BN refines local partition geometry in piecewise-affine networks during training.
problem Understanding the effect of BN on the function realized during training in piecewise-affine networks.
method Analyzing the geometry of switching hyperplanes and affine-region partition conditioned on a mini-batch.
result BN increases expected local partition refinement in ReLU and piecewise-affine networks.
We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.
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.
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.
SPLK reduces Kriging complexity for large spatial datasets with exogenous variables.
problem Efficiently modeling large-scale spatial systems with exogenous variables.
method Sparse Pseudo-input Local Kriging (SPLK) using hyperplanes for domain partitioning and sparse approximation.
result SPLK outperforms or matches existing methods for spatial datasets.
Oblique BART improves tree-based predictions.
problem Axis-aligned decision rules in BART can be suboptimal.
method Developed an oblique version of BART using data-adaptive hyperplane partitions.
result Oblique BART outperformed axis-aligned BART and other tree methods on benchmarks.
Proof of K(π,1) conjecture for affine Artin groups.
problem Asphericality of complements of affine hyperplane arrangements.
method Combinatorics of noncrossing partition posets, dual Artin groups, and topological models.
result Affine Artin groups are aspherical.
Locality-sensitive hashing converts high-dimensional feature vectors, such as image and speech, into bit arrays and allows high-speed similarity calculation with the Hamming distance. There is a hashing scheme that maps feature vectors to bit arrays depending on the signs of the inner products between feature vectors a…
For a given lattice, we establish an equivalence involving a closed zone of the corresponding Voronoi polytope, a lamina hyperplane of the corresponding Delaunay partition and a quadratic form of rank 1 being an extreme ray of the corresponding L-type domain.
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.
This paper solves a variation of the isoperimetric problem in higher dimensions.
problem Minimizing a weighted perimeter functional with given half-space volumes.
method Introduced a weighted perimeter functional with three weights.
result Characterized two types of minimizers made of spherical domes.
A new neural network model for optimal treatment assignment.
problem Learning optimal treatment assignment from observational data.
method Prescriptive ReLU network (P-ReLU) model that balances performance and interpretability.
result P-ReLU can be converted into an equivalent prescriptive tree with hyperplane splits for interpretability.
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.
The BSP-Forest extends 2D BSP-Tree to dD for regression tasks.
problem Extending BSP-Tree to higher dimensions for regression.
method Generate cutting hyperplanes parallel to d-2 dimensions, sample two free dimensions.
result BSP-Forest achieves similar performance with fewer cuts and outperforms other regression forests.
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.
The complement of a complex hyperplane arrangement is known to be homotopic to a minimal CW complex. There are several approaches to the minimality. In this paper, we restrict our attention to real two dimensional cases, and introduce the "dual" objects so called minimal stratifications. The strata are explicitly descr…
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.
Unified approach to verify NN properties using ReLU's unique polytope structure.
problem Lack of robustness and interpretability in ReLU NNs for risk-sensitive applications.
method Identifying and traversing the local polytopes of ReLU NNs, developing an algorithm to verify properties.
result Unified approach to examine network behavior in risk-sensitive settings.
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.
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.
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…
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.
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…
A new framework for verifying robustness of neural networks.
problem Verifying the robustness of neural networks against adversarial attacks.
method LayerCert framework exploiting the nested hyperplane arrangement structure of ReLU networks.
result LayerCert reduces the number and size of convex programs needed for robustness verification.
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.
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.
Optimizes projections for binary data separation using spectral connectivity.
problem Maximizing separability of binary partitions in unlabelled datasets.
method Minimizes the second eigenvalue of the graph Laplacian, using spectral connectivity.
result Optimal projections converge to the maximum margin hyperplane as scaling parameter approaches zero.
We examine the existence of tangent hyperplanes to subriemannian balls. Strictly abnormal shortest paths are allowed
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.
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.
The study proves hyperplanes are the only torically maximal hypersurfaces.
problem Characterizing torically maximal hypersurfaces.
method Analyzing the logarithmic Gauss map and properties of projective spaces.
result Hyperplanes are the only torically maximal hypersurfaces.
The paper proves a margin inequality for separating hyperplanes, useful for analyzing algorithmic bias.
problem Analyzing the implicit bias of algorithms in machine learning.
method Proves a nonsmooth Kurdyka-Lojasiewicz inequality for margin function.
result The bias of algorithm iterates converges at least as fast as the square-root of the margin convergence rate.
Study Coxeter groups over fusion rings and their geometric realisations.
problem Understanding Coxeter groups and their embeddings.
method Investigate faithful realisations and Vinberg systems.
result Induce embeddings of hyperplane complements.
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.
Upper bounds on fixed points in PWL neural networks with hyperplane analysis.
problem Analyzing the number of fixed points in neural networks with PWL activation.
method Hyperplane arrangements to bound the number of fixed points.
result Upper bounds on the number of fixed points for PWL networks, showing exponential growth in layers.