Number Of Hyperplanes Research Articles

The resultant method in higher dimensions

Stability results play an important role in Galois geometries. The famous resultant method, developed by Szőnyi and Weiner [12], [11], became very fruitful and resulted in many stability theorems in the last two decades. This method is based on some bivariate polynomials associated to point sets. In this paper we generalize the method for the multidimensional case and show some new applications. We build up the multivariate polynomial machinery and apply it for Rédei polynomials. We can prove a high dimensional analogue of the result of Szőnyi-Weiner [9], concerning the number of hyperplanes being skew to a point set of the space. We prove general results on “partial blocking sets”, using the tools we have developed.

Random concave functions

Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave functions on the unit simplex measure the concentration of capital, and their gradient maps define novel investment strategies. The gradient maps may also be regarded as optimal transport maps on the simplex. In this paper we construct and study probability measures supported on spaces of concave functions. These measures may serve as prior distributions in Bayesian statistics and Cover’s universal portfolio, and induce distribution-valued random variables via optimal transport. The random concave functions are constructed on the unit simplex by taking a suitably scaled (mollified, or soft) minimum of random hyperplanes. Depending on the regime of the parameters, we show that as the number of hyperplanes tends to infinity there are several possible limiting behaviors. In particular, there is a transition from a deterministic almost sure limit to a nontrivial limiting distribution that can be characterized using convex duality and Poisson point processes.

Non-Gaussian hyperplane tessellations and robust one-bit compressed sensing

We show that a tessellation generated by a small number of random affine hyperplanes can be used to approximate Euclidean distances between any two points in an arbitrary bounded set T , where the random hyperplanes are generated by subgaussian or heavy-tailed normal vectors and uniformly distributed shifts. The number of hyperplanes needed for constructing such tessellations is determined by natural metric complexity measures of the set T and the wanted approximation error. In comparison, previous results in this direction were restricted to Gaussian hyperplane tessellations of subsets of the Euclidean unit sphere. As an application, we obtain new reconstruction results in memoryless one-bit compressed sensing with non-Gaussian measurement matrices: by quantizing at uniformly distributed thresholds, it is possible to accurately reconstruct low-complexity signals from a small number of one-bit quantized measurements, even if the measurement vectors are drawn from a heavy-tailed distribution. These reconstruction results are uniform in nature and robust in the presence of pre-quantization noise on the analog measurements as well as adversarial bit corruptions in the quantization process. Moreover, if the measurement matrix is subgaussian then accurate recovery can be achieved via a convex program.

Convex Hulls and the size and the Size of the Hidden Layer in a MLP Based Classifier

Designing a feedforward neural network has always posed many questions regarding the number of hidden layers and the number of neurons on each hidden layer. While there is no unique and global solution, it is known that the particularities of the problem to solve, especially for classification, offer some guidance about how to solve these questions. One heuristic approach, when only a hidden layer is involved, analyzes how the involved classes are separated from each other using a finite number of hyperplanes, thereby defining the size of the hidden layer on the network. On this article, using computational geometry concepts, an automated and time efficient method is presented and discussed for estimating the quantity of neurons in the hidden layer by computing the number of hyperplanes separating the classes, based on convex hulls and approximation to alpha shapes. Examples on different situations that may arise and the results on using it are illustrated. It can be seen from the results that the proposed method gives very good estimation for the number of hidden neurons.

Hardy inequalities for the Heisenberg Laplacian on convex bounded polytopes

We prove a Hardy-type inequality for the gradient of the Heisenberg Laplacian on open bounded convex polytopes on the first Heisenberg group. The integral weight of the Hardy inequality is given by the distance function to the boundary measured with respect to the Carnot-Carathéodory metric. The constant depends on the number of hyperplanes, given by the boundary of the convex polytope, which are not orthogonal to the hyperplane $x_3=0$.

Darboux theory of integrability for polynomial vector fields on 𝕊n

ABSTRACTThis is a survey on the Darboux theory of integrability for polynomial vector fields, first in and second in the n-dimensional sphere . We also provide new results about the maximum number of invariant parallels and meridians that a polynomial vector field on can have in function of its degree. These results in some sense extend the known result on the maximum number of hyperplanes that a polynomial vector field in can have in function of the degree of .

