We calculate Euclidean distance degrees for common manifold optimization types.
problem Optimizing on manifold structures.
method Closed-form expressions for stationary points of Euclidean distance function.
result Closed-form expressions for all stationary points on manifold optimization.
Study infinite Euclidean distance discriminants of algebraic varieties.
problem Understanding the structure of data points with infinitely many critical points in Euclidean distance correspondence.
method Developed computer code to compute discriminants and proved properties of fibers.
result Infinite Euclidean distance discriminants contain all data points with infinitely many critical points for the nearest-point problem.
Paper proposes a method to recover point configurations from noisy distance data.
problem Recovering point configurations from noisy distance data.
method Robust Euclidean Distance Geometry via Dual Basis (RoDEoDB) algorithm.
result Exact recovery guarantees for point configuration and Gram matrix under mild conditions.
Matrix profile has been recently proposed as a promising technique to the problem of all-pairs-similarity search on time series. Efficient algorithms have been proposed for computing it, e.g., STAMP, STOMP and SCRIMP++. All these algorithms use the z-normalized Euclidean distance to measure the distance between subsequ…
The study compares Euclidean and cosine distances in medical drug prescription prediction.
problem Comparing Euclidean and cosine distances in medical drug prescription prediction.
method Established geometric properties and compared distances in real-world medical data.
result Different distances lead to different optimizing nonlinear kernel embedding frameworks.
Revisits Isomap, showing it constructs Euclidean representations of geodesic structure.
problem Nonlinear dimension reduction of manifold data.
method Revisits Isomap's rationale, clarifying its approach to constructing Euclidean representations of geodesic structure.
result Convexity is not required for shortest path distances to converge to Riemannian distances.
Extends manifold learning to non-Euclidean metrics.
problem Applying manifold learning to data in non-Euclidean spaces.
method Generalizes manifold learning to metric spaces and studies conditions for convergence.
result Conditions for the convergence of graph Laplacian in metric spaces.
Euclidean nets reveal properties of higher-dimensional manifolds.
problem Characterize the geometry of manifolds based on discrete Euclidean distances.
method Isometric embeddings and properties of geodesics.
result Manifolds share properties with Euclidean space in terms of geodesics and distances.
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.
We address noisy Euclidean distances in high dimensions, estimating noise levels and correcting distances.
problem Distorted pairwise Euclidean distances due to heteroskedastic noise.
method Developed a hyperparameter-free approach to jointly estimate noise magnitudes and correct distances.
result Our method provides accurate noise magnitude estimates and corrected distances in high-dimensional settings.
Smooth maps preserve distances on specific revolution surfaces.
problem Existence of smooth maps on revolution surfaces.
method Proving existence of maps preserving distances on meridians and parallels.
result Smooth maps exist from revolution surfaces to Euclidean plane.
For time series comparisons, it has often been observed that z-score normalized Euclidean distances far outperform the unnormalized variant. In this paper we show that a z-score normalized, squared Euclidean Distance is, in fact, equal to a distance based on Pearson Correlation. This has profound impact on many distanc…
Paper tackles robust Euclidean distance estimation with sparse outliers.
problem Estimating point positions from corrupted distance measurements.
method Proposes a novel algorithm using Nyström method and robust PCA.
result Achieves accurate recovery with minimal anchors and sparse outliers.
Introduces Grassmann Distance Complexity to measure algebraic set nearest point problems.
problem Measuring complexity of finding nearest points in Grassmannian space.
method Uses Lipschitz critical point theory and o-minimal geometry.
result Establishes fundamental properties of GDC, including bounds and finiteness conditions.
Study rigidity by logarithmic capacity and related functions.
problem Rigidity phenomena in kernel functions and capacities.
method Exploration of Bergman kernel, logarithmic capacity, Green's function, and Euclidean distance/volume.
result Established rigidity theorems by logarithmic capacity.
A new method compares unaligned datasets using log-Euclidean signatures of SPD matrices.
problem Efficiently comparing datasets with unknown alignment.
method Diffusion operators, Riemannian geometry, log-Euclidean metric.
result LES distance recovers meaningful structural differences, outperforming existing methods.
The paper studies convexity of products of squared Euclidean distances.
problem Convexity of products of squared Euclidean distances.
method Proved a convexity principle and applied it to products of squared distances, computed Hessian-positive regions and exact convexity levels.
result Computed exact convexity and quasiconvexity truncation levels for the two-centre model.
We study the use of power weighted shortest path distance functions for clustering high dimensional Euclidean data, under the assumption that the data is drawn from a collection of disjoint low dimensional manifolds. We argue, theoretically and experimentally, that this leads to higher clustering accuracy. We also pres…
We define a class of Euclidean distances on weighted graphs, enabling to perform thermodynamic soft graph clustering. The class can be constructed form the "raw coordinates" encountered in spectral clustering, and can be extended by means of higher-dimensional embeddings (Schoenberg transformations). Geographical flow …
Generating point clouds, e.g., molecular structures, in arbitrary rotations, translations, and enumerations remains a challenging task. Meanwhile, neural networks utilizing symmetry invariant layers have been shown to be able to optimize their training objective in a data-efficient way. In this spirit, we present an ar…
This paper addresses Gaussian Process regression over probability measures, revealing a non-stationarity issue between Euclidean and Wasserstein kernels.
problem Non-stationarity issue between Euclidean and Wasserstein kernels in Gaussian Process regression over probability measures.
method Assuming Euclidean input space, applying algebraic transformation based on uncovered non-stationarity relationship to create a non-stationary and Wasserstein-based Gaussian Process model.
result An algebraic transformation simplifies learning a non-stationary Gaussian Process model over probability measures.
