Neuc-MDS extends MDS for non-Euclidean data.
problem Limitations of classical MDS with non-Euclidean data.
method Generalizes inner product to symmetric bilinear forms, optimizes eigenvalues of dissimilarity Gram matrix.
result Optimizes STRESS for non-Euclidean data.
ClustGeo uses Ward-like clustering with spatial constraints in R.
problem Hierarchical clustering with spatial/geographical constraints.
method Ward-like hierarchical clustering algorithm with two dissimilarity matrices and a mixing parameter.
result Determines optimal spatial contiguity without sacrificing variable quality.
We consider general non-Euclidean distance measures between real world objects that need to be classified. It is assumed that objects are represented by distances to other objects only. Conditions for zero-error dissimilarity based classifiers are derived. Additional conditions are given under which the zero-error deci…
This paper considers the problem of low-dimensional visualisation of very high dimensional information sources for the purpose of situation awareness in the maritime environment. In response to the requirement for human decision support aids to reduce information overload (and specifically, data amenable to inter-point…
Topolow embeds dissimilarity data into Euclidean space robustly against non-metricity and sparsity.
problem Embedding dissimilarity data into Euclidean space when dissimilarities are non-metric or sparse.
method Topolow uses a physics-inspired, gradient-free optimization framework to maximize likelihood under a Laplace error model.
result Topolow outperforms standard MDS methods in reconstructing sparse and non-Euclidean data.
New clustering method promotes diversity while maintaining fairness.
problem Developing clustering methods that enhance diversity with respect to protected attributes.
method Attraction-repulsion dissimilarities to penalize homogeneity with respect to protected attributes.
result Improves diversity in clusters without compromising fairness.
Virtual reality explores non-Euclidean Sol geometry.
problem Exploring non-Euclidean geometries in virtual reality.
method Developed a VR software for Sol geometry.
result Demonstrated the feasibility of non-Euclidean geometry in VR.
Study uses crochet to visualize non-Euclidean geometry.
problem Understanding non-Euclidean surfaces through physical models.
method Parametrization of crochet models to represent Lobachevskian surface.
result Crochet models reflect non-Euclidean geometry characteristics.
We introduce a new nearest-prototype classifier, the prototype vector machine (PVM). It arises from a combinatorial optimization problem which we cast as a variant of the set cover problem. We propose two algorithms for approximating its solution. The PVM selects a relatively small number of representative points which…
Unified framework for non-Euclidean CPD under scalable stochastic mirror descent.
problem Handling non-Euclidean losses in tensor decomposition.
method Tensor fiber sampling strategy-based stochastic mirror descent.
result Global convergence to a stationary point under reasonable conditions.
Paper uses non-Euclidean analysis to classify brain structure variations.
problem Classifying joint variations in multi-object brain structures.
method Combines non-Euclidean statistics and non-parametric integrative analysis.
result Effective, robust, and interpretable joint structure found.
Hybrid clustering combines partitional and hierarchical clustering for computational effectiveness and versatility in cluster shape. In such clustering, a dissimilarity measure plays a crucial role in the hierarchical merging. The dissimilarity measure has great impact on the final clustering, and data-independent prop…
Paper improves SOMs for non-Euclidean data modeling.
problem Traditional SOMs assume Euclidean data, limiting their applicability.
method Introduces topology-related extensions to traditional SOM algorithm.
result Improves SOMs for non-Euclidean data, enhancing data modeling.
DMAE uses neural networks to cluster data with flexible dissimilarity functions.
problem Clustering data with complex dissimilarity functions.
method Integrates a dissimilarity mixture model into deep learning architectures.
result DMAE achieves competitive clustering accuracy compared to other methods.
This foreword discusses the contributions of Bolyai, Gauss, and Lobachevsky to non-Euclidean geometry.
problem The development of non-Euclidean geometries by Bolyai, Gauss, and Lobachevsky.
method Historical review of the contributions of these mathematicians.
result The foundational work on non-Euclidean geometries by Bolyai, Gauss, and Lobachevsky.
