Study curve flows with global forcing terms using a distance comparison principle.
problem Analyse the behavior of curves under curve flows with global forcing terms.
method Prove a distance comparison principle for curve shortening flow with arbitrary global forcing terms.
result Established a distance comparison principle for curve flows with global forcing terms.
Study develops geodesic theory for foliations, proving Laplacian comparison theorems.
problem Comparing Laplacians on totally geodesic Riemannian foliations.
method Variational theory of geodesics, limit of Riemannian distance approximations.
result Sharp comparison theorems for sub-Riemannian distance in Sasakian foliations.
New distance comparison principle for curve shortening flow in higher dimensions.
problem Understanding curve shortening flow in higher dimensions.
method Established a variant of Huisken's distance comparison principle.
result Symmetric curve shortening flow with one-to-one convex projection develops Type I singularities and becomes asymptotically circular.
There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into ac…
A comparison-based algorithm finds nearest neighbors in metric spaces.
problem Finding nearest neighbors without direct distance information.
method Recursive splitting using random pivot points to form a comparison tree.
result The height of the comparison tree is logarithmic in the number of points, leading to efficient search performance.
Enhances graph comparison by incorporating edge features using Fused Gromov-Wasserstein distance.
problem Graph distances overlook edge attributes, limiting their effectiveness.
method Introduced Fused Gromov-Wasserstein distance for graph comparison with edge features. Proposed algorithms for distance and barycenter computation.
result Empirically validated the effectiveness of the novel distance in graph learning tasks.
A new metric HCP distance for comparing distributions.
problem Comparing high-dimensional probability distributions efficiently.
method Hilbert curve projection to low-dimensional coupling, followed by transport distance calculation.
result HCP distance is a proper metric for probability measures with bounded supports.
The paper extends the collar theorem to non-compact surfaces using new comparison theorems.
problem Proving the collar theorem for non-compact surfaces.
method Developed new Toponogov-type triangle comparison theorems.
result Eliminated the compactness hypothesis for the collar theorem.
In this paper, we use the distance comparison principle, first been developed by G. Huisken, to study the spatial curve shortening flow. We have got the result that if the initial curve is the helix, then the local minimum of the ratio of the extrinsic and intrinsic distance is non-decreasing. And we have proved a Gray…
In this work, we will verify some comparison results on Kahler manifolds. They are complex Hessian comparison for the distance function from a closed complex submanifold of a Kahler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scala…
TristouNet improves speaker comparison using neural networks and triplet loss.
problem Speaker comparison and change detection in short speech turns.
method Triplet loss for training neural network to project speech sequences into fixed-dimensional space.
result Significant improvements over state-of-the-art techniques for speaker comparison and change detection.
The study examines how quadratic inequalities affect distances in length spaces.
problem Effects of quadratic inequalities on distances in length spaces.
method Analyzes quadratic inequalities on distances between points in quadruples.
result Quadratic inequalities significantly alter distances in length spaces.
Paper tackles noisy comparison oracle for robust clustering algorithms.
problem Finding robust clustering algorithms under noisy comparison oracle.
method Develops algorithms for k-center clustering and agglomerative hierarchical clustering using noisy comparison oracle.
result Proves robust algorithms achieve good approximation guarantees with high probability.
A new sliced IGW distance for Gromov-Wasserstein alignment.
problem Scalability issues in Gromov-Wasserstein alignment for high-dimensional problems.
method Proposed a sliced IGW distance with rotational invariance.
result Natural rotational invariance of the sliced IGW distance.
Study curve flows with a global forcing term, proving distance comparison and convexity.
problem Analyzing the behavior of curves under curve shortening flow with a global forcing term.
method Distance comparison principle, finite time exclusion of singularities, convexity and convergence analysis.
result Convexity and smooth exponential convergence to a circle for closed curves.
An algorithm learns a kernel matrix from relative-distance constraints for semi-supervised clustering.
problem Learning metrics from relative-distance constraints to capture finer structures.
method Log determinant divergence for kernel matrix learning with relative-distance constraints.
result Kernels learned from relative-distance constraints yield better clusterings than existing methods.