Integral points off hyperplanes in positive characteristic

We give a natural sufficient condition which ensures the finiteness of the number of integral points on a projective space omitting finitely many hyperplanes defined over a one-dimensional function field of positive characteristic. In this setting, when the number of hyperplanes is 2n+2, where n is the dimension of the ambient projective space, we use this natural condition to explain and sharpen an earlier result of the second named author, which provides a concrete condition on the coefficients of the linear forms producing hyperplanes such that the above finiteness property is satisfied.

An optimal generalization of the Colorful Carathéodory theorem

The Colorful Carathéodory theorem by Bárány (1982) states that given d+1 sets of points in Rd, the convex hull of each containing the origin, there exists a simplex (called a ‘rainbow simplex’) with at most one point from each point set, which also contains the origin. Equivalently, either there is a hyperplane separating one of these d+1 sets of points from the origin, or there exists a rainbow simplex containing the origin. One of our results is the following extension of the Colorful Carathéodory theorem: given ⌊d/2⌋+1 sets of points in Rd and a convex object C, then either one set can be separated from C by a constant (depending only on d) number of hyperplanes, or there is a ⌊d/2⌋-dimensional rainbow simplex intersecting C.

Reliability sensitivity analysis based on multi-hyperplane combination method

Abstract For a nonlinear limit state function, the first-order reliability method (FORM) may cause large errors in the computation of not only the reliability index or failure probability but also the reliability sensitivity. In order to obtain more the accurate results of the reliability sensitivity analysis, a number of hyperplanes are built near the design point by first-order Tayler series expansion, which replace the known nonlinear limit state hypersurface, and an equivalent computational method is utilized to construct an equivalent hyperplane of the obtained hyperplanes. And the reliability sensitivities can be estimated more accurately by the derived equations based on the equivalent hyperplane. An example shows that the method is effective and feasible.

Star configurations in [formula omitted

Star configurations are certain unions of linear subspaces of projective space. They have appeared in several different contexts: the study of extremal Hilbert functions for fat point schemes in the plane; the study of secant varieties of some classical algebraic varieties; the study of the resurgence of projective schemes. In this paper we study some algebraic properties of the ideals defining star configurations, including getting partial results about Hilbert functions, generators and minimal free resolutions of the ideals and their symbolic powers. We also show that their symbolic powers define arithmetically Cohen–Macaulay subschemes and we obtain results about the primary decompositions of the powers of the ideals. As an application, we compute the resurgence for the ideal of the codimension n−1 star configuration in Pn in the monomial case (i.e., when the number of hyperplanes is n+1).

Separation in totally-sewn 4-polytopes with the decreasing universal edge property

The separation number s(O, P) of a d-polytope with respect to a point O in the interior of P is the minimum number of hyperplanes necessary to strictly separate O from any facet of P. According to the Separation Conjecture, s(O, P) ≤ 2 d for d-dimensional convex polytopes in \({\mathbb{R}^d}\) . We verify the Conjecture for totally-sewn 4-polytopes with the property that the number of universal edges does not increase as the polytopes are constructed.

Punctured combinatorial Nullstellensätze

In this article we extend Alon’s Nullstellensatz to functions which have multiple zeros at the common zeros of some polynomials g 1,g 2, …, g n , that are the product of linear factors. We then prove a punctured version which states, for simple zeros, that if f vanishes at nearly all, but not all, of the common zeros of g 1(X 1), …,g n (X n ) then every residue of f modulo the ideal generated by g 1, …, g n , has a large degree. This punctured Nullstellensatz is used to prove a blocking theorem for projective and affine geometries over an arbitrary field. This theorem has as corollaries a theorem of Alon and Furedi which gives a lower bound on the number of hyperplanes needed to cover all but one of the points of a hypercube and theorems of Bruen, Jamison and Brouwer and Schrijver which provides lower bounds on the number of points needed to block the hyperplanes of an affine space over a finite field.