Computes cohomology groups for NEC groups, focusing on Fuchsian groups.
problem Understanding the cohomology of non-Euclidean crystallographic groups.
method Computes cohomology groups for geometrically finite NEC groups, and determines the ring structure for Fuchsian groups.
result Determination of cohomology groups and ring structures for Fuchsian groups.
New method for predicting portfolio dynamics using non-Euclidean geometry.
problem Predicting efficient portfolios with geometric structure.
method Non-Euclidean conditional expectation and filtering equations.
result Accurate numerical forecasts of portfolio dynamics.
Non-Euclidean BPM extends optimization theory to non-Euclidean norms.
problem Extending BPM's convergence theory to non-Euclidean norms.
method Iteratively minimizing over norm balls in non-Euclidean geometry.
result Most BPM guarantees carry over to non-Euclidean norms.
Algorithm improves SVM classification in non-Euclidean spaces.
problem Limitations of traditional SVM in non-Euclidean spaces.
method Covariance-adjusted SVM using Cholesky Decomposition.
result Cholesky-SVM outperforms traditional SVM in non-Euclidean spaces.
Random Forest proximity measures for multi-view classification.
problem Combining multiple heterogeneous data views for classification.
method Building dissimilarity representations for each view, fusing them dynamically.
result Dynamic View Selection improves multi-view classification performance.
New method clusters stationary stochastic processes using covariance-based dissimilarity.
problem Clustering wide-sense stationary ergodic stochastic processes.
method Covariance-based dissimilarity measure with consistent algorithms for offline and online clustering.
result Asymptotically consistent algorithms for efficient clustering.
New RDPC dissimilarity measure improves time series clustering.
problem Improving time series clustering methods for diverse data.
method Combining weighted Pearson correlation with largest element-wise differences.
result RDPC outperforms existing methods in complex datasets.
The study connects curvature to the elastic energy of non-Euclidean thin bodies.
problem Understanding the elastic energy scaling of non-Euclidean thin bodies.
method Calculating the Γ-limit for the elastic energies of small balls, proving the scaling is \(h^4\).
result The natural scaling for non-Euclidean rods is \(h^4\), confirming previous claims.
These lecture notes are based on [arXiv: math/0702714, 0907.4469, 0907.4470]. We introduce and study basic aspects of non-Euclidean geometries from a coordinate-free viewpoint.
Study of pulleys and gears in spherical and hyperbolic geometries.
problem Understanding mechanical systems in non-Euclidean spaces.
method Analysis of pulley and gear systems in spherical and hyperbolic geometries.
result Similar laws governing movement in non-Euclidean geometries.
We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulations of three-dimensional hyperbolic space are available at http://h3.hypernom.com.
This paper deals with various topics in analysis on hyperbolic spaces. It surveys some recent progress in non-Euclidean Fourier Analysis and proves some new results for the geodesic Radon transform on hyperbolic spaces.
The study extends inscription problems to non-Euclidean geometries.
problem Generalizing inscription problems to non-Euclidean geometries.
method Symplectic and Riemannian geometry techniques.
result Proved generalized inscription theorems for hyperbolic and spherical surfaces.
Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.
problem Approximating local extrinsic curvature on discrete manifolds.
method Constructing discrete curvature forms on piecewise flat manifolds, using weighted sums of hinge angles.
result Converges to smooth curvature values as mesh refinement occurs, favorably comparing with other discrete approaches.
This paper tightens the generalization error bound for graph embedding in non-Euclidean spaces.
problem High generalization error in non-Euclidean graph embedding, preventing practical applications.
method Novel upper bound of graph embedding's generalization error using local Rademacher complexity.
result The new bound is tighter and faster, allowing better performance in non-Euclidean spaces.
The paper proposes scalable methods for selecting prototypes from large dissimilarity datasets.
problem Selecting good prototypes from large dissimilarity datasets.
method Genetic algorithms, dissimilarity-based hashing, unsupervised and supervised criteria.
result The methods select good prototypes efficiently from large datasets.
The author suggests using non-Euclidean geometry for psychometric models.
problem Current psychometric models lack geometric insights.
method Illustrates how non-Euclidean geometry can be applied to psychometrics.
result Geometric concepts may improve psychometric model understanding.
