Parallel Hyperplanes Research Articles

Latent variable model inversion for intervals. Application to tolerance intervals in class-modelling situations, and specification limits in process control

The paper deals with the inversion of intervals when a PLS (Partial Least Squares) model is used. However, instead of discretizing the interval, it is proved that the region resulting from the inversion of a PLS model is a convex set bounded by two parallel hyperplanes, each corresponding to the direct inversion of each endpoint of the given interval.When the domain of the input variables is a convex set, any feasible solution with predictions within the interval set in the response can be obtained as a convex combination of a point on each of the two hyperplanes. In this way, the new solutions preserve the internal structure of the input variables.This methodology can be of interest in several domains where the response under study is defined in terms of an interval of admissible values, such as specifications for a product in an industrial process, or tolerance intervals for computing compliant class-models.The inversion of the corresponding fitted model defines a region in the input space (predictor variables) whose predictions fall within the specified interval. Then, estimating and exploring this region will increase the information about the problem under study.

Kreĭn twin support vector machines for imbalanced data classification

Conventional classification assumes a balanced sample distribution among classes. However, such a premise leads to biased performance over the majority class (with the highest number of instances). The Twin Support Vector Machines (TWSVM) obtained great prominence due to their low computational burden compared to the standard SVM. Besides, traditional machine learning seeks methods whose solution depends on a convex problem or semi-positive definite similarity matrices. Yet, this kind of matrix cannot adequately represent many real-world applications. The above defines the need to use non-negative measures as an indefinite function in a Reproducing Kernel Kreĭn Space (RKKS). This paper proposes a novel approach called Kreĭn Twin Support Vector Machines (KTSVM), which appropriately incorporates indefinite kernels within a TWSVM-based gradient optimization. To code pertinent input patterns within an imbalanced data discrimination, our KTSVM employs an implicit mapping to a RKKS. Also, our approach takes advantage of the TWSVM scheme’s benefits by creating two parallel hyperplanes. This promotes the KTSVM optimization in a gradient-descent framework. Results obtained on synthetic and real-world datasets demonstrate that our solution performs better in terms of imbalanced data classification than state-of-the-art techniques.

Robust and compact maximum margin clustering for high-dimensional data

In the field of machine learning, clustering has become an increasingly popular research topic due to its critical importance. Many clustering algorithms have been proposed utilizing a variety of approaches. This study focuses on clustering of high-dimensional data using the maximum margin clustering approach. In this paper, two methods are introduced: The first method employs the classical maximum margin clustering approach, which separates data into two clusters with the greatest margin between them. The second method takes cluster compactness into account and searches for two parallel hyperplanes that best fit to the cluster samples while also being as far apart from each other as possible. Additionally, robust variants of these clustering methods are introduced to handle outliers and noise within the data samples. The stochastic gradient algorithm is used to solve the resulting optimization problems, enabling all proposed clustering methods to scale well with large-scale data. Experimental results demonstrate that the proposed methods are more effective than existing maximum margin clustering methods, particularly in high-dimensional clustering problems, highlighting the efficacy of the proposed methods.

Proximity and Flatness Bounds for Linear Integer Optimization

This paper deals with linear integer optimization. We develop a technique that can be applied to provide improved upper bounds for two important questions in linear integer optimization. Given an optimal vertex solution for the linear relaxation, how far away is the nearest optimal integer solution (if one exists; proximity bounds)? If a polyhedron contains no integer point, what is the smallest number of integer parallel hyperplanes defined by an integral, nonzero, normal vector that intersect the polyhedron (flatness bounds)? This paper presents a link between these two questions by refining a proof technique that has been recently introduced by the authors. A key technical lemma underlying our technique concerns the areas of certain convex polygons in the plane; if a polygon [Formula: see text] satisfies [Formula: see text], where τ denotes [Formula: see text] counterclockwise rotation and [Formula: see text] denotes the polar of K, then the area of [Formula: see text] is at least three. Funding: J. Paat was supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2021-02475]. R. Weismantel was supported by the Einstein Stiftung Berlin.

