Survey of spectral, probabilistic, and deep metric learning methods.
problem Developing effective distance metrics for various machine learning tasks.
method Divided into spectral, probabilistic, and deep approaches, covering various techniques and their applications.
result Comprehensive overview of metric learning methods, including new developments and applications.
The paper studies Kropina metrics with a specific curvature property.
problem Characterizing Kropina metrics with isotropic scalar curvature.
method Tensor analysis to derive expressions and characterize metrics.
result Characterization of Kropina metrics with isotropic scalar curvature.
New elastic metrics for surface shape analysis.
problem Analyzing shapes of surfaces in 3D space.
method Introducing a family of elastic metrics on surface spaces, computing geodesics, and comparing results.
result New metrics generalize SRNF and include geodesics for comparison.
Recently, metric learning and similarity learning have attracted a large amount of interest. Many models and optimisation algorithms have been proposed. However, there is relatively little work on the generalization analysis of such methods. In this paper, we derive novel generalization bounds of metric and similarity …
New metrics on curve spaces improve shape analysis.
problem Discretization of curve spaces and metric completeness.
method Sobolev metrics on discrete regular curves, completeness analysis.
result The finite-dimensional Riemannian manifolds are complete.
Injectivity of X-ray transform proven for non-smooth metrics.
problem Injectivity of X-ray transform on non-smooth metrics.
method Microlocal analysis of the normal operator, establishing ellipticity and smoothing properties.
result Injectivity of X-ray transform on L2 for metrics with finitely differentiable tensor. New TDA approach using Finsler metrics.
problem Traditional TDA concepts and methods.
method Introducing Finsler metrics for TDA.
result Relevance of Finsler metrics to TDA.
This paper explores the impact of metric choice on Fréchet regression.
problem Choosing the right metric for Fréchet regression in complex data.
method Review and extensive numerical studies of existing dimension reduction methods.
result Different metrics significantly affect the estimation of central and central mean space.
New connection between complex analysis and PDE for spherical conical metrics.
problem Understanding spectral properties of spherical conical metrics.
method Analyzing monodromy, eigenfunctions, and developing maps.
result Monodromy reducibility implies real-valued eigenfunction with eigenvalue 2.
Introduces a metric on vector-valued one-forms for functional data analysis.
problem Metric on vector-valued one-forms for functional data analysis.
method Diffeomorphism-invariant Riemannian metric calculation and geodesic equations.
result Geodesically and metrically incomplete space with specific curvature properties.
Unified treatment of elastic metrics for curves in any dimension.
problem Defining metrics on spaces of Euclidean curves for statistical analysis.
method Developing a unified approach to elastic metrics, extending results on existence of solutions and algorithms for computing distances and geodesics.
result Unified treatment of elastic metrics for all parameter choices, extending previous work.
Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.
problem Analysis of geometric properties on asymmetric metric measure spaces.
method Introduction of upper gradients, q-Laplacian, and q-heat flow in asymmetric settings. result Extension of concepts from symmetric to asymmetric metric measure spaces.
This tutorial explains distance metric learning, its algorithms, and evaluates their performance.
problem Improving similarity-based algorithms by learning distances from data.
method Describes the problem, mathematical foundations, and evaluates popular algorithms.
result Outstanding algorithms identified for distance metric learning.
A new metric measures saturation of neural network layers.
problem Analyzing the quality of latent representations in neural networks.
method Layer Saturation metric based on spectral analysis.
result Saturation is related to generalization and predictive performance.
The paper explores metrics and models for analyzing biological shapes.
problem Analyzing biological shapes using mathematical metrics.
method Review of Riemannian metrics and evolution equations, focusing on diffeomorphic shape analysis.
result Introduction of a new class of metrics involving optimization of a growth tensor.
Study Riemannian metric bundles and their connections to K-theory.
problem Understanding geometry and topology of manifolds with Riemannian metrics.
method Develop rigorous theory of Riemannian metric bundles and apply to K-theory.
result Contribute to deeper understanding of manifold geometry and topology.
Metrics stabilize persistent homology in data analysis.
problem Stabilizing invariants for characterizing connectivity structures in data.
method Using contour functions to define metrics for rank invariants.
result Optimal contours provide robust descriptors of spatial patterns.
A novel criterion selects optimal distance metrics for cell profile analysis.
problem Determining the most accurate distance metric for high-dimensional cell profiles.
method Generalized proposition and corollaries to evaluate and select distance metrics.
result Wasserstein and cosine similarity metrics are optimal for general cases.
Two new proofs provide Eguchi-Hanson metrics as ALE bubbles for Kummer constructions of K3 metrics.
problem Constructing Ricci-flat Kähler metrics on the K3 surface with special holonomy.
method Singular perturbation and weighted function space analysis.
result Large families of compact hyper-Kähler orbifolds as volume non-collapsed limits of Kummer constructions.
Study heat kernel on manifolds with fibred boundary metrics.
problem Analyzing spectral problems in manifolds with fibred boundary metrics.
method Construct heat kernel as polyhomogeneous conormal distribution.
result Fundamental step towards analysis of Ray-Singer torsion, eta-invariants and index theorems.
New metrics for SPD matrices explore affine invariance and symmetry principles.
problem Choosing appropriate metrics for SPD matrices based on invariance principles.
method Investigates power-affine and deformed-affine metrics within a continuum of SPD metrics.
result Introduces new families of metrics based on affine invariance and symmetry.
Researchers prove existence of a special Einstein metric on a 12-dimensional sphere.
problem Proving the existence of a non-round Einstein metric invariant under a specific group action.
method Numerical analysis techniques were used to produce an approximate Einstein metric, which was then perturbed into a true Einstein metric.
result A novel O(3)imesO(10)-invariant Einstein metric on S12 was successfully constructed. The paper analyzes the generalization of deep neural networks for metric and similarity learning.
problem Lack of rigorous understanding of generalization performance in metric and similarity learning.
method Derive explicit form of true metric, construct structured deep ReLU neural network, establish excess risk bounds.
result Explicit excess risk bounds for metric and similarity learning are derived.
