Classifies pairs of hyperplanes in Einstein universe, contributing to crooked surfaces classification.
problem Classifying pairs of hyperplanes in the Einstein universe.
method Algebraic invariant and symplectic splittings model Einstein hyperplanes.
result Contributes to a complete disjointness criterion for crooked surfaces.
A novel hyperplane classifier minimizes probability density.
problem Clustering and semi-supervised classification of unlabelled data.
method Minimum density hyperplane that minimizes the integral of empirical probability density.
result The minimum density hyperplane is asymptotically equivalent to maximum margin hyperplane.
Simple construction for classifying spaces of projections of immersions.
problem Classifying spaces for projections of immersions with controlled singularities.
method Explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions.
result Structure theorems for these classifying spaces.
Researchers classify K-stability of log Fano hyperplane arrangements.
problem Determining K-stability of log Fano hyperplane arrangements.
method Comprehensive analysis of K-stability conditions.
result Classification of K-stability for log Fano hyperplane arrangements.
A new method combines simple binary classifiers to build complex multiclass classifiers, achieving performance limits in a Gaussian setting.
problem Building a sophisticated multiclass classifier from simple binary decisions.
method Combining O(logK) simple binary classifiers to form a K-class classifier. result Explicit performance bounds across various decoding and dimensional regimes for a stylized Gaussian setting.
Paper presents a new method for multiclass classification using hyperplane arrangements.
problem Developing efficient multiclass classifiers.
method Mixed integer programming formulations with hyperplane arrangements, kernel trick adaptation, and dimensionality reductions.
result Our proposal outperforms other methods in multiclass classification tasks.
Study on recovering sparse linear classifiers from mixed binary responses.
problem Learning a mixture of sparse linear classifiers from binary responses.
method Query-based approach to identify all sparse vectors from a set.
result Upper bounds on the number of queries required for recovery.
New NHCAs improve multi-category classification efficiency.
problem Efficient multi-category classification for real-world problems.
method Twin SVM (TWSVM), Generalized eigenvalue proximal SVM (GEPSVM), Regularized GEPSVM (RegGEPSVM), and Improved GEPSVM (IGEPSVM) with OAA, BT, and TDS approaches.
result TDS-TWSVM outperforms other methods in classification accuracy.
We give a general treatment of the somewhat unfamiliar operation on manifolds called Connected Sum at Infinity, or CSI for short. A driving ambition has been to make the geometry behind the well definition and basic properties of CSI as clear and elementary as possible. CSI then yields a very natural and elementary pro…
Unsupervised classifier performs as well as supervised ones on ImageNet dataset.
problem Achieving performance of supervised learning classifiers without labeled data.
method Incremental shift and rotation operations on selected hyperplanes.
result 6.2% Top 3 probability of error on ImageNet dataset.
Tropical SVM tackles phylogenomics by classifying multi-locus data.
problem Classifying multi-locus data sets for phylogenetic analysis.
method Proposes tropical support vector machines (SVMs) for phylogenomics, formulated as linear programming problems.
result Developed methods for hard and soft margin tropical SVMs, proving necessary and sufficient conditions for separation.
Linear classifiers separate the data with a hyperplane. In this paper we focus on the novel method of construction of multithreshold linear classifier, which separates the data with multiple parallel hyperplanes. Proposed model is based on the information theory concepts -- namely Renyi's quadratic entropy and Cauchy-S…
Extends cobordism groups of immersions to projections with new results.
problem Understanding cobordism groups of projected immersions.
method Classifying space construction by Szűcs and Terpai, Salomonsen's exact sequence.
result Obtains results on cobordism groups in small and large codimensional cases.
Study of CR twistor model Q2,2 and its sections.
problem Classify and describe projective lines and hyperplane sections of the CR twistor model.
method Explicit projective methods, classification of lines and sections, use of involution j. result Complete relative classification of smooth quadric sections and explicit non-spherical CR structures.
Modified SVM improves classification accuracy for imbalanced classes.
problem Improper SVM handling of class variances leads to misclassification.
method Adjust SVM margins to reflect class variances, proportional to standard deviation.
result Improved predictive performance for imbalanced classes.
Improved upper bound for mass partitioning problem using Gray codes.
problem Partitioning masses on a high-dimensional space with hyperplanes.
method Classifying Gray code patterns and utilizing group actions.
result An improved bound for the Grünbaum-Hadwiger-Ramos problem.
