptions on a given $d_{N}$-monotone transference plan $\pi$ which imply, respectively, strong consistency of disintegration, continuity of the conditional probabilities of the first marginal, and a regularity property for the geometry of chain of transport rays. We show that this regularity is sufficient for the construction of a transport map with the same transport cost of $\pi$. We apply these results to the Monge problem in $\mathbb{R}^{d}$ with smooth, convex, and compact obstacle obtaining the existence of an optimal map, provided the first marginal is absolutely continuous with respect to the d-dimensional Lebesgue measure.

A fixed point result for multivalued generalized contraction maps

Using the concept of u-distance, a fixed point problem in metric spaces for closed valued maps, is solved. Consequently, several known fixed point results are either improved or generalized.