Regular Simplex Research Articles

Regular simplices in three-dimensional normed spaces

We give a new proof for the following result of Petty: Every 3-dimensional normed space contains a regular 3-simplex.

Scissors Congruence, the Golden Ratio and Volumes in Hyperbolic 5-Space

By different scissors congruence techniques a number of dissection identities are presented between certain quasi-Coxeter polytopes, whose parameters are related to the golden section, and an ideal regular simplex in hyperbolic 5-space. As a consequence, several hyperbolic polyhedral 5-volumes can be computed explicitly in terms of Apery’s constant ζ(3) and the trilogarithmic value **image**.

Constructing symmetric informationally complete positive-operator-valued measures in Bloch space

Here we study the construction of symmetric informationally complete positive-operator-valued measures (SIC-POVMs) in the Bloch space. In this space an SIC-POVM corresponds to a regular simplex, that is a set of real equiangular, unitary vectors. Since the Bloch space also contains vectors which do not describe quantum states, it is necessary to add an extra condition to enforce the members of the simplex to describe pure quantum states. We show that in the case of a three-dimensional quantum systems it is possible to find such a simplex in an analytical way. The solution turns out to be unitarily equivalent to a Weyl–Heisenberg covariant SIC-POVM.

Simplex and Diamond Hierarchies: Models and Applications

AbstractHierarchical spatial decompositions are a basic modelling tool in a variety of application domains. Several papers on this subject deal with hierarchical simplicial decompositions generated through regular simplex bisection. Such decompositions, originally developed for finite elements, are extensively used as the basis for multi‐resolution models of scalar fields, such as terrains, and static or time‐varying volume data. They have also been used as an alternative to quadtrees and octrees as spatial access structures. The primary distinction among all such approaches is whether they treat the simplex or clusters of simplices, called diamonds, as the modelling primitive. This leads to two classes of data structures and to different query approaches. We present the hierarchical models in a dimension‐independent manner, and organize the description of the various applications, primarily interactive terrain rendering and isosurface extraction, according to the dimension of the domain.

Steiner equiangular tight frames

We provide a new method for constructing equiangular tight frames (ETFs). The construction is valid in both the real and complex settings, and shows that many of the few previously-known examples of ETFs are but the first representatives of infinite families of such frames. It provides great freedom in terms of the frame’s size and redundancy. This method also explicitly constructs the frame vectors in their native domain, as opposed to implicitly defining them via their Gram matrix. Moreover, in this domain, the frame vectors are very sparse. The construction is extremely simple: a tensor-like combination of a Steiner system and a regular simplex. This simplicity permits us to resolve an open question regarding ETFs and the restricted isometry property (RIP): we show that the RIP behavior of some ETFs is unfortunately no better than their coherence indicates.

Sums of Laplace eigenvalues: Rotations and tight frames in higher dimensions

The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be maximal among simplexes for the regular simplex (the regular tetrahedron, in three dimensions), maximal among parallelepipeds for the hypercube, and maximal among ellipsoids for the ball, provided the volume and moment of inertia of an "inverse" body are suitably normalized. This result holds for Dirichlet, Robin and Neumann eigenvalues. Additionally, the cubical torus is shown to be maximal among flat tori. The method of proof involves tight frames for euclidean space generated by the orbits of the rotation group of the extremal domain. The ball is conjectured to maximize sums of Neumann eigenvalues among general convex domains, provided the volume and moment of inertia of the polar dual of the domain are suitably normalized.

Dynamic programming in the presence of indeterminacy

480 Indeterminacy must be addressed in optimization in many scientific and technical fields, such as the reg ulation of production processes and the creation of new materials and technologies. The required out comes are optimal parameter values, optimal trajecto ries toward the goal, and often models of the process (response surfaces). Traditional dynamic program ming permits the derivation of optimal trajectories, but only where a model of the process exists—that is, only in conditions of complete determinacy. Optimi zation methods of search type permit the determina tion of optimal parameter values but do not provide optimal trajectories for their attainment. Such meth ods do not provide means of deriving a model of the process. To derive optimal parameter values, optimal trajec tories, and a model of the process, we propose integra tion of dynamic programming, evolutionary planning (search based optimization), and adaptive identifica tion. The proposed method is based on a successive iterative approach: formulation of experiments within a small region of the response surface; construction of a mathematical model in that region; extrapolation of the model to the whole response surface; identifica tion of the optimal trajectory on the extrapolated sur face; determination of the trajectory within a small specified vicinity; subsequent repetition of these oper ations for a new region of the response surface in the direction of motion of the optimal trajectory; and con tinuation until the optimal trajectories and optimal parameters have been established. So as to be specific, we consider examples in which the response surface is a smooth single extremum sur face (Fig. 1a) or has a more complex configuration with gullies (Fig. 1b). First we select the coordinates of the initial point A0 in Fig. 2. Around this point, we construct a regular simplex and conduct experiments to determine the values of the response surface function. From the results, by the least squares method, we construct a model of the form y = a1x1 + a2x2 + a0. The model is extended over the whole region of investigation. Then, by dynamic programming, we determine the optimal trajectory from point A1 to point A0 in the plane y I = ⎯1.9x1 – 1.9x2 + 1.8. The model is extended to a distance specified by the simplex increment (points 2 and 3 in Fig. 3a). The resulting trajectory is fixed in region I (seg ment A0–A2 in Fig. 2). Around point A2, which is the final point of the trajectory, we again construct a regu Dynamic Programming in the Presence of Indeterminacy

A Software to Generate Evenly Spaced Points on the Unit N-Simplex

In a number of quantitative applications, a set of non-negative variables summing to a constant value is involved. A software able to generate evenly spaced points on the regular simplex (that is N-tuples summing to one) has been realized and it is freely available to anybody who might be interested.