Stability of hypersurfaces with constant mean curvature trapped between two parallel hyperplanes

Static equilibrium configurations of continua supported by surface tension are given by constant mean curvature (CMC) surfaces which are critical points of a variational problem to extremize the area while keeping the volume fixed. CMC surfaces are used as mathematical models of a variety of continua, such as tiny liquid drops, stars, and nuclei, to play important roles in both mathematics and physics. Therefore, the geometry of CMC surfaces and their properties such as stability are of special importance in differential geometry and in a variety of physical sciences. In this paper we examine the stability of CMC hypersurfaces in arbitrary dimensions, possibly having boundaries on two parallel hyperplanes, by investigating the second variation of the area. We determine the stability of non-uniform liquid bridges or unduloids for the first time in all dimensions and all parameter (the ratio of the neck radius to bulge radius) regimes. The analysis is assisted by numerical computations.

Absolute minima of potentials of certain regular spherical configurations

We use methods of approximation theory to find the absolute minima on the sphere of the potential of spherical (2m−3)-designs with a non-trivial index 2m that are contained in a union of m parallel hyperplanes, m≥2, whose locations satisfy certain additional assumptions. The interaction between points is described by a function of the dot product, which has positive derivatives of orders 2m−2, 2m−1, and 2m. This includes the case of the classical Coulomb, Riesz, and logarithmic potentials as well as a completely monotone potential of the distance squared. We illustrate this result by showing that the absolute minimum of the potential of the set of vertices of the icosahedron on the unit sphere S2 in R3 is attained at the vertices of the dual dodecahedron and the one for the set of vertices of the dodecahedron is attained at the vertices of the dual icosahedron. The absolute minimum of the potential of the configuration of 240 minimal vectors of E8 root lattice normalized to lie on the unit sphere S7 in R8 is attained at a set of 2160 points on S7 which we describe.

Stability of hypersurfaces of constant mean curvature with free boundary in two parallel hyperplanes

Stability of hypersurfaces of constant mean curvature with free boundary in two parallel hyperplanes

Sets in [formula omitted] with doubling 2k + δ are near convex progressions

For δ>0 sufficiently small and A⊂Zk with |A+A|≤(2k+δ)|A|, we show either A is covered by mk(δ) parallel hyperplanes, or satisfies |coˆ(A)∖A|≤ckδ|A|, where coˆ(A) is the smallest convex progression (convex set intersected with an affine sub-lattice) containing A. This generalizes the Freiman-Bilu 2k theorem, Freiman's 3|A|−4 theorem, and recent sharp stability results of the present authors for sumsets in Rk conjectured by Figalli and Jerison.

Covering by Planks and Avoiding Zeros of Polynomials

Abstract We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the celebrated Bang plank covering theorem. We prove a tight polynomial analog of the Bang theorem for the Euclidean ball and an even stronger polynomial version for the complex projective space. Specifically, for the ball, we show that for every real nonzero $d$-variate polynomial $P$ of degree $n$, there exists a point in the unit $d$-dimensional ball at distance at least $1/n$ from the zero set of the polynomial $P$. Using the polynomial approach, we also prove the strengthening of the Fejes Tóth zone conjecture on covering a sphere by spherical segments, closed parts of the sphere between two parallel hyperplanes. In particular, we show that the sum of angular widths of spherical segments covering the whole sphere is at least $\pi $.

Research on Twin Extreme Learning Fault Diagnosis Method Based on Multi-Scale Weighted Permutation Entropy.

Due to the complicated engineering operation of the check valve in a high−pressure diaphragm pump, its vibration signal tends to show non−stationary and non−linear characteristics. These leads to difficulty extracting fault features and, hence, a low accuracy for fault diagnosis. It is difficult to extract fault features accurately and reliably using the traditional MPE method, and the ELM model has a low accuracy rate in fault classification. Multi−scale weighted permutation entropy (MWPE) is based on extracting multi−scale fault features and arrangement pattern features, and due to the combination of extracting a sequence of amplitude features, fault features are significantly enhanced, which overcomes the deficiency of the single−scale permutation entropy characterizing the complexity of vibration signals. It establishes the check valve fault diagnosis model from the twin extreme learning machine (TELM). The TELM fault diagnosis model established, based on MWPE, aims to find a pair of non−parallel classification hyperplanes in the equipment state space to improve the model’s applicability. Experiments show that the proposed method effectively extracts the characteristics of the vibration signal, and the fault diagnosis model effectively identifies the fault state of the check valve with an accuracy rate of 97.222%.

Lifting Methods in Mass Partition Problems

Abstract Many results about mass partitions are proved by lifting $\mathds {R}^d$ to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other results, we prove the existence of equipartitions of $d+1$ measures in $\mathds {R}^d$ by parallel hyperplanes and of $d+2$ measures in $\mathds {R}^d$ by concentric spheres. For measures whose supports are sufficiently well separated, we prove results where one can cut a fixed (possibly different) fraction of each measure either by parallel hyperplanes, concentric spheres, convex polyhedral surfaces of few facets, or convex polytopes with few vertices.

