arXiv research

The paper establishes distance estimates for manifolds with lower scalar curvature bounds.

We consider the problem of metric learning subject to a set of constraints on relative-distance comparisons between the data items. Such constraints are meant to reflect side-information that is not expressed directly in the feature vectors of the data items. The relative-distance constraints used in this work are part…

This work tightens generalization error bounds using Wasserstein distance.

Two new algorithms select matrix rows and columns to preserve distances.

We show that sub-surfaces of a Heegaard surface for which the relative Hempel distance of the splitting is sufficiently high have to appear in any Heegaard surface of genus bounded by half that distance.

We give an alternative definition of relative hyperbolicity based on properties of closest-point projections on peripheral subgroups. We also derive a distance formula for relatively hyperbolic groups, similar to the one for mapping class groups.

We present a case-study demonstrating the usefulness of Bayesian hierarchical mixture modelling for investigating cognitive processes. In sentence comprehension, it is widely assumed that the distance between linguistic co-dependents affects the latency of dependency resolution: the longer the distance, the longer the …

Introduces new Wasserstein distances for more intrinsic metrics.

QCML improves bond similarity learning in illiquid markets.

Bayesian nonparametric models improve OOD detection, especially with complex covariance structures.

Kim-Milman flow map stable under regular target measures

In recent years it has become popular to study machine learning problems in a setting of ordinal distance information rather than numerical distance measurements. By ordinal distance information we refer to binary answers to distance comparisons such as $d(A,B)<d(C,D)$. For many problems in machine learning and statist…

The set of Clifford bundles of bounded geometry over open manifolds can be endowed with a metrizable uniform structure. For one fixed bundle $E$ we define the generalized component $\gencomp (E)$ as the set of Clifford bundles $E'$ which have finite distance to $E$. If $D$, $D'$ are the associated generalized Dirac ope…

Paper analyzes neural network distances and stability, leading to a new learning rule.

We deal with hypersurfaces in the framework of the $n$-dimensional relative differential geometry. We consider a hypersurface $\varPhi$ of $\mathbb{R}^{n+1}$ with position vector field $\mathbf{x}$, which is relatively normalized by a relative normalization $\mathbf{y}$. Then $\mathbf{y}$ is also a relative normalizati…

Quantum channels' contraction under privacy constraints studied.

Paper classifies critical points in half-space with new distance function.

The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, i…

We introduce two versions of a new sketch for approximately embedding the Gaussian kernel into Euclidean inner product space. These work by truncating infinite expansions of the Gaussian kernel, and carefully invoking the RecursiveTensorSketch [Ahle et al. SODA 2020]. After providing concentration and approximation pro…

As a highlighting research topic in the multimedia area, cross-media retrieval aims to capture the complex correlations among multiple media types. Learning better shared representation and distance metric for multimedia data is important to boost the cross-media retrieval. Motivated by the strong ability of deep neura…

Given a compact Alexadrov $n$-space $Z$ with curvature curv $\ge κ$, and let $f: Z\to X$ be a distance non-increasing onto map to another Alexandrov $n$-space with curv $\ge κ$. The relative volume rigidity conjecture says that if $X$ achieves the relative maximal volume i.e. $vol(Z)=vol(X)$, then $X$ is isometric to $…

Study finite-energy metrics over complex manifold degenerations.

Deep learning approximates shortest path distances in large graphs.

Distance metric learning is an important component for many tasks, such as statistical classification and content-based image retrieval. Existing approaches for learning distance metrics from pairwise constraints typically suffer from two major problems. First, most algorithms only offer point estimation of the distanc…

We deal with hypersurfaces in the framework of the relative differential geometry in $\mathbb{R}^4$. We consider a hypersurface $\varPhi$ in $\mathbb{R}^4$ with position vector field $\vect{x}$ which is relatively normalized by a relative normalization $\vect{y}$. Then $\vect{y}$ is also a relative normalization of eve…

Relatively extremal knots are the relative minima of the ropelength functional in C^1 topology. On the set curves of fixed length, they are the relative maxima of thickness (normal injectivity radius) functional, including the ideal knots. We prove that a C^{1,1} relatively extremal knot in R^n has thickness equal to h…

Person re-identification (Re-ID) aims at matching images of the same person across disjoint camera views, which is a challenging problem in multimedia analysis, multimedia editing and content-based media retrieval communities. The major challenge lies in how to preserve similarity of the same person across video footag…

This handbook translates lead time analysis into R code.

Diffusion-weighted MR imaging (DWI) is the only method we currently have to measure connections between different parts of the human brain in vivo. To elucidate the structure of these connections, algorithms for tracking bundles of axonal fibers through the subcortical white matter rely on local estimates of the fiber …

Model-based clustering is widely-used in a variety of application areas. However, fundamental concerns remain about robustness. In particular, results can be sensitive to the choice of kernel representing the within-cluster data density. Leveraging on properties of pairwise differences between data points, we propose a…

Enhances LDL by integrating distance and directional information for more robust label feature representation.

The isotropic 3-space I^3 which is one of the Cayley--Klein spaces is obtained from the Euclidean space by substituting the usual Euclidean distance with the isotropic distance. In the present paper, we give several classifications on the surfaces in I^3 with the constant relative curvature (analogue of the Gaussian cu…

Inequalities linking entropy, Fisher info, Stein discrepancy, and Wasserstein distance on Riemannian manifolds.

We introduce a pathwise approach to analyze the relative performance of an equity portfolio with respect to a benchmark market portfolio. In this energy-entropy framework, the relative performance is decomposed into three components: a volatility term, a relative entropy term measuring the distance between the portfoli…

Improved sampling from complex distributions with reduced bias.

New method uses Cantor embeddings and Wasserstein distances to analyze predictive states in time series data.

Survey on a nonlinear d'Alembertian from general relativity.

The paper introduces a statistical test to assess and rank distance measures.

Study robust hypothesis testing under Hellinger distance, proving lower bounds and providing tests.

Geometric trick simplifies link homotopy and concordance.

dtSNE preserves local densities in low-dimensional embeddings.

Optimized parallel algorithms for identifying strong ties in data.

A left-invariant sub-Riemannian metric $d$ on the shortened Lorentz group $SO_0(2,1)$ under the condition that $d$ is right-invariant relative to the orthogonal Lie subgroup $1\otimes SO(2)$ is studied. The distance between arbitrary two elements, the cut locus (as the union of the subgroup $1\otimes SO(2)$ with the an…

Recent work in distance metric learning has focused on learning transformations of data that best align with provided sets of pairwise similarity and dissimilarity constraints. The learned transformations lead to improved retrieval, classification, and clustering algorithms due to the better adapted distance or similar…

Multi-label classification is a type of supervised learning where an instance may belong to multiple labels simultaneously. Predicting each label independently has been criticized for not exploiting any correlation between labels. In this paper we propose a novel approach, Nearest Labelset using Double Distances (NLDD)…

A timelike space is a Hausdorff topological space equipped with a partial order relation $<$ and a distance function $ρ$ satisfying a collection of axioms including a set of compatibility conditions between the partial order relation and the distance function. The distance function is defined only on a subset of the pr…

Defines a distance function on a manifold using symplectic embeddings and recovers the metric.

Cluster analysis is one of the essential tasks in data mining and knowledge discovery. Each type of data poses unique challenges in achieving relatively efficient partitioning of the data into homogeneous groups. While the algorithms for numeric data are relatively well studied in the literature, there are still challe…