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.
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.
We study large-scale spatial systems that contain exogenous variables, e.g. environmental factors that are significant predictors in spatial processes. Building predictive models for such processes is challenging because the large numbers of observations present makes it inefficient to apply full Kriging. In order to r…
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…
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.
The Binary Space Partitioning~(BSP)-Tree process is proposed to produce flexible 2-D partition structures which are originally used as a Bayesian nonparametric prior for relational modelling. It can hardly be applied to other learning tasks such as regression trees because extending the BSP-Tree process to a higher dim…
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 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…
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 ℓ1 non-convex problem associated with DPCP, we develop a geo…
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 study the problem of determining the optimal low dimensional projection for maximising the separability of a binary partition of an unlabelled dataset, as measured by spectral graph theory. This is achieved by finding projections which minimise the second eigenvalue of the graph Laplacian of the projected data, whic…
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}.
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.
We prove that the hyperplanes parallel to en+1 are the unique examples of translating solitons C1−asymptotic to two half-hyperplanes outside a vertical cylinder in Rn+1.
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.
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 deep neural network (DNN) with piecewise linear activations can partition the input space into numerous small linear regions, where different linear functions are fitted. It is believed that the number of these regions represents the expressivity of the DNN. This paper provides a novel and meticulous perspective to l…
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.
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.
We examine the existence of tangent hyperplanes to subriemannian balls. Strictly abnormal shortest paths are allowed
In this paper we show that an immersed nontrivial translating soliton for mean curvature flow in Rn+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.
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.
Considering the classification problem, we summarize the nonparallel support vector machines with the nonparallel hyperplanes to two types of frameworks. The first type constructs the hyperplanes separately. It solves a series of small optimization problems to obtain a series of hyperplanes, but is hard to measure the …
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…
In this article, we completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.
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.
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…
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.
We prove that horospheres, hyperspheres and hyperplanes in a hyperbolic space H n , n ≥ 3, admit no perturbations with compact support which increase their mean curvature. is is an extension of the analogous result in the Euclidean spaces, due to M. Gromov, which states that a hyperplane in a Euclidean space R n ad…
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.
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⊂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 …
Paper proves a Cohen-Dimca-Orlik type theorem for Z-local systems of hyperplane arrangements.
problem Proving a Cohen-Dimca-Orlik type theorem for Z-local systems. method Analyzing local system cohomology groups of hyperplane arrangements complements.
result Proves a Cohen-Dimca-Orlik type theorem for Z-local systems. Proves a limit on hyperplanes in complex manifolds.
problem Limiting the number of hyperplanes in complex manifolds.
method Effective density theorem for periodic orbits, Margulis functions, restricted projection theorem, equidistribution result.
result Proves a quantitative finiteness theorem for hyperplanes.
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…