Study classifies graphs in Euclidean and non-Euclidean spaces with specific curvature conditions.
problem Classifying graphs with prescribed curvature in various spaces.
method Proves rigidity and classification results for graphs in Riemannian manifolds, focusing on R2 and R3. result Provides general splitting theorems for graphs in these settings.
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…
We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulation of the product of two-dimensional hyperbolic space with one-dimensional euclidean space is available at http://h2xe.hypernom.com.
New method turns optimization algorithms into uniformly stable learning algorithms for non-Euclidean norms.
problem Non-Euclidean norms in binary classification problems.
method Black-box reduction method using uniformly convex regularizers.
result Achieves optimal statistical risk bounds on excess risk for non-Euclidean norms.
This paper considers networks where relationships between nodes are represented by directed dissimilarities. The goal is to study methods that, based on the dissimilarity structure, output hierarchical clusters, i.e., a family of nested partitions indexed by a connectivity parameter. Our construction of hierarchical cl…
Gradient descent near stability threshold shows sharpness oscillations.
problem Understanding sharpness and stability in non-Euclidean norms during gradient descent.
method Interpreted EoS through Directional Smoothness, defined generalized sharpness for arbitrary norms.
result Non-Euclidean GD exhibits sharpness oscillations around the stability threshold.
New optimization method combines gradient clipping and non-Euclidean smoothness.
problem Improving optimization in non-Euclidean spaces for machine learning.
method Hybrid of steepest descent and conditional gradient, incorporating weight decay.
result Achieves optimal convergence rate and demonstrates effectiveness in deep learning.
Improved k-medoids clustering with faster algorithms for big data.
problem High runtime cost of k-medoids clustering algorithms.
method Modifications to PAM, CLARA, and CLARANS algorithms to achieve O(k) runtime improvement.
result 458x and 1191x speedup in runtime for k=100,200 respectively.
DID measures similarity invariant to diffeomorphisms.
problem Measuring similarity invariance to diffeomorphisms.
method DID measures similarity as the solution to an optimization problem in a Reproducing Kernel Hilbert Space.
result DID is invariant to diffeomorphisms and can be efficiently approximated.
Gradient descent near stability threshold exhibits sharpness oscillations.
problem Understanding sharpness behavior near stability threshold in non-Euclidean norms.
method Interpreted EoS through Directional Smoothness and generalized sharpness under arbitrary norms.
result Non-Euclidean GD with generalized sharpness shows sharpness oscillations near 2/η. Paper introduces a new method for learning with distributions using dissimilarity measures.
problem Learning with probability distributions using dissimilarity measures.
method Introduces embeddings based on dissimilarity of distributions to templates, extending similarity theory to population distributions.
result Proves that dissimilarity theory holds for empirical distributions and shows better performance of Wasserstein distance embedding.
Faster k-Medoids Clustering improves PAM, CLARA, CLARANS algorithms.
problem Efficiently clustering non-Euclidean data with high k.
method Modifications to PAM, CLARA, CLARANS algorithms to achieve O(k)-fold speedup.
result 200-fold speedup observed on real data with k=100.
New dissimilarity measures enhance affinity propagation for complex network clustering.
problem Improving community detection in complex networks using affinity propagation.
method Leverage network latent geometry to design dissimilarity matrices.
result Affinity propagation outperforms state-of-the-art methods in community detection.
Improves Gower's similarity for mixed-type variables with automatic weighting.
problem Handling missing values and unbalanced variable contributions in Gower's similarity for mixed-type data.
method Automatic weighting scheme minimizing differences in correlation between contributing dissimilarities and weighted Gower's dissimilarity.
result Improved performance in classification and imputation of missing values.
A method for classification using pairwise similarities and unlabeled data.
problem Handling pairwise similarities and unlabeled data for classification.
method Empirical risk minimization approach to create an unbiased risk estimator.
result Derives an unbiased risk estimator for handling both similarities and unlabeled data.
FPI methods compute barycenters of Gaussian sets for various dissimilarity measures.
problem Efficiently compute barycenters of Gaussian sets for multiple dissimilarity measures.
method Fixed-Point Iterations (FPI) for several dissimilarity measures.
result FPI provides a useful toolbox for fusion/reduction of Gaussian sets.