English
Related papers

Related papers: Subdivision Schemes in Metric Spaces

200 papers

Clustering is a fundamental unsupervised learning approach. Many clustering algorithms -- such as $k$-means -- rely on the euclidean distance as a similarity measure, which is often not the most relevant metric for high dimensional data…

Machine Learning · Computer Science 2019-10-22 Aude Genevay , Gabriel Dulac-Arnold , Jean-Philippe Vert

Here, a non-linear analysis method is applied rather than classical one to study projective Finsler geometry. More intuitively, by means of an inequality on Ricci-Finsler curvature, a projectively invariant pseudo-distance is introduced and…

Differential Geometry · Mathematics 2013-10-03 M. Sepasi , B. Bidabad

Spaces of quasi-invariant measures supplied with different topologies are studied. Their embeddings, projective decompositions, conditions for their metrizability are investigated. Theorems about convergence of nets of quasi-invariant…

Probability · Mathematics 2016-06-08 Sergey Victor Ludkowski

The MM principle is a device for creating optimization algorithms satisfying the ascent or descent property. The current survey emphasizes the role of the MM principle in nonlinear programming. For smooth functions, one can construct an…

Optimization and Control · Mathematics 2015-07-29 Kenneth Lange , Kevin L. Keys

In recent work we have shown how an accurate reduced model can be utilized to perform mesh refinement in random space. That work relied on the explicit knowledge of an accurate reduced model which is used to monitor the transfer of activity…

Numerical Analysis · Mathematics 2016-09-21 Jing Li , Panos Stinis

Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We overview the theory of classical MDS, along with its…

Statistics Theory · Mathematics 2020-07-14 Henry Adams , Mark Blumstein , Lara Kassab

Single-cell omics enable the profiles of cells, which contain large numbers of biological features, to be quantified. Cluster analysis, a dimensionality reduction process, is used to reduce the dimensions of the data to make it…

Genomics · Quantitative Biology 2024-06-06 Okezue Bell , Arthur Lee , Elizabeth Engle

This paper develops a theory of clustering and coding which combines a geometric model with a probabilistic model in a principled way. The geometric model is a Riemannian manifold with a Riemannian metric, ${g}_{ij}({\bf x})$, which we…

Machine Learning · Computer Science 2024-05-14 L. Thorne McCarty

We present and analyse a numerical framework for the approximation of nonlinear degenerate elliptic equations of the Stefan or porous medium types. This framework is based on piecewise constant approximations for the functions, which we…

Numerical Analysis · Mathematics 2019-12-20 Jerome Droniou , Robert Eymard

We present a general framework for analyzing the complexity of subdivision-based algorithms whose tests are based on the sizes of regions and their distance to certain sets (often varieties) intrinsic to the problem under study. We call…

Symbolic Computation · Computer Science 2018-01-19 Michael Burr , Shuhong Gao , Elias Tsigaridas

We present two practical improvement techniques for unsupervised segmentation learning. These techniques address limitations in the resolution and accuracy of predicted segmentation maps of recent state-of-the-art methods. Firstly, we…

Computer Vision and Pattern Recognition · Computer Science 2024-12-02 Alp Eren Sari , Francesco Locatello , Paolo Favaro

A new family of combined subdivision schemes with one tension parameter is proposed by the interpolatory and approximating subdivision schemes. The displacement vectors between the points of interpolatory and approximating subdivision…

Numerical Analysis · Mathematics 2019-04-30 Rabia Hameed

We derive general structure and rigidity theorems for submetries $f: M \to X$, where $M$ is a Riemannian manifold with sectional curvature $\sec M \ge 1$. When applied to a non-trivial Riemannian submersion, it follows that $diam X \leq…

Differential Geometry · Mathematics 2014-04-16 Xiaoyang Chen , Karsten Grove

Several multiscale methods account for sub-grid scale features using coarse scale basis functions. For example, in the Multiscale Finite Volume method the coarse scale basis functions are obtained by solving a set of local problems over…

Machine Learning · Computer Science 2017-11-15 Shing Chan , Ahmed H. Elsheikh

One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in…

Numerical Analysis · Mathematics 2016-12-23 Licia Lenarduzzi , Robert Schaback

We study a generalized framework for structured sparsity. It extends the well-known methods of Lasso and Group Lasso by incorporating additional constraints on the variables as part of a convex optimization problem. This framework provides…

Machine Learning · Computer Science 2011-06-28 Andreas Argyriou , Luca Baldassarre , Jean Morales , Massimiliano Pontil

Distances are pervasive in machine learning. They serve as similarity measures, loss functions, and learning targets; it is said that a good distance measure solves a task. When defining distances, the triangle inequality has proven to be a…

Machine Learning · Computer Science 2020-07-08 Silviu Pitis , Harris Chan , Kiarash Jamali , Jimmy Ba

In general, objects can be distinguished on the basis of their features, such as color or shape. In particular, it is assumed that similarity judgments about such features can be processed independently in different metric spaces. However,…

Machine Learning · Computer Science 2025-02-13 Yoshiyuki Ohmura , Wataru Shimaya , Yasuo Kuniyoshi

Diversities are an extension of the concept of a metric space which assign a non-negative value to every finite set of points, rather than just pairs. A general theory of diversities has been developed which exhibits many deep analogies to…

Metric Geometry · Mathematics 2026-03-04 David Bryant , Paul Tupper

We introduce a framework for quasi-Newton forward--backward splitting algorithms (proximal quasi-Newton methods) with a metric induced by diagonal $\pm$ rank-$r$ symmetric positive definite matrices. This special type of metric allows for a…

Optimization and Control · Mathematics 2018-11-27 Stephen Becker , Jalal Fadili , Peter Ochs