Unique hyperplanes are the only translating solitons asymptotic to half-hyperplanes.
problem Characterizing translating solitons asymptotic to half-hyperplanes.
method Analyzing the geometry of translating solitons in Rn+1. result Hyperplanes are the only examples of translating solitons asymptotic to two half-hyperplanes.
Softplus regressions use multiple hyperplanes to classify data.
problem Classifying data with flexible nonlinear decision boundaries.
method Softplus function based regression models convolving gamma distributions.
result Softplus regressions achieve comparable classification accuracy to SVM but with less computation.
A new algorithm finds a separating hyperplane with fewer updates.
problem Finding a separating hyperplane with minimal updates.
method Optimistic Perceptron algorithm.
result The Optimistic Perceptron finds a separating hyperplane with no more than $ rac{1}{γ}$ updates.
The study investigates tangent hyperplanes to subriemannian balls, including abnormal paths.
problem Existence of tangent hyperplanes to subriemannian balls.
method Examines strict abnormal shortest paths.
result Includes abnormal paths in the analysis of tangent hyperplanes.
Paper proposes a new classifier for hyperbolic spaces using horospherical boundaries.
problem Optimization of large margin classifiers in hyperbolic spaces.
method Horospherical decision boundaries for geodesically convex optimization.
result Geodesically convex optimization leads to globally optimal solutions.
New algorithm clusters hyperplanes with improved accuracy.
problem Clustering data from a union of hyperplanes.
method Dual Principal Component Pursuit (DPCP) with geometric analysis.
result DPCP can uniquely identify the dominant hyperplane under certain conditions.
Survey on hyperplane arrangements and their topology.
problem Topology of hyperplane arrangements.
method Focus on the relationship between topology and real structure.
result Relationship between topology and real structure of hyperplane arrangements.
Segre varieties' hyperplane sections are unstable under certain conditions.
problem Stability of hyperplane sections of Segre varieties under different conditions.
method Proving instability with respect to any polarization for non-smooth or meqn cases. result Normal hyperplane sections of Segre varieties are K-unstable under specified conditions.
The study classifies minimal helix submanifolds and Riemannian foliations.
problem Classifying minimal helix submanifolds and foliations in Euclidean space.
method Analyzing ruled minimal helix submanifolds and using Corollary 1.3 to classify Riemannian foliations.
result Minimal helix submanifolds are either cylinders or extrinsic products with a complex line.
Research examines arrangements of hyperplanes in real projective spaces, focusing on specific cases.
problem Analyzing the structure of hyperplane arrangements in real projective spaces.
method Investigates arrangements of m hyperplanes in the n-dimensional real projective space, with a focus on m=n+3 and n=3 or n=4. result Provides insights into the structure of chambers cut out by these specific hyperplane arrangements.
The paper characterizes grim hyperplanes for translating solitons in mean curvature flow.
problem Characterizing grim hyperplanes for translating solitons in mean curvature flow.
method Analyzing translating solitons with nonnegative scalar curvature and mean curvature that do not change signs on each end.
result An embedded translating soliton is either a hyperplane or a grim hyperplane if it has nonnegative scalar curvature and mean curvature that do not change signs on each end.
Horospheres, hyperspheres, and hyperplanes in hyperbolic spaces are rigid in terms of mean curvature.
problem Proving rigidity of geometric shapes in hyperbolic spaces.
method Analyzing perturbations of horospheres, hyperspheres, and hyperplanes to show they cannot increase their mean curvature.
result Horospheres, hyperspheres, and hyperplanes in hyperbolic spaces H n are rigid in terms of mean curvature.
Many examples of nonpositively curved closed manifolds arise as blow-ups of projective hyperplane arrangements. If the hyperplane arrangement is associated to a finite reflection group W, and the blow-up locus is W-invariant, then the resulting manifold M will admit a cell decomposition whose maximal cells are all comb…
The Lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to describe attaching maps of these cells for the complement of a complex hyperplane arrangement defined over real …
We prove that the topological complexity of (a motion planning algorithm on) the complement of generic complex essential hyperplane arrangement of n hyperplanes in an r-dimensional linear space is min{n+1,2r}.
We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincare polynomial, and Tutte polynomial. We consider basic algebraic properties of such chain complexes, including long-exact sequences associated to deletio…
Study of Alexander modules for hyperplane arrangements, distinguishing complements.
problem Distinguishing homotopy equivalent but non-homeomorphic hyperplane arrangement complements.
method Analysis of twisted Alexander modules and polynomials.
result Distinguish non-homeomorphic homotopy equivalent arrangement complements.