Practical volume approximation of high-dimensional convex bodies, applied to modeling portfolio dependencies and financial crises

We examine volume computation of general-dimensional polytopes and more general convex bodies, defined by the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest for the public in general and for policy makers in particular. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be estimated by computing volumes of convex bodies.We design and implement practical algorithms in the exact and approximate setting, and experimentally juxtapose them in order to study the trade-off of exactness and accuracy for speed. We also experimentally find an efficient parameter-tuning to achieve a sufficiently good estimation of the probability density of each copula. Our C++ software, based on Eigen and available on github, is shown to be very effective in up to 100 dimensions. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises.

Point-hyperplane Incidence Geometry and the Log-rank Conjecture

We study the log-rank conjecture from the perspective of point-hyperplane incidence geometry. We formulate the following conjecture: Given a point set in ℝ d that is covered by constant-sized sets of parallel hyperplanes, there exists an affine subspace that accounts for a large (i.e., 2 –polylog( d ) ) fraction of the incidences, in the sense of containing a large fraction of the points and being contained in a large fraction of the hyperplanes. In other words, the point-hyperplane incidence graph for such configurations has a large complete bipartite subgraph. Alternatively, our conjecture may be interpreted linear-algebraically as follows: Any rank- d matrix containing at most O (1) distinct entries in each column contains a submatrix of fractional size 2 –polylog( d ) , in which each column is constant. We prove that our conjecture is equivalent to the log-rank conjecture; the crucial ingredient of this proof is a reduction from bounds for parallel k -partitions to bounds for parallel ( k -1)-partitions. We also introduce an (apparent) strengthening of the conjecture, which relaxes the requirements that the sets of hyperplanes be parallel. Motivated by the connections above, we revisit well-studied questions in point-hyperplane incidence geometry without structural assumptions (i.e., the existence of partitions). We give an elementary argument for the existence of complete bipartite subgraphs of density Ω (ε 2 d / d ) in any d -dimensional configuration with incidence density ε, qualitatively matching previous results proved using sophisticated geometric techniques. We also improve an upper-bound construction of Apfelbaum and Sharir [ 2 ], yielding a configuration whose complete bipartite subgraphs are exponentially small and whose incidence density is Ω (1/√ d ). Finally, we discuss various constructions (due to others) of products of Boolean matrices which yield configurations with incidence density Ω (1) and complete bipartite subgraph density 2 -Ω (√ d ) , and pose several questions for this special case in the alternative language of extremal set combinatorics. Our framework and results may help shed light on the difficulty of improving Lovett’s Õ(√ rank( f )) bound [ 20 ] for the log-rank conjecture. In particular, any improvement on this bound would imply the first complete bipartite subgraph size bounds for parallel 3-partitioned configurations which beat our generic bounds for unstructured configurations.

Binary scalar products

Let A,B⊆Rd both span Rd such that 〈a,b〉∈{0,1} holds for all a∈A, b∈B. We show that |A|⋅|B|≤(d+1)2d. This allows us to settle a conjecture by Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich (2015) concerning 2-level polytopes. Such polytopes have the property that for every facet-defining hyperplane H there is a parallel hyperplane H′ such that H∪H′ contain all vertices. The authors conjectured that for every d-dimensional 2-level polytope P the product of the number of vertices of P and the number of facets of P is at most d2d+1, which we show to be true.

An intelligent fault diagnosis method for roller bearings using an adaptive interactive deviation matrix machine

Support matrix machines (SMMs) take a matrix as the modeled element and can fully mine the structural information of matrix samples. However, relying solely on a pair of parallel hyperplanes limits the performance of SMMs in classifying complex data. Therefore, this paper proposes an adaptive interactive deviation matrix machine (AIDMM). In the AIDMM, a sensitive margin parameter is introduced to construct two deviation hyperplanes, so that the parameter margin between the two deviation hyperplanes becomes flexible. Compared to the original fixed maximum-margin method, the parameter-margin AIDMM can better adjust the boundary of the deviation hyperplane according to the data, which contributes to improving insensitivity to noise and enhancing robustness. In addition, a multi-rank projection matrix is introduced to obtain a low-rank solution, which gives AIDMM a better fitting ability and avoids the problem of large training errors. Two roller bearing fault datasets are applied for experimental verification, and the experimental results show that AIDMM has excellent classification performance in roller bearing fault diagnosis.

