arXiv research

New method learns local metrics for k-NN classification using sample similarity.

Distance metric learning is a successful way to enhance the performance of the nearest neighbor classifier. In most cases, however, the distribution of data does not obey a regular form and may change in different parts of the feature space. Regarding that, this paper proposes a novel local distance metric learning met…

Similarity/Distance measures play a key role in many machine learning, pattern recognition, and data mining algorithms, which leads to the emergence of metric learning field. Many metric learning algorithms learn a global distance function from data that satisfy the constraints of the problem. However, in many real-wor…

Extends manifold learning to non-Euclidean metrics.

We show that the volume of a simple Riemannian metric on $D^n$ is locally monotone with respect to its boundary distance function. Namely if $g$ is a simple metric on $D^n$ and $g'$ is sufficiently close to $g$ and induces boundary distances greater or equal to those of $g$, then $vol(D^n,g')\ge vol(D^n,g)$. Furthermor…

Improved image ranking model using ordinal distance metric learning and multidimensional scaling.

Assigns compact set distance-like functions to non-compact geodesic spaces.

Study validates Michor-Mumford conjecture in infinite dimensional Hilbertian H-type groups.

In this paper, we present a novel two-stage metric learning algorithm. We first map each learning instance to a probability distribution by computing its similarities to a set of fixed anchor points. Then, we define the distance in the input data space as the Fisher information distance on the associated statistical ma…

Distance metric learning can be viewed as one of the fundamental interests in pattern recognition and machine learning, which plays a pivotal role in the performance of many learning methods. One of the effective methods in learning such a metric is to learn it from a set of labeled training samples. The issue of data …

Large scale agglomerative clustering is hindered by computational burdens. We propose a novel scheme where exact inter-instance distance calculation is replaced by the Hamming distance between Kernelized Locality-Sensitive Hashing (KLSH) hashed values. This results in a method that drastically decreases computation tim…

A new distance metric derived from information theory and estimation theory.

Null distance metric studies spacetime convergence.

Tree Mover's Distance measures graph attributes and improves GNN performance.

This study improves estimation of locally stationary functional time series using NW method.

We show that a small perturbation of the boundary distance function of a simple Finsler metric on the $n$-disc is also the boundary distance function of some Finsler metric. (Simple metric form an open class containing all flat metrics.) The lens map is map that sends the exit vector to the entry vector as a geodesic c…

The study compares Euclidean and cosine distances in medical drug prescription prediction.

Research shows that certain metric spaces cannot contain rigid structures and provides evidence for loose embeddings into Euclidean spaces.

Estimates path-valued data using signature metrics and local kernels.

We prove that every Kaehler metric, whose potential is a function of the time-like distance in the flat Kaehler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kaehler manifolds with the above mentioned metrics. New examples o…

Study rough Riemannian metrics on manifolds, proving their connectedness and completeness.

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 …

New null distance bounds confirm Big Bang singularity in cosmological models.

The paper proves uniform Temple charts and applies them to null distance metrics.

This work extends stochastic localization to joint probability measures for data analysis.

We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of definin…

Study geodesic structure of compact balls space and find explicit isometry.

For a compact Riemannian manifold with boundary, we want to find the metric structure from knowledge of distances between boundary points. This is called the "boundary rigidity problem". If the boundary is not concave, which means locally not all shortest paths lie entirely in the boundary, then we are able to find the…

The authors compute distances between arbitrary elements of Lie groups SU(2) and SO(3) for special left-invariant sub-Riemannian metrics $ρ$ and $d$. To compute distances for the second metric, we essentially use the fact that canonical two-sheeted covering epimorphism $Ω$ of the Lie group SU(2) onto the Lie group SO(3…

Survey of spectral, probabilistic, and deep metric learning methods.

Landmark-based node embeddings approximate shortest path distances in random graphs.

Quantum Earth Mover's distance improves stability and efficiency in quantum learning.

The need for appropriate ways to measure the distance or similarity between data is ubiquitous in machine learning, pattern recognition and data mining, but handcrafting such good metrics for specific problems is generally difficult. This has led to the emergence of metric learning, which aims at automatically learning…

Study geodesic extendibility on metric spaces and map them to a half-space.

We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have almost any other compact set of the cube as a tangent at all points or just in a den…

CAMEL enhances manifold embedding and learning with curvature metrics.

New method calculates Ricci curvature from distances between weighted volumes.

For many tasks and data types, there are natural transformations to which the data should be invariant or insensitive. For instance, in visual recognition, natural images should be insensitive to rotation and translation. This requirement and its implications have been important in many machine learning applications, a…

R-PCA extends PCA to Riemannian manifolds for structured data.

New method estimates geodesic distances using spherelets.

We study left-invariant distances on Lie groups for which there exists a one-parameter family of homothetic automorphisms. The main examples are Carnot groups, in particular the Heisenberg group with the standard dilations. We are interested in criteria implying that, locally and away from the diagonal, the distance is…

Metrics specifying distances between data points can be learned in a discriminative manner or from generative models. In this paper, we show how to unify generative and discriminative learning of metrics via a kernel learning framework. Specifically, we learn local metrics optimized from parametric generative models. T…

Quantitative metric spaces study function shapes and sphere diameters.

This paper is connected with the problem of describing path metric spaces that are homeomorphic to manifolds and biLipschitz homogeneous, i.e., whose biLipschitz homeomorphism group acts transitively. Our main result is the following. Let $X = G/H$ be a homogeneous manifold of a Lie group $G$ and let $d$ be a geodesic …

In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition $BE (K, N)$. The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells …

The article deals with intrinsic metrics, Dirac operators and spectral triples induced by regular Dirichlet and resistance forms. We show, in particular, that if a local resistance form is given and the space is compact in resistance metric, then the intrinsic metric yields a geodesic space. Given a regular Dirichlet f…

Study of geodesics on Riemannian stacks, measuring distances on orbit spaces.

Proposes a new graph kernel framework using regularized Wasserstein distances.