EKH adds metrics to knot theory, enabling more detailed analysis.
problem Lack of quantitative data in knot theory.
method Integrates metric into knot theory with evolutionary Khovanov homology (EKH).
result EKH reveals non-trivial knot invariants at appropriate scales.
Paper examines stability of Bayesian posterior measures using integral probability metrics.
problem Stability of Bayesian inference in large-scale inverse problems.
method New families of integral probability metrics for likelihood and prior perturbations.
result Constructs new stability results for Bayesian posterior measures.
Following a review of metric, ultrametric and generalized ultrametric, we review their application in data analysis. We show how they allow us to explore both geometry and topology of information, starting with measured data. Some themes are then developed based on the use of metric, ultrametric and generalized ultrame…
Enhanced AI analysis predicts S&P 500 stock dynamics using various financial metrics.
problem Predicting S&P 500 stock performance with complex interplay of factors.
method Advanced financial metrics, machine learning, and integration of traditional and modern analytics.
result Enhanced predictive accuracy in market behavior and investment strategies.
We classify radial scalar flat metrics with constant third coeffcient of its TYZ expansion. As a byproduct of our analysis we provide a characterization of Simanca's scalar flat metric.
Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics, but their discretization is stil…
The paper connects different convergence concepts in geometric analysis.
problem Comparing convergence concepts in geometric analysis.
method Relating Lp convergence and volume convergence to Intrinsic Flat and Gromov-Hausdorff convergence. result Conditions for convergence of Riemannian manifolds under specific conditions.
A new method, InfoGuide, improves automatic clustering analysis.
problem Lack of automatic clustering analysis frameworks.
method Capturing traces of information gain between clustering retrievals.
result InfoGuide can enable more automatic clustering analysis.
Geometric analysis on diffeomorphism groups for fluid dynamics and information geometry.
problem Geometric analysis of fluid flows and optimal mass transport.
method Review of metrics and topology on diffeomorphism groups.
result Introduction of new metrics and topology for diffeomorphism groups.
The paper gauges AGI's impact on GDP growth using mathematical metrics.
problem Determining the economic effect of AGI on GDP growth.
method Analysis of historical data, development of a new mathematical algorithm, regression analysis.
result There is a positive correlation between AGI growth and real GDP growth.
A simpler metric for latent space geometry.
problem Complexity in capturing geometric structure of data manifolds.
method Prior-based approximate latent Riemannian metric.
result The proposed metric is simple, efficient, and robust.
The space of embedded submanifolds plays an important role in applications such as computational anatomy and shape analysis. We can define two different classes on Riemannian metrics on this space: so-called outer metrics are metrics that measure shape changes using deformations of the ambient space and they find appli…
The paper shows non-balanced condition for a specific metric.
problem Analyzing the balanced condition for the Eguchi-Hanson metric.
method Examining the blow-up of C2 at the origin and showing non-balanced condition for a specific metric. result The metric mgEH is not balanced for any positive integer m. Magnitude is not continuous but may be stable for most finite metric spaces.
problem Stability of magnitude invariant in finite metric spaces.
method Investigates the continuity properties of magnitude with respect to Gromov-Hausdorff topology.
result Magnitude is nowhere continuous but may be generically continuous.
Unified approach to shape matching using optimal control.
problem Shape registration of curves and surfaces.
method Unified Riemannian metrics, optimal control, chordal distances.
result Unified framework for shape matching.
Hierarchical geodesic model for analyzing shapes on manifolds.
problem Analyzing temporal observations on manifold-valued data.
method Adapted functional-based metric for efficiency; variational time discretization of geodesics.
result Performed hypothesis tests and estimated mean trends in longitudinal analysis.
Study on Frechet distance properties for paths and graphs.
problem Understanding topological properties of Frechet distance spaces.
method Proving path-connectedness of Frechet distance spaces and metric balls.
result Spaces of paths and graphs under Frechet distance are path-connected.
Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves Imm(S1,Rd) and on its Sobolev completions Iq(S1,Rd). We prove local well-posedness of the ge…
Geometric analysis of normal distributions using Fisher and Killing metrics.
problem Quantifying the difference between Fisher and Killing metrics on the space of normal distributions.
method Riemannian geometry, Fisher information metric, Killing metric, asymptotic geodesics.
result Approximation of Fisher metric by Killing metric for long distances is justified.
Causal analysis predicts market trends using time series data.
problem Predicting financial market trends using diverse time series data.
method Causal analysis based on lagged Pearson correlation applied to financial metrics.
result Discrimination of causal connections between different types of market data.
Wasserstein archetypal analysis finds optimal data summaries using Wasserstein metric.
problem Finding optimal data summaries using Wasserstein metric.
method Alternative formulation of archetypal analysis based on Wasserstein metric, with regularization and gradient-based computational approach.
result Existence and consistency of solutions for the regularized problem.
We obtain a compactness result for various classes of Riemannian metrics in dimension four; in particular our method applies to anti-self-dual metrics, Kahler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can be compactified by adding m…
Compact metrics from singular conformal deformations with curvature bounds.
problem Compactness of conformal metrics under curvature bounds.
method Use of A∞-weights from harmonic analysis. result Pre-compactness of metric spaces in Gromov-Hausdorff topology.
We investigate metric learning in the context of dynamic time warping (DTW), the by far most popular dissimilarity measure used for the comparison and analysis of motion capture data. While metric learning enables a problem-adapted representation of data, the majority of methods has been proposed for vectorial data onl…