A theorem divides hyperplanes evenly with a line through the origin.
problem Dividing hyperplanes evenly with a line.
method Direct proof using measures on hyperplanes.
result A line through the origin divides hyperplanes evenly.
We show some characterizations of hyperspheres in the (n+1)-dimensional Euclidean space En+1 with intrinsic and extrinsic properties such as the n-dimensional area of the sections cut off by hyperplanes, the (n+1)-dimensional volume of regions between parallel hyperplanes, and the n-dimensional surf…
Two SVM frameworks are introduced for classification problems.
problem Classifying data with nonparallel hyperplanes.
method Two SVM frameworks: separate and simultaneous construction of hyperplanes.
result Max-min distance-based NSVM for multiclass classification.
We compute the cohomology with group ring coefficients of the complement of a finite collection of affine hyperplanes in a finite dimensional complex vector space. It is nonzero in exactly one degree, namely the degree equal to the rank of the hyperplane arrangement.
Study on recovering supports of multiple sparse vectors from mixed linear measurements.
problem Recovering supports of multiple sparse vectors from a mixture of linear measurements.
method Developed algorithms to identify the support of all component vectors using polynomial and quasi-polynomial number of measurements.
result Polynomial and quasi-polynomial number of measurements sufficient for recovering the supports of all component vectors.
The paper studies the envelope of mid-hyperplanes of a hypersurface and its properties.
problem Understanding the properties of the envelope of mid-hyperplanes of a hypersurface.
method Analyzes mid-hyperplanes of a smooth hypersurface and their envelopes, proving properties and conditions.
result The envelope of mid-hyperplanes consists of centers of conics with contact of order at least 3 with the hypersurface.
Paper provides first theoretical guarantees for hyperbolic space learning.
problem Learning a classifier in hyperbolic space for hierarchical data.
method Efficient algorithm for large-margin hyperplane learning in hyperbolic space.
result The low embedding dimension in hyperbolic space leads to superior classifier learning guarantees.
The paper classifies isoparametric hypersurfaces in conic Finsler spaces.
problem Identifying new isoparametric hypersurfaces in conic Finsler spaces.
method Introduced isoparametric functions and hypersurfaces in conic Finsler spaces, classified them in specific spaces.
result Found additional isoparametric hypersurfaces in conic Minkowski spaces, such as helicoids.
Study of first homology group of Milnor fiber boundary for generic hyperplane arrangements in C^3.
problem Computing the first homology group of the Milnor fiber boundary for generic hyperplane arrangements.
method Analyzing the Milnor fiber boundary for hyperplane arrangements in C^3.
result Affirmative answer to the conjecture of Suciu and example of arrangements with non-trivial torsion.
Study hyperplanes in abelian groups and their signatures for manifold identification.
problem Identifying manifolds based on their homology groups and coordinate hyperplanes.
method Investigates isomorphisms preserving coordinate hyperplanes in products of cyclic groups.
result Recovering coordinate hyperplanes from their union and applying to manifold identification.
Efficiently clusters large datasets using low-density hyperplanes.
problem Clustering large datasets efficiently.
method Incremental estimation of low-density hyperplanes using stochastic gradient descent.
result The method automatically selects an appropriate number of clusters.
We consider a twisted version of the Hurewicz map on the complement of a hyperplane arrangement. The purpose of this paper is to prove surjectivity of the twisted Hurewicz map under some genericity conditions. As a corollary, we also prove that a generic section of the complement of a hyperplane arrangement has non-tri…
New study finds non-trivial integer torsion in Milnor fibers of hyperplane arrangements.
problem Torsion in homology of Milnor fibers of hyperplane arrangements.
method Techniques from recent paper [arXiv:1209.3414]
result Homology groups of Milnor fibers can have non-trivial integer torsion.
The Sample Compression Conjecture of Littlestone & Warmuth has remained unsolved for over two decades. This paper presents a systematic geometric investigation of the compression of finite maximum concept classes. Simple arrangements of hyperplanes in Hyperbolic space, and Piecewise-Linear hyperplane arrangements, are …
We show that certain aspherical manifolds arising from hyperplane arrangements in negatively curved manifolds have relatively hyperbolic fundamental group.