A new method embeds distributions in a common space for optimal transport comparison.
problem Comparing distributions in different metric spaces.
method Sub-embedding robust Wasserstein (SERW) distance.
result SERW mimics GW distance properties and provides a cost relation.
New slicing methods speed up Gaussian mixture Wasserstein distance computations.
problem High computational cost of the mixture Wasserstein distance.
method Slicing-based approximations to reduce computational complexity.
result Significant reduction in computational complexity while preserving key properties.
Study metric learning from limited preference comparisons, showing how low-dimensional structure can still reveal metric information.
problem Learning metric from limited pairwise preference comparisons.
method Ideal point model, divide-and-conquer approach for low-dimensional structure.
result Metric can be jointly identified even with limited comparisons when items exhibit low-dimensional structure.
This paper compares two clustering evaluation metrics, revealing their differences and properties.
problem Understanding the differences between misclassification error distance and adjusted Rand index.
method Population origins, data analysis examples, detailed case studies, and simulation study.
result Reveals previous misconceptions about the two metrics and their distributions.
Polynomial-time relaxation improves Gromov-Hausdorff metric computation.
problem Computing Gromov-Hausdorff distance is intractable.
method Semidefinite programming relaxation of Gromov-Hausdorff metric.
result Relaxed distance can be computed in polynomial time.
The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.
problem Proving comparison theorems for sub-Laplacian in Riemannian foliations with minimal leaves.
method Using Riemannian foliations with minimal leaves, the paper proves comparison theorems for the sub-Laplacian.
result The comparison theorems yield a Bonnet-Myers type theorem, stochastic completeness, and Lipschitz regularization property for the sub-Riemannian semigroup.
A new dataset distance using optimal transport, agnostic of model and label sets.
problem Quantifying task similarity across datasets without model dependence.
method Optimal transport for model-agnostic dataset comparison.
result The new distance correlates with transfer learning difficulty across various datasets.
A new graph kernel uses LCS and Wasserstein distance for better graph comparisons.
problem Graph learning methods can be limited by information from distant vertices and path length constraints.
method Proposes a Graph Kernel based on LCS similarity and Wasserstein distance in a novel metric space.
result The new kernel emphasizes comparisons between similar paths and reduces information loss.
This note explores comparison geometry concepts and theorems.
problem Exploring various comparison theorems in geometry.
method Analyzes Rauch and Toponogov theorems and their applications.
result Introduction of Gromov-Hausdorff convergence and Alexandrov Spaces.
Develops a hypothesis testing framework for generalized Thurstone models.
problem Determining whether pairwise comparison data fits a generalized Thurstone model.
method Introduces separation distance and derives upper and lower bounds for testing.
result Critical threshold for testing depends on observation graph topology and scales as Θ((nk)−1/2) for complete graphs. In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some esti…
New method embeds DNA sequences for faster, more informative gene comparison.
problem Slow and costly sequence comparison methods for genes without exact matches.
method Recurrent neural networks to embed sequences in a low-dimensional space.
result Embedding allows for better comparison of genes without exact matches.
Here, a non-linear analysis method is applied rather than classical one to study projective Finsler geometry. More intuitively, by means of an inequality on Ricci-Finsler curvature, a projectively invariant pseudo-distance is introduced and an analogous of Schwarz' lemma in Finsler geometry is proved. Next, the Schwarz…
The paper analyzes prediction and recovery bounds for noisy ordinal embedding.
problem Predicting and recovering embeddings from noisy distance comparisons.
method Derives prediction error bounds, investigates Maximum Likelihood estimator, proposes new algorithms.
result Relates prediction errors to embedding accuracy through a nonlinear map.
A new numerical framework simplifies elastic surface matching and comparison.
problem Challenging problem in surface comparison and matching in computer vision.
method Relaxing the geodesic boundary constraint using a varifold fidelity metric.
result Flexibility to deal with arbitrary topologies and sampling patterns, scalability to large meshes.
