arXiv research

We extend the theoretical analysis of a recently proposed single subspace learning algorithm, called Dual Principal Component Pursuit (DPCP), to the case where the data are drawn from of a union of hyperplanes. To gain insight into the properties of the $\ell_1$ non-convex problem associated with DPCP, we develop a geo…

Paper proposes a method to solve log-optimal portfolios under ambiguous return distributions.

Efficiently clusters large datasets using low-density hyperplanes.

New findings on great circle fibrations and contact structures on odd spheres.

Unique hyperplanes are the only translating solitons asymptotic to half-hyperplanes.

A new reinforcement learning method improves Max-Cut solutions without needing training data.

Survey on hyperplane arrangements and their topology.

This paper studies ruled real hypersurfaces in indefinite complex projective space.

A new method combines simple binary classifiers to build complex multiclass classifiers, achieving performance limits in a Gaussian setting.

Segre varieties' hyperplane sections are unstable under certain conditions.

Horospheres, hyperspheres, and hyperplanes in hyperbolic spaces 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$\{n+1,2r\}$.

To construct flexible nonlinear predictive distributions, the paper introduces a family of softplus function based regression models that convolve, stack, or combine both operations by convolving countably infinite stacked gamma distributions, whose scales depend on the covariates. Generalizing logistic regression that…

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 theorem divides hyperplanes evenly with a line through the origin.

We show some characterizations of hyperspheres in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$ 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…

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.

Two SVM frameworks are introduced for classification problems.

We examine the existence of tangent hyperplanes to subriemannian balls. Strictly abnormal shortest paths are allowed

PET-TURTLE improves clustering accuracy for imbalanced data.

In this paper we show that an immersed nontrivial translating soliton for mean curvature flow in $\mathbb{R}^{n+1}$($n=2,3)$ is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most quadratic growth. For an embedded translating soliton $Σ$ with nonnegative scalar curva…

Study of first homology group of Milnor fiber boundary for generic hyperplane arrangements in C^3.

Study hyperplanes in abelian groups and their signatures for 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…

We show that certain aspherical manifolds arising from hyperplane arrangements in negatively curved manifolds have relatively hyperbolic fundamental group.

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 …

In hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a 4-dimensional real symplectic vector space model Einstein hyperplanes and the inv…

The paper proves a margin inequality for separating hyperplanes, useful for analyzing algorithmic bias.

In this article, we completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.

Associating distinct groups of objects (clusters) with contiguous regions of high probability density (high-density clusters), is central to many statistical and machine learning approaches to the classification of unlabelled data. We propose a novel hyperplane classifier for clustering and semi-supervised classificati…

The paper constructs random concave functions on the unit simplex.

Study Coxeter groups over fusion rings and their geometric realisations.

Upper bounds on fixed points in PWL neural networks with hyperplane analysis.

Hyperplane hashing aims at rapidly searching nearest points to a hyperplane, and has shown practical impact in scaling up active learning with SVMs. Unfortunately, the existing randomized methods need long hash codes to achieve reasonable search accuracy and thus suffer from reduced search speed and large memory overhe…

We use Morse theory to prove that the Lefschetz Hyperplane Theorem holds for compact smooth Deligne-Mumford stacks over the site of complex manifolds. For $Z \subset X$ a hyperplane section, $X$ can be obtained from $Z$ by a sequence of deformation retracts and attachments of high-dimensional finite disc quotients. We …

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…

Paper proves a Cohen-Dimca-Orlik type theorem for Z-local systems of hyperplane arrangements.

Proves a limit on hyperplanes in complex manifolds.

Bordifications of hyperplane arrangements yield complexes with homotopy type of wedges of spheres.

We study torsion properties of the twisted Alexander modules of the affine complement $M$ of a complex essential hyperplane arrangement, as well as those of punctured stratified tubular neighborhoods of complex essential hyperplane arrangements. We investigate divisibility properties between the twisted Alexander polyn…

The complement of an arrangement A of a finite number of affine hyperplanes in complex n-space has the structure of a poset of spaces indexed by the intersection poset, L(A). The space corresponding to G in L(A) is homotopy equivalent to the complement of the hyperplanes in the central arrangement A_G normal to G. This…

New geometric mechanism solves four envelope problems.

New method constructs asymptotic convex hypersurfaces via equidistant hyperplanes.

We compute the l^2-Betti numbers of the complement of a finite collection of affine hyperplanes in complex space. At most one of the l^2-Betti numbers is non-zero.

Study proves minimality of certain hyperplane intersections in wide cones.

Proves non-positivity of Hirzebruch form on stable weights and connects to flat logarithmic connections.

The paper examines stable CMC hypersurfaces with boundaries on parallel hyperplanes.