Survey and Review

In this issue, we feature two very different articles. The first, “Estimation of Subspace Arrangements with Applications in Modeling and Segmenting Mixed Data” by Yi Ma, Allen Yang, Harm Derksen, and Robert Fossum, is an excellent example of interdisciplinary mathematical research. The work shows how sophisticated mathematical ideas can be used to address a problem which arises in many applications. Moreover, it points to new mathematical research directions. The basic problem the article addresses is a very important problem in analysis of high dimensional data. Suppose you have a cloud of points in a large dimensional space; the points could represent consumer preferences, for example. The question is to find patterns in the data by identifying a set of hyperplanes which “fit” the data. The challenge is that the number of hyperplanes and their dimensions are to be learned from the data. Thus the task involves “grouping the points,” i.e., segmentation, and determining the best approximation for that group of points, i.e., model selection. What makes this hard is that these two steps are tightly coupled. In their paper, the authors describe a novel method, which they call GPCA (generalized principal component analysis), that solves this problem. The method is a remarkable result based on ideas from statistics, and abstract and computational algebra. Moreover, the approach is justified using statistical arguments. Examples from computer vision and image compression are given. The subject of the paper is a very “hot” area of research. While there are many other approaches to this problem, this self-contained introduction to GPCA and its mathematical foundation is a must-read. The second article, “Singularly Perturbed Linear Two-Point Boundary Value Problems” by Robert O'Malley, Jr., is about a classical problem in the study of differential equations involving a small parameter. Problems such as this arise in fluid dynamics, for example, in the study of air flow over an airplane wing. The small parameter in this case is viscosity. The flow field is described by the differential equation, while the boundary conditions correspond to the nature of the flow on the wing and away in the undisturbed parts of the air. This problem, in its canonical form, is the subject of the survey. What is remarkable is the rich and complex behavior exhibited by this innocent-looking linear problem when a coefficient in the differential equation is allowed to change signs. The asymptotic analysis described in the paper is important because it can provide a closed-form approximate solution. But it is also important because it guides the development of computational methods. Through the asymptotics, the nature of the solution is revealed. In particular, the solution can be exponentially small and exponentially large, as well as having two spatial scales. This behavior is a harbinger of computational nightmares if one were to approach the problem numerically without any consideration to the wild behavior. Thorough understanding of the qualitative nature of the asymptotic solution, revealed by the analysis described, is the foundation of accurate and stable computational schemes for such problems. The article is a pleasure to read, just like the author's classic text Introduction to Singular Perturbations. It gives a nice historical perspective and illustrates the main ideas with concrete examples.

Free arrangements over finite field

The freeness of hyperplane arrangements in a three dimensional vector space over finite field is discussed. We prove that if the number of hyperplanes is greater than some bound, then the freeness is determined by the characteristic polynomial.

Multiobjective GAs, Quantitative Indices, and Pattern Classification

The concept of multiobjective optimization (MOO) has been integrated with variable length chromosomes for the development of a nonparametric genetic classifier which can overcome the problems, like overfitting/overlearning and ignoring smaller classes, as faced by single objective classifiers. The classifier can efficiently approximate any kind of linear and/or nonlinear class boundaries of a data set using an appropriate number of hyperplanes. While designing the classifier the aim is to simultaneously minimize the number of misclassified training points and the number of hyperplanes, and to maximize the product of class wise recognition scores. The concepts of validation set (in addition to training and test sets) and validation functional are introduced in the multiobjective classifier for selecting a solution from a set of nondominated solutions provided by the MOO algorithm. This genetic classifier incorporates elitism and some domain specific constraints in the search process, and is called the CEMOGA-Classifier (constrained elitist multiobjective genetic algorithm based classifier). Two new quantitative indices, namely, the purity and minimal spacing, are developed for evaluating the performance of different MOO techniques. These are used, along with classification accuracy, required number of hyperplanes and the computation time, to compare the CEMOGA-Classifier with other related ones.

Optimal signed-rank tests based on hyperplanes

For analysing k-variate data sets, Randles (J. Amer. Statist. Assoc. 84 (1989) 1045) considered hyperplanes going through k - 1 data points and the origin. He then introduced an empirical angular distance between two k-variate data vectors based on the number of hyperplanes (the so-called interdirections) that separate these two points, and proposed a multivariate sign test based on those interdirections. In this paper, we present an analogous concept (namely, lift-interdirections) to measure the regular distances between data points. The empirical distance between two k-variate data vectors is again determined by the number of hyperplanes that separate these two points; in this case, however, the considered hyperplanes are going through k distinct data points. The invariance and convergence properties of the empirical distances are considered. We show that the lift-interdirections together with Randles’ interdirections allow for building hyperplane-based versions of the optimal testing procedures developed in Hallin and Paindaveine (Ann. Statist. 30 (2002a) 1103, Statistical Data Analysis Based on the L 1 -Norm and Related Procedures (2002b) Birkhäuser, Basel, pp. 271; Bernoulli 8 (2002c) 787, Ann. Statist. (2004a) to appear) for a broad class of location and time series problems. The resulting procedures, which generalize the univariate signed-rank procedures, are affine-invariant and asymptotically invariant under a group of monotone radial transformations (acting on the standardized residuals). Consequently, they are asymptotically distribution-free under the class of elliptical distributions. They are optimal under correctly specified radial densities and, in several cases, enjoy a uniformly good efficiency behavior. These asymptotic properties are confirmed by a Monte-Carlo study, and, finally, a simple robustness study is conducted. It is remarkable that, in the test construction, the value of the test statistic depends on the data cloud only through the geometrical notions of data vectors and oriented hyperplanes, and their relations “above” and “below”.