Maximally Efficient Modeling of DNA Sequence Motifs at All Levels of Complexity

Identification of transcription factor binding sites is necessary for deciphering gene regulatory networks. Several new methods provide extensive data about the specificity of transcription factors but most methods for analyzing these data to obtain specificity models are limited in scope by, for example, assuming additive interactions or are inefficient in their exploration of more complex models. This article describes an approach--encoding of DNA sequences as the vertices of a regular simplex--that allows simultaneous direct comparison of simple and complex models, with higher-order parameters fit to the residuals of lower-order models. In addition to providing an efficient assessment of all model parameters, this approach can yield valuable insight into the mechanism of binding by highlighting features that are critical to accurate models.

A geometric comparison of entanglement and quantum nonlocality in discrete systems

We compare entanglement with quantum nonlocality employing a geometric structure of the state space of bipartite qudits. The central object is a regular simplex spanned by generalized Bell states. The Collins–Gisin–Linden–Massar–Popescu–Bell inequality is used to reveal states of this set that cannot be described by local-realistic theories. Optimal measurement settings necessary to ascertain nonlocality are determined by means of a recently proposed parameterization of the unitary group combined with robust numerical methods. The main results of this paper are descriptive geometric illustrations of the state space that emphasize the difference between entanglement and quantum nonlocality. Namely, it is found that the shape of the boundaries of separability and Bell inequality violation are essentially different. Moreover, it is also shown that for mixtures of states sharing the same amount of entanglement, Bell inequality violations and entanglement measures are non-monotonically related.

Iterative Optimization of an ESI IT Mass Spectrometer Using Regular Simplex and a Multivariate Target Function Representing the S/N Ratio

Standard automatic tuning for mass spectrometry usually exploits a one-variable-at-a-time approach. This method suffers from important drawbacks: the target function selected for optimization improves the signal of a single channel or a pool of channels without considering noise; the interactions between the parameters are not evaluated. The optimization of the experimental settings of an ESI IT mass spectrometer is carried out here by a multivariate procedure exploiting a target function representing the S/N ratio calculated by principal component analysis and a regular simplex optimization algorithm. A preliminary feasibility study was performed since the target function must be sensitive to the changes in the experimental conditions applied during the iterative tuning and be free from drifts. The feasibility study was carried out to evaluate: the presence of memory effects; the size of the variations in the S/N ratio; the number of scans needed to generate a reliable S/N ratio; the concentration of the multi-standard mixture to use during tuning; the experimental duration required to achieve S/N stability when the experimental settings are modified. The feasibility study led to the identification of the best protocol to accomplish the tuning, while simplex optimization allowed the S/N ratio to be improved by about 70% with respect to the default conditions suggested by the manufacturer.

An Application of Lagrange Multiplier for integer Linear Programming in production- Transportation with Flexible Transportation Cost

The Transportation problem is a classic Operations Research problem where the objective is to determine the schedule for transporting goods from source to destination in a way that minimizes the shipping cost while satisfying supply and demand constraints. Although it can be solving as a linear programming problem. Linear programming makes use of the simplex method, an algorithm invented to solve a linear program by progressing from one extreme point of the feasible polyhedron to an adjacent one. The algorithm contains tactics like pricing and pivoting. For a transportation problem, a simplified version of the regular simplex method can be used, known as the transportation simplex method. In this paper will discuss the functionality of both of these algorithms, and compared their optimized values with non-linear method called the Lagrange Multiplier Method. Lagrange Multiplier is an algorithm that uses different mechanisms to choose the best optimal solutions. This method based on transforming the linear structure transportation problem into the nonlinear structure and solved it directly, by the techniques. The objective of the study was to find out how these algorithms behave in terms of accuracy and speed when a large-scale problem is being solved.

Regular Simplex Fingerprints and Their Optimality Properties

This paper addresses the design of additive fingerprints that are maximally resilient against linear collusion attacks on a focused correlation detector, as defined below. Let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> be the length of the host vector and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> les <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> + 1 the number of users. The focused detector performs a correlation test in order to decide whether a user of interest is among the colluders. Both the fingerprint embedder and the colluders are subject to squared-error distortion constraints. We show that simplex fingerprints maximize a geometric figure of merit for this detector. In that sense they outperform orthogonal fingerprints but the advantage vanishes as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> rarr infin. They are also optimal in terms of minimizing the probability of error of the focused detector when the attack is a uniform averaging of the marked copies followed by the addition of white Gaussian noise. Reliable detection is guaranteed provided that the number of colluders <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> Lt radic( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ). Moreover, we study the probability of error performance of simplex fingerprints for the focused correlation detector when the colluders use nonuniform averaging plus white Gaussian noise attacks.

