An empirical method to determine kernel definiteness.
problem Determining definiteness of kernels is time-consuming and requires effort.
method Empirical approach using sampling and optimization with evolutionary algorithms.
result Empirical method can disprove definiteness and estimate likelihood of indefinite matrices.
This paper proposes a new differential privacy definition using Rao distance.
problem Improving differential privacy definitions for better sequential composition.
method Using Rao distance instead of divergences of densities to define privacy.
result Proposed definition shares interpretation with previous definitions but improves sequential composition.
New distances defined between space-times, proving some definite.
problem Defining distances between space-times.
method Introducing causal-null-compactifiable space-times and using cosmological time and null distance.
result Various definite distances defined, proving convergence of space-times.
A framework for measuring differences in categorical data.
problem Measuring differences between categorical variables.
method Introduces a general framework for efficient and flexible implementation of distances for categorical variables.
result Framework allows for incorporation of existing distances and introduces new formulations.
Defines null distance function from spacetime time function.
problem Describes null distance function from spacetime time function.
method Defines null distance function from spacetime time function.
result Null distance function encodes causal structure of spacetime.
A new tree-sliced Wasserstein distance improves optimal transport computations.
problem Computational and statistical drawbacks in optimal transport.
method Introducing tree metrics and averaging Wasserstein distances using random tree metrics.
result Tree-sliced Wasserstein distance outperforms other methods on benchmarks.
Paper interprets EMD for sets and proposes EMI, showing EMD's inferiority.
problem Definiteness issues in EMD for set comparisons.
method Set-theoretic interpretation of EMD, proposing EMI, analyzing definiteness, and comparing EMD to EMI.
result EMD is inferior to EMI in computer vision tasks.
Gaussian kernels on complex manifolds are never positive definite.
problem Analyzing positive definiteness of Gaussian kernels on non-simply-connected Riemannian manifolds.
method Combining recent preprint analysis and classical Riemannian geometry comparison theorems.
result Gaussian kernels are never positive definite on non-simply-connected closed Riemannian manifolds.
Positive definite matrices abound in a dazzling variety of applications. This ubiquity can be in part attributed to their rich geometric structure: positive definite matrices form a self-dual convex cone whose strict interior is a Riemannian manifold. The manifold view is endowed with a "natural" distance function whil…
Paper introduces a new distance measure for Gaussian Mixture Models.
problem Developing a new distance measure for Gaussian Mixture Models.
method Embedding K-component Gaussian Mixture Models into the manifold of symmetric positive definite matrices and calculating a lower bound for the Fisher-Rao metric.
result Demonstrated effectiveness through experiments on standard datasets.
Study of metrics on positive-definite matrices from power potential, linking to power means.
problem Understanding metrics on positive-definite matrices derived from power potential.
method Explicit expressions for geodesics and distance function derived from Hessian of power potential.
result Geodesics and distance function converge to weighted matrix geometric mean as β tends to zero.
We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.
Gradient descent algorithms on manifolds solve control and mean computation problems.
problem Control and mean computation on positive definite Hermitian matrices.
method Riemannian and natural gradient algorithms applied to geodesic distance.
result Efficient algorithms for control and mean computation demonstrated.
A projective parameter of a geodesic on a Finsler space is defined to be solution of a certain ODE. Using projective parameter and Funk metric, one can construct a projectively invariant intrinsic pseudo-distance on a Finsler space. In the present work, solutions of the projective parameter's ODE are characterized with…
New distances measure mixtures of Gaussians, useful in machine learning.
problem Comparing distributions with disjoint supports.
method Schoenberg-Rao distances based on concave Rao's entropy.
result Closed-form distances for mixtures of Gaussians.
The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and v…
New Sliced-Wasserstein distances for non-Euclidean data.
problem Computational burden of Wasserstein distance on non-Euclidean manifolds.
method Derive Sliced-Wasserstein distances and flows on Cartan-Hadamard manifolds.
result General constructions and non-parametric schemes for minimizing new distances.
Proposes a new Sliced-Wasserstein distance for covariance matrices in M/EEG signals.
problem Efficiently dealing with distributions of covariance matrices in M/EEG multivariate time series.
method Defines a Sliced-Wasserstein distance for symmetric positive definite matrices and applies it to brain-age prediction and Brain Computer Interface applications.
result Demonstrates computational efficiency and strong theoretical guarantees for the proposed distance.
We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, maximum mean discrepancies (MMD), that is, distances between embeddings of distributions to reproduc…
In a previous paper we introduced a notion of "genericity" for countable sets of curves in the curve complex of a surface S, based on the Lebesgue measure on the space of projective measured laminations in S. With this definition we prove that for each fixed g > 1 the set of irreducible genus g Heegaard splittings of h…
A new metric for comparing probability measures on graphs, scalable and negative definite.
problem Optimal transport's high complexity and indefiniteness for kernel machines.
method Sobolev transport metric for graph metrics, closed-form formula, negative definiteness.
result Sobolev transport yields a scalable and negative definite metric.
The paper tightens bounds on distances between Reeb graphs.
problem Certifying quasi-universality of distances between Reeb graphs.
method Establishes tight bi-Lipschitz bounds for various distances.
result Proves strict universality of the functional contortion distance for contour trees and coincides with interleaving distance for merge trees.
We propose an SDP relaxation for the Gromov-Wasserstein distance, providing globally optimal solutions.
problem Matching objects between incomparable spaces using the Gromov-Wasserstein distance.
method Semi-definite programming (SDP) relaxation of the GW distance.
result The SDP relaxation provides globally optimal solutions for the GW distance in some instances.