Genetic classifiers for remotely sensed images: Comparison with standard methods

In this article the effectiveness of some recently developed genetic algorithm-based pattern classifiers was investigated in the domain of satellite imagery which usually have complex and overlapping class boundaries. Landsat data, SPOT image and IRS image are considered as input. The superiority of these classifiers over k-NN rule, Bayes' maximum likelihood classifier and multilayer perceptron (MLP) for partitioning different landcover types is established. Results based on producer's accuracy (percentage recognition score), user's accuracy and kappa values are provided. Incorporation of the concept of variable length chromosomes and chromosome discrimination led to superior performance in terms of automatic evolution of the number of hyperplanes for modelling the class boundaries, and the convergence time. This non-parametric classifier requires very little a priori information, unlike k-NN rule and MLP (where the performance depends heavily on the value of k and the architecture, respectively), and Bayes' maximum likelihood classifier (where assumptions regarding the class distribution functions need to be made).

Multidimensional signal space partitioning using a minimal set of hyperplanes for detecting ISI-corrupted symbols

A signal space partitioning technique is presented for detecting symbols transmitted through intersymbol interference channels. The decision boundary is piecewise linear and is made up of several hyperplanes. The goal here is to minimize the number of hyperplanes for a given performance measure, namely, the minimum distance between any signal and the decision boundary, Unlike in Voronoi partitioning, individual hyperplanes are chosen to separate signal clusters rather than signal pairs. The convex regions associated with individual signals, which together form the overall decision region, generally overlap or coincide among in-class signals. The technique leads to an asymptotically optimum detector when the target distance is set at half the minimum distance associated with the maximum-likelihood sequence detector. Complexity and performance can be easily traded as the target distance is a flexible design parameter.

Partitioning the feature space of a classifier with linear hyperplanes

We describe the design and use of linear hyperplanes to partition the feature space of a classifier. The objective of the partitioning is to minimize the average entropy of the class distribution in the final partitions. The hyperplanes are characterized by a vector /spl nu//sub n/ and scalar h/sub n/, which are computed with the objective of maximizing the mutual information associated with the partitioning. We show that the problem of designing the /spl nu//sub n/ can be simplified to an approximately equivalent problem, the solution of which, interestingly enough, turns out to be the linear discriminant of the data. We also describe a decision-tree based technique that partitions the feature space in a hierarchical fashion where each node in the decision tree represents a linear hyperplane that further partitions the feature space into two regions. The end result of the tree-growing process is that the entire feature space is partitioned into nonoverlapping regions where each region is bounded by a number of hyperplanes. As the criterion for the design of the hyperplanes is the minimization of the average class entropy of the regions, each region is characterized by the occurrence of only a small number of classes. The partitioning information provided by the decision-tree can then be used to eliminate a number of classes from being considered, and hence simplifies the job of the classifier. We show the application of the decision-tree as a preprocessor to a classifier in the speech recognition problem. Here the classes are modeled with mixtures of tens to hundreds of thousands of Gaussians, and the use of the partitioning information reduces the computation associated with the classification process by factors larger than 20 with negligible degradation in the word error rate (note, however, that the overall decoding process comprises other steps in addition to the classification-the computational speedup mentioned above does not affect the computation of these other steps). We also compare the performance of this quantization scheme with standard Gaussian clustering schemes and show that the decision-tree based quantization provides better performance in most cases.

On the number of points of bounded height¶on arithmetic projective spaces

Let K be a number field and \(\) its ring of integers. Let \(\) be a Hermitian vector bundle over \(\). In the first part of this paper we estimate the number of points of bounded height in \(\) (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d>1 in \(\) of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of ℙNK blown up at a linear subspace of codimension two.