Covering Point-Sets with Parallel Hyperplanes and Sparse Signal Recovery

We give a new deterministic construction of integer sensing matrices that can be used for the recovery of integer-valued signals in compressed sensing. This is a family of \(n \times d\) integer matrices, \(d \ge n\), with bounded sup-norm and the property that no \(\ell \) column vectors are linearly dependent, \(\ell \le n\). Further, if \(\ell \le o(\log n)\) then \(d/n \rightarrow \infty \) as \(n \rightarrow \infty \). Our construction comes from particular sets of difference vectors of point-sets in \({\mathbb {R}}^n\) that cannot be covered by few parallel hyperplanes. We construct examples of such sets on the \(0, \pm 1\) grid and use them for the matrix construction. We also show a connection of our constructions to a simple version of the Tarski’s plank problem.

Multi-view support vector ordinal regression with data uncertainty

Ordinal regression (OR) is a paradigm which learns a prediction model on the data with ordered classes. Despite much progress in OR, the existing OR works learn the classifier from only one view and the multi-view learning in OR has not been considered. What is more, there may exist uncertain information in the multi-view OR data. In this paper, we put forward a novel approach, called multi-view support vector ordinal regression with uncertain data (MORU), which can improve the OR classifier by incorporating the multi-view information and handling the data uncertainty. In our method, a series of parallel hyperplanes are applied to separate the multi-view ordered data, and the uncertain information is considered in the input data. Then, we adopt a heuristic framework to solve the OR learning problem. Experimental results have illustrated that our method obtains superior performance to the existing OR techniques.

Multi-class fuzzy support matrix machine for classification in roller bearing fault diagnosis

As a new classification method with the matrix as the input, support matrix machine (SMM) makes full use of the structured information between rows and columns of the input matrix to establish an accurate prediction model, which has been widely used in the field of fault diagnosis. However, the principle of SMM is to construct two parallel hyperplanes to complete the segmentation between different types of samples. When there are noise and outliers in the sample data, it is difficult for SMM to construct an ideal parallel hyperplane. In view of this, this paper proposes a multi-class fuzzy support matrix machine (MFSMM) by establishing nonparallel hyperplane objective function and integrating fuzzy attributes. In MFSMM, MFSMM establishes two nonparallel fuzzy hyperplanes by objective function, which maximizes the interval between any two fuzzy hyperplanes while considering the sample structure information. Meanwhile, fuzzy plane assigns different membership degrees to different training samples, which greatly reduces the influence of noise on the construction of optimal classification hyperplane. By analyzing two kinds of roller bearing experimental data, the results show that MFSMM has higher classification accuracy and stronger fault tolerance for samples with uncertain information.

A multi-plank generalization of the Bang and Kadets inequalities

If a convex body K ⊂ ℝn is covered by the union of convex bodies C1, …, CN, multiple subadditivity questions can be asked. Two classical results regard the subadditivity of the width (the smallest distance between two parallel hyperplanes that sandwich K) and the inradius (the largest radius of a ball contained in K): the sum of the widths of the Ci is at least the width of K (this is the plank theorem of Thøger Bang), and the sum of the inradii of the Ci is at least the inradius of K (this is due to Vladimir Kadets).We adapt the existing proofs of these results to prove a theorem on coverings by certain generalized non-convex “multi-planks”. One corollary of this approach is a family of inequalities interpolating between Bang’s theorem and Kadets’s theorem. Other corollaries include results reminiscent of the Davenport-Alexander problem, such as the following: if an m-slice pizza cutter (that is, the union of m equiangular rays in the plane with the same endpoint) in applied N times to the unit disk, then there will be a piece of the partition of inradius at least \({{\sin \pi /m} \over {N + \sin \pi /m}}\).

An approximation theorem of Runge type for kernels of certain non-elliptic partial differential operators

For a constant coefficient partial differential operator P(D) with a single characteristic direction, such as the time-dependent free Schrödinger operator as well as non-degenerate parabolic differential operators like the heat operator, we characterize when open subsets X1⊆X2 of Rd form a P-Runge pair. The presented condition does not require any kind of regularity of the boundaries of X1 nor X2. As part of our result we prove that for a large class of non-elliptic operators P(D) there are smooth solutions u of the equation P(D)u=0 on Rd with support contained in an arbitarily narrow slab bounded by two parallel characteristic hyperplanes for P(D).