The paper describes distances on Sol-type groups using novel geometric techniques.
problem Understanding distances on Sol-type groups.
method New technique of Euclidean curve surgery to describe uniformly roughly geodesic paths.
result The rough isometry type of distances on Sol-type groups is determined by a specific metric restriction.
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.
Graph-based methods provide a powerful tool set for many non-parametric frameworks in Machine Learning. In general, the memory and computational complexity of these methods is quadratic in the number of examples in the data which makes them quickly infeasible for moderate to large scale datasets. A significant effort t…
APGD algorithm reconstructs point set from partial distance measurements.
problem Reconstructing point set configuration from partial Euclidean distance measurements.
method Asymmetric Projected Gradient Descent (APGD) for EDMC problem.
result Global convergence and exact recovery with O(μ2r3κ2nlogn) observations. Permutation invariant network learns Wasserstein metrics.
problem Understanding the space of probability measures and comparing distributions.
method Permutation invariant network mapping samples to a low-dimensional space.
result Network can generalize to compute distances between unseen densities and learn moments.
New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.
problem Finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows with surgery for inversive distance circle packings.
result Longtime existence and global convergence of combinatorial curvature flows with surgery.
Learning a distance function or metric on a given data manifold is of great importance in machine learning and pattern recognition. Many of the previous works first embed the manifold to Euclidean space and then learn the distance function. However, such a scheme might not faithfully preserve the distance function if t…
A fast binary embedding method preserves Euclidean distances in high-dimensional data.
problem Preserving Euclidean distances in high-dimensional datasets.
method Stable noise-shaping quantization of Ax with A a sparse Gaussian random matrix, followed by a linear transformation. result Euclidean distances are approximated by the ℓ1 norm on binary sequences, leading to accurate binary codes. A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.
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…
In this paper we demonstrate how the geometrically motivated algorithm to determine whether a two generator real Mobius group acting on the Poincare plane is or is not discrete can be interpreted as a non-Euclidean Euclidean algorithm. That is, the algorithm can be viewed as an application of the Euclidean division alg…
The article generalizes Clairaut's formula for geodesics on submanifolds.
problem Conditions for geodesics on specific submanifolds.
method Study of geodesics on submanifolds involving Euclidean distance.
result Generalization of Clairaut's formula for higher dimensions.
Although recovering an Euclidean distance matrix from noisy observations is a common problem in practice, how well this could be done remains largely unknown. To fill in this void, we study a simple distance matrix estimate based upon the so-called regularized kernel estimate. We show that such an estimate can be chara…
Principal Component Analysis (PCA) is one of the most important methods to handle high dimensional data. However, most of the studies on PCA aim to minimize the loss after projection, which usually measures the Euclidean distance, though in some fields, angle distance is known to be more important and critical for anal…
FastMap-D embeds directed graphs using potential fields.
problem Embedding directed graphs in Euclidean space.
method Generalization of FastMap to handle directed graphs using a potential field and machine learning.
result FastMap-D outperforms other approaches in embedding directed graphs.
A new tensorial metric describes geometry in 4D space.
problem Understanding the structure of hypercomplex space.
method Developed a new geometry group in R^4 with a tensorial metric.
result Riemannian and Euclidean distances are special cases of the Alpha Group's metric.
Upper bound for max-sliced 2-Wasserstein distance between measures.
problem Estimating distance between probability measures and their empirical counterparts.
method Same technique as previous work, upper bound approach.
result Upper bound for expected max-sliced 2-Wasserstein distance.
A scalable version of MADD improves big-data classification speed.
problem High computational complexity of MADD in big data.
method Selecting a representative set and using Random Fourier Features.
result Achieves similar performance to MADD but at a fraction of the computing time.
We study surfaces with decorations and prove uniformization in non-Euclidean geometries.
problem Discrete conformal equivalence in non-Euclidean geometries.
method Variational principle and continuous deformation.
result One master theory of discrete conformal equivalence across different geometries.
Most random ReLU networks are vulnerable to small, Euclidean adversarial perturbations.
problem Vulnerability of ReLU networks to adversarial attacks.
method Analysis of random ReLU networks with decreasing dimensions, using gradient flow and descent.
result Most examples can be perturbed by small Euclidean distances via gradient methods.
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…
LOT framework speeds up event distance computation in collider physics.
problem Computational inefficiency in quantifying event distances.
method Linearized Optimal Transport (LOT) for efficient computation.
result LOT significantly reduces computational cost without sacrificing accuracy.
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…
The Procrustes distance is used to quantify the similarity or dissimilarity of (3-dimensional) shapes, and extensively used in biological morphometrics. Typically each (normalized) shape is represented by N landmark points, chosen to be homologous (i.e. corresponding to each other), as far as possible, and the Procrust…
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…
New neural nets respect triangle inequality, improving graph and reinforcement learning performance.
problem Neural nets lack inductive bias for certain subadditive distances.
method Introduced novel architectures that universally approximate norm-induced metrics.
result Neural nets with triangle inequality inductive bias outperform existing approaches.
Study the hanging chain shape around a circle.
problem Finding the shape of a curve extremizing potential energy to a circle.
method Analyzes curves minimizing potential energy to a circle, considering both inside and outside.
result Describes shapes of curves for different powers of distance to the circle.