Related papers: Multidimensional Scaling on Metric Measure Spaces

Latent space models assume that network ties are more likely between nodes that are closer together in an underlying latent space. Euclidean space is a popular choice for the underlying geometry, but hyperbolic geometry can mimic more…

Large collections of high-dimensional data have become nearly ubiquitous across many academic fields and application domains, ranging from biology to the humanities. Since working directly with high-dimensional data poses challenges, the…

The cognitive framework of conceptual spaces proposes to represent concepts as regions in psychological similarity spaces. These similarity spaces are typically obtained through multidimensional scaling (MDS), which converts human…

Developments in dynamical systems theory provides new support for the macroscale modelling of pdes and other microscale systems such as Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically resolving subgrid…

The definition of weighted distance measure involves weights. The paper proposes a weighted distance measure without the help of weights. Here, weights are intrinsically added to the measure, and for this, the concept of metric space is…

Real-world data such as digital images, MRI scans and electroencephalography signals are naturally represented as matrices with structural information. Most existing classifiers aim to capture these structures by regularizing the regression…

Modern sample points in many applications no longer comprise real vectors in a real vector space but sample points of much more complex structures, which may be represented as points in a space with a certain underlying geometric structure,…

We consider the problem of subspace estimation in a Bayesian setting. Since we are operating in the Grassmann manifold, the usual approach which consists of minimizing the mean square error (MSE) between the true subspace $U$ and its…

Low-dimensional embeddings are essential for machine learning tasks involving graphs, such as node classification, link prediction, community detection, network visualization, and network compression. Although recent studies have identified…

The medial axis of a smoothly embedded surface in $\mathbb{R}^3$ consists of all points for which the Euclidean distance function on the surface has at least two minima. We generalize this notion to the mid-sphere axis, which consists of…

Multidimensional scaling of gene sequence data has long played a vital role in analysing gene sequence data to identify clusters and patterns. However the computation complexities and memory requirements of state-of-the-art dimensional…

We show that Euclidean geometry in suitably high dimension can be expressed as a theory of orthogonality of subspaces with fixed dimensions and fixed dimension of their meet.

We provide a general framework to study convergence properties of families of maps. For manifolds $M$ and $N$ where $M$ is equipped with a volume form $\mathcal{V}$ we consider families of maps in the collection $\{(\phi, B) : B \subset M,…

Metric learning aims to embed one metric space into another to benefit tasks like classification and clustering. Although a greatly distorted metric space has a high degree of freedom to fit training data, it is prone to overfitting and…

Let $\mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $\mathfrak{M}$-universal if every $X\in\mathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find…

Suppose that a metric space $X$ is the union of two metric subspaces $A$ and $B$ that embed into Euclidean space with distortions $D_A$ and $D_B$, respectively. We prove that then $X$ embeds into Euclidean space with a bounded distortion…

Reparametrization invariant Sobolev metrics on spaces of regular curves have been shown to be of importance in the field of mathematical shape analysis. For practical applications, one usually discretizes the space of smooth curves and…

This is mainly a survey on the properties of Strong Metric Regularity (SMR) and Strong Metric subRegularity (SMsR) of mappings representing first order optimality conditions (so-called optimality mappings) of optimization problems in…

In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by…

Learning well-separated features in high-dimensional spaces, such as text or image embeddings, is crucial for many machine learning applications. Achieving such separation can be effectively accomplished through the dispersion of…