Partial soft-matching distance improves neural representation comparison by allowing some neurons to remain unmatched.
problem Neural representations are noisy and contain outliers, making traditional matching methods unreliable.
method Extends soft-matching distance to a partial optimal transport setting, allowing some neurons to remain unmatched.
result Partial soft-matching provides robust correspondences that are more reliable under noise and outliers.
DEOT method compares distributions across agents with privacy and efficiency.
problem Comparing distributions across agents in a distributed system.
method Decentralized entropic optimal transport with mini-batch randomized block-coordinate descent and decentralized kernel approximation.
result The method provides a privacy-preserving and communication-efficient solution to distributed distribution comparison.
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.
Study compares sub-Riemannian curvature to optimal control variational problems.
problem Comparing sub-Riemannian curvature to optimal control variational problems.
method Introducing sub-Riemannian Bakry-Émery curvature and proving sub-Laplacian comparison theorems.
result Established sharp measure contraction property for 3-Sasakian manifolds.
We present a criterion for the stochastic completeness of a submanifold in terms of its distance to a hypersurface in the ambient space. This relies in a suitable version of the Hessian comparison theorem. In the sequel we apply a comparison principle with geometric barriers for establishing mean curvature estimates fo…
The study explores how to infer the geometry of space forms from similarity comparisons.
problem Inferring the geometry of space forms from unreliable similarity measurements.
method Introducing ordinal capacity and spread, proving their relation to space form properties, and using statistical analysis of similarity measurements.
result The statistical behavior of ordinal spread variables can identify the underlying space form.
Survey of distance computations in noncommutative geometry, linking to optimal transport.
problem Computing distances in noncommutative geometry.
method Review of explicit computations in various noncommutative geometries.
result Connes distance as a noncommutative version of Monge-Kantorovich metric.
We analyze (the harmonic map representation of) static solutions of the Einstein Equations in dimension three from the point of view of comparison geometry. We find simple monotonic quantities capturing sharply the influence of the Lapse function on the focussing of geodesics. This allows, in particular, a sharp estima…
Paper shows z-score normalized Euclidean distance equals Pearson correlation, impacting clustering methods.
problem Theoretical and practical impact of Euclidean distance vs. Pearson correlation in time series analysis.
method Demonstrates equivalence between z-score normalized Euclidean distance and Pearson correlation, and modifies k-Means algorithm.
result Standard k-Means algorithm produces similar results to modified version, but interpretation is strictly Pearson correlation.
New stable distance for classifying materials from point cloud data.
problem Classifying materials from noisy and sparse data.
method A new distance on persistence diagrams for matching and comparing topological features.
result Stability of the new distance provides theoretical justification for its use in materials classification.
The problem of filtering information from large correlation matrices is of great importance in many applications. We have recently proposed the use of the Kullback-Leibler distance to measure the performance of filtering algorithms in recovering the underlying correlation matrix when the variables are described by a mu…
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.
The paper studies curvature bounds for manifolds with density.
problem Curvature bounds for Riemannian manifolds with density.
method Develops new tools for studying weighted sectional curvature bounds, including a weighted Rauch comparison theorem and a modified convexity notion.
result Improves results for spaces of positive weighted sectional curvature and symmetry.
s-OTDD compares datasets efficiently without training, robust to class variations.
problem Efficiently compare datasets without training or class variations.
method Moment Transform Projection (MTP) and sliced optimal transport.
result s-OTDD correlates with optimal transport and transfer learning performance.
COPT optimizes graph distances via simultaneous optimal transport.
problem Learning graph representations unsupervisedly.
method Simultaneous optimization of dual transport plans between vertices and graph signals.
result COPT preserves spectral information and outperforms state-of-the-art methods.
Efficiently augments triplet data for better data analytics.
problem Lack of direct pairwise distance information for data analysis.
method Triplets augmentation to infer hidden information from existing data.
result Improves quality of kernel-based and kernel-free data analytics.
New formulations for comparing metric measure spaces with arbitrary positive measures.
problem Comparing metric measure spaces with arbitrary positive measures.
method Two novel formulations: a divergence and a conic lifting approach.
result Efficiently solvable formulations for comparing metric spaces with arbitrary positive measures.