Related papers: Classification in asymmetric spaces via sample com…
We solve the problem of best approximation by partial isometries of given rank to an arbitrary rectangular matrix, when the distance is measured in any unitarily invariant norm. In the case where the norm is strictly convex, we parametrize…
Recent literature has shown that symbolic data, such as text and graphs, is often better represented by points on a curved manifold, rather than in Euclidean space. However, geometrical operations on manifolds are generally more complicated…
We consider nonconforming methods for symmetric elliptic problems and characterize their quasi-optimality in terms of suitable notions of stability and consistency. The quasi-optimality constant is determined and the possible impact of…
The concept of a quasi-metric space arises by relaxing the requirement of the symmetry axiom in the definition of a metric. This small variation alters several structural properties possessed by a standard metric space. This article aims to…
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…
The concept of quasi-partial b-metric-like spaces is being introduced and studied with the help of topology. Examples are also discussed to support the results. Some fixed point theorems are proved in the setting of quasi-partial…
Metric learning seeks perceptual embeddings where visually similar instances are close and dissimilar instances are apart, but learned representations can be sub-optimal when the distribution of intra-class samples is diverse and distinct…
Recent research has seen numerous supervised learning-based methods for 3D shape segmentation and remarkable performance has been achieved on various benchmark datasets. These supervised methods require a large amount of annotated data to…
In this paper we develop proximal methods for statistical learning. Proximal point algorithms are useful in statistics and machine learning for obtaining optimization solutions for composite functions. Our approach exploits closed-form…
Nearest neighbor methods are a popular class of nonparametric estimators with several desirable properties, such as adaptivity to different distance scales in different regions of space. Prior work on convergence rates for nearest neighbor…
We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a…
Let $\mathrm{Sym}_q(m)$ be the space of symmetric matrices in $\mathbb{F}_q^{m\times m}$. A subspace of $\mathrm{Sym}_q(m)$ equipped with the rank distance is called a symmetric rank-metric code. In this paper we study the covering…
Learning the embedding space, where semantically similar objects are located close together and dissimilar objects far apart, is a cornerstone of many computer vision applications. Existing approaches usually learn a single metric in the…
Asymmetric information distances are used to define asymmetric norms and quasimetrics on the statistical manifold and its dual space of random variables. Quasimetric topology, generated by the Kullback-Leibler (KL) divergence, is considered…
The core idea of metric-based few-shot image classification is to directly measure the relations between query images and support classes to learn transferable feature embeddings. Previous work mainly focuses on image-level feature…
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)…
Distance metric learning (DML) approaches learn a transformation to a representation space where distance is in correspondence with a predefined notion of similarity. While such models offer a number of compelling benefits, it has been…
We propose an efficient algorithm for learning mappings between two metric spaces, $\X$ and $\Y$. Our procedure is strongly Bayes-consistent whenever $\X$ and $\Y$ are topologically separable and $\Y$ is "bounded in expectation" (our term;…
We study quasi-modular pseudometric spaces as asymmetric refinements of modular metric structures. To each such space we associate canonical forward and backward quasi-uniformities and the corresponding directional topologies. We introduce…
Classifying large-scale image data into object categories is an important problem that has received increasing research attention. Given the huge amount of data, non-parametric approaches such as nearest neighbor classifiers have shown…