This work generalizes Log-Determinant divergences to infinite-dimensional settings.
problem Generalizing Log-Determinant divergences to infinite-dimensional spaces.
method Introducing a parametrized family of divergences, Alpha-Beta Log-Determinant divergences, for positive definite unitized trace class operators.
result The Alpha-Beta Log-Det divergences encompass various divergences and metrics, including the affine-invariant Riemannian distance and symmetric Stein divergence.
New distances for comparing multivariate normal distributions.
problem Comparing multivariate normal distributions efficiently and accurately.
method Approximated Fisher-Rao distance and pullback SPD cone distances.
result Efficient computation of distances between normal distributions.
The paper explores null distance convergence for warped product spacetimes.
problem Defining convergence for sequences of spacetimes as metric spaces.
method Using the null distance to define convergence of spacetimes.
result Optimal convergence theorem for warped product spacetimes.
The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. This link makes possible a better understanding of both theories. The concepts of the theory of affine spheres lead to a new definition of an …
New kernel improves MMDs with theoretical guarantees for gradient flows.
problem Non-smoothness of negative distance kernel in MMDs.
method Smoothed 1D absolute value function followed by fractional integral transform.
result Improved theoretical guarantees for Wasserstein gradient flows.
Fundamental weight systems identified as quantum states.
problem Identifying which weight systems are quantum states.
method Analyzing the Cayley distance kernel on the symmetric group and its positivity.
result All fundamental gl(n)-weight systems are quantum states.
Review of distance-based methods for time series classification.
problem Challenges in classifying time series data.
method Distance-based approaches for time series classification.
result New methods exploit distances to improve classification performance.
The economy globalization measure problem is discussed. Four macroeconomic indices of twenty among the "richest" countries are examined. Four types of "distances" are calculated.Two types of networks are next constructed for each distance measure definition. It is shown that the globalization process can be best charac…
J. Hempel's definition of the distance of a Heegaard surface generalizes to a complexity for a knot which is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essenti…
Paper introduces a generalized Bures-Wasserstein geometry for SPD matrices.
problem Understanding the geometry of SPD matrices for machine learning.
method Proposes a generalized Bures-Wasserstein geometry parameterized by a symmetric positive definite matrix.
result The GBW geometry outperforms the BW geometry in machine learning applications.
SIPS extends graph embedding by approximating more types of similarities.
problem Graph embedding's limitation in approximating certain types of similarities.
method Shifted inner-product similarity (SIPS) with bias terms.
result SIPS can approximate PD and CPD similarities, improving graph embedding performance.
Paper develops Gaussian process for distributions using Wasserstein distances.
problem Forecasting Gaussian processes indexed by probability distributions.
method Developed positive definite kernels based on Wasserstein distances.
result Efficient forecasting of Gaussian processes indexed by distributions.
Study of strictly accretive matrices using Finsler geometry.
problem Characterize the set of strictly accretive matrices.
method Introduced Finsler metrics and characterized geodesics and distance.
result Geodesic distance applied to matrix approximation problem.
Exploring distance functions on spacetime models.
problem No canonical distance function exists for Lorentzian manifolds.
method Comparing Riemannianization and null distance function approaches.
result Concrete comparison of distance functions in GRW setting.
Clarifies definitions of global hyperbolicity in various spaces.
problem Clarifying terminology in recent literature on global hyperbolicity.
method Comparing definitions in Lorentzian length spaces, optimal transport, and topological preordered spaces.
result The causal relation is a closed order and preserves compactness in all cases.
Proof of existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.
problem Existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.
method Construct an isotopic map instead of edge-flipping algorithm, generalizing Dyer et al's method.
result Strict proof of existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.
Paper proposes a new Wasserstein distance for mixtures of radially contoured distributions.
problem Generalization of Wasserstein distance to non-elliptically contoured distributions.
method Relaxed formulation for mixtures of radially contoured distributions without marginal consistency.
result The new distance yields more stable error and better color distribution in image transfer tasks.
Characterizes convexity of distance functions on Riemannian manifolds.
problem Understanding convexity of distance functions on Riemannian manifolds.
method Characterization of proximal normal cones, separation theorems, and analysis of convex subsets' boundaries.
result Convexity of distance functions for various boundary conditions on Riemannian manifolds.
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.
New geometric structures defined on SPD matrices for better understanding.
problem Understanding SPD matrices and their geometric properties.
method Introducing Finslerian and dual information-geometric structures on James' bicone domain.
result Geodesics correspond to straight lines in coordinate systems, and new dissimilarities generalize existing ones.
Defines a measure of knot concordance using cobordism distance.
problem Measuring how close knots are to being linearly dependent.
method Cobordism distance on cyclic subgroups of knot concordance group.
result Projective space of knot concordance group with integer-valued metric.
The paper explores a new type of kernel using Wasserstein distance for better classification of shapes.
problem Improving kernel methods for shape classification.
method Defined and studied exponential kernels based on regularized Wasserstein distance.
result Wasserstein squared exponential kernels perform better on small shape datasets.
WIPS optimizes inner product weights to approximate various similarities.
problem Learning high-quality node representations and accurate similarities.
method Weighted inner product similarity (WIPS) with adjustable weights.
result WIPS can approximate arbitrary general similarities including positive definite and indefinite kernels.
This note improves correlation stress tests using geodesic distance.
problem Improving financial risk management through better covariance stress tests.
method Proposes a new geometrically invariant definition of correlation stress tests.
result Demonstrates a submanifold approach to stress testing covariance matrices.
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…