相关论文: Computing medians and means in Hadamard spaces
Volume approximation is an important problem found in many applications of computer graphics, vision, and image processing. The problem is about computing an accurate and compact approximate representation of 3D volumes using some simple…
The Fr\'echet mean (or barycenter) generalizes the expectation of a random variable to metric spaces by minimizing the expected squared distance to the random variable. Similarly, the median can be generalized by its property of minimizing…
We present an algorithm to compute the geometric median of shapes which is based on the extension of median to high dimensions. The median finding problem is formulated as an optimization over distances and it is solved directly using the…
We study equilibrium problems in Hadamard spaces, which extend variational inequalities and many other problems in nonlinear analysis. In this paper, first we study the existence of solutions of equilibrium problems associated with…
Geodesic metric spaces support a variety of averaging constructions for given finite sets. Computing such averages has generated extensive interest in diverse disciplines. Here we consider the inverse problem of recognizing computationally…
There are several applications of stochastic optimization where one can benefit from a robust estimate of the gradient. For example, domains such as distributed learning with corrupted nodes, the presence of large outliers in the training…
We obtain Sion's minimax theorem in Hadamard spaces and discuss its applications. Among other things, we study several fundamental properties of resolvents of saddle functions in Hadamard spaces. An application to the proximal point…
This paper revisits an adaptation of the random forest algorithm for Fr\'echet regression, addressing the challenge of regression in the context of random objects in metric spaces. Recognizing the limitations of previous approaches, we…
In this paper, the proximal point algorithm for quasi-convex minimization problem in nonpositive curvature metric spaces is studied. We prove $\Delta$-convergence of the generated sequence to a critical point (which is defined in the text)…
The $k$-Means clustering problem on $n$ points is NP-Hard for any dimension $d\ge 2$, however, for the 1D case there exists exact polynomial time algorithms. Previous literature reported an $O(kn^2)$ time dynamic programming algorithm that…
We study the sublinear multivariate mean estimation problem in $d$-dimensional Euclidean space. Specifically, we aim to find the mean $\mu$ of a ground point set $A$, which minimizes the sum of squared Euclidean distances of the points in…
The backward-backward algorithm is a tool for finding minima of a regularization of the sum of two convex functions in Hilbert spaces. We generalize this setting to Hadamard spaces and prove the convergence of an error-tolerant version of…
In this paper, we extend the proximal point algorithm for vector optimization from the Euclidean space to the Riemannian context. Under suitable assumptions on the objective function the well definition and full convergence of the method to…
Recent interest in treespaces as well-founded mathematical domains for phylogenetic inference and statistical analysis for populations of anatomical trees has motivated research into efficient and rigorous methods for optimization problems…
In this paper, we consider the equilibrium problems and also their regularized problems under the setting of Hadamard spaces. The solution to the regularized problem is represented in terms of resolvent operators. As an essential machinery…
We develop two simple and efficient approximation algorithms for the continuous $k$-medians problems, where we seek to find the optimal location of $k$ facilities among a continuum of client points in a convex polygon $C$ with $n$ vertices…
The Fr\'{e}chet mean is a fundamental notion of central tendency defined as a minimizer of a sum of squared distances in a general metric space. In this paper, we study Fr\'{e}chet means in tropical geometry -- a piecewise linear,…
In the past few years powerful generalizations to the Euclidean k-means problem have been made, such as Bregman clustering [7], co-clustering (i.e., simultaneous clustering of rows and columns of an input matrix) [9,18], and tensor…
We investigate the asymptotic behavior of sequences generated by the proximal point algorithm for convex functions in complete geodesic spaces with curvature bounded above. Using the notion of resolvents of such functions, which was…
New algorithms are devised for finding the maxima of multidimensional point samples, one of the very first problems studied in computational geometry. The algorithms are very simple and easily coded and modified for practical needs. The…