Simplex based space filling designs

Space filling designs are important for deterministic computer experiments. Even a single experiment can be very time consuming and can have many input parameters. Furthermore the underlying function generating the output is often nonlinear. Thus, the computer experiment has to be designed carefully. There exist many design criteria, which can be numerically optimized. Here, a method is developed, which does not need algorithmic optimization. A mesh of nearly regular simplices is constructed and the vertices of the simplices are used as potential design points. The extraction of a design from these meshes is very fast and easy to implement once the underlying mesh has been constructed. The extracted designs are highly competitive regarding the maximin design criterion and it is easy to extract designs for nonstandard design spaces.

Diagnosing patients with a combination of principal component analysis and case based reasoning

This paper addresses the application of a principal component analysis (PCA) of categorical data prior to diagnosing a patients dataset using a case-based reasoning (CBR) system. The particularity is that the standard PCA techniques are designed to deal with numerical attributes, but our medical dataset contains many categorical data and alternative methods such as RS-PCA are required. Thus, we propose to hybridize RS-PCA (regular simplex PCA) and a simple CBR system. Results show how the hybrid system, when diagnosing a medical dataset, produces results similar to the ones obtained when using the original attributes. These results are quite promising since they allow diagnosis with less computation effort and memory storage.

Special issue: Data mining and hybrid intelligent systems

This special issue encompasses four papers devoted to the recent developments in the field of Data Mining. The issue originated from presentations at the “8th International Conference on Hybrid Intelligent Systems” held at Technical University of Catalonia, in Barcelona (Spain), September 10–12th, 2008. Seven selected papers were submitted to the special issue, every paper was revised by at least three referees and finally four of them were accepted according to the referees’ evaluations. This special issue is focused on data mining and the use of hybrid intelligent systems. The submissions embrace theoretical models dealing with classification problems as well as applications. The first paper, “AD-SVMs: A Light Extension of SVMs for Multicategory Classification”, is devoted to the support vector machines and their use for multicategory classification. Nanculef et al. generalize the geometrical formulation, obtaining a method that preserves the form and complexity of the binary case optimizing a single convex quadratic program where each new class introduces just one additional constraint. The second paper, “Analysis and Improvement of the Genetic Discovery Component of XCS”, is devoted to the analysis of the well known evolutionary learning algorithm XCS that uses genetic algorithms to evolve a population of classifiers online. Morales-Ortigosa et al. conduct a systematic experimental analysis of the effect of the different genetic operators. The overall analysis enables them to provide important insights into the behavior of different operators and to improve the learning of interval-based rules in real-world domains on average. Two other papers are related to the application of case-based reasoning systems to two medical problems. The first one, “Case-based reasoning as a decision support system for cancer diagnosis: A case study”, is devoted to automatic classification of leukemia patients from microarray data. De Paz et al. present a case-based reasoning system for automatic classification. The system incorporates novel algorithms for data mining that allow filtering, classification, and knowledge extraction. The second one, “Diagnosing Patients with a Combination of Principal Component Analysis and Case Based Reasoning” is focused on diagnosing patients. Pous et al. address the application of a principal component analysis of categorical data prior to diagnosing a patients dataset using a case-based reasoning system. They propose the hybridization of a regular simplex principal component analysis and a simple case based reasoning systems. Finally, as Guest Editors of this special issue, we would like to thank all the authors for their contributions and the referees for their outstanding cooperation and constructive feedback.

Large regular simplices contained in a hypercube

We prove that the n-dimensional unit hypercube contains an n-dimensional regular simplex of edge length c√n, where c > 0 is a constant independent of n.

On one extremal property of a regular simplex

In this paper, we show that the Lp-error of asymmetric linear approximation of the quadratic function Q(x) = ∑d j=1 x 2 j on simplices in R d of fixed volume is minimized on regular simplices.

Spherical two-distance sets

A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b so that the inner products of distinct vectors of S are either a or b. It is known that the largest cardinality g ( n ) of spherical two-distance sets does not exceed n ( n + 3 ) / 2 . This upper bound is known to be tight for n = 2 , 6 , 22 . The set of mid-points of the edges of a regular simplex gives the lower bound L ( n ) = n ( n + 1 ) / 2 for g ( n ) . In this paper using the so-called polynomial method it is proved that for nonnegative a + b the largest cardinality of S is not greater than L ( n ) . For the case a + b < 0 we propose upper bounds on | S | which are based on Delsarte's method. Using this we show that g ( n ) = L ( n ) for 6 < n < 22 , 23 < n < 40 , and g ( 23 ) = 276 or 277.

Local ridge regression for face recognition

Ridge regression (RR) for classification is a regularized least square method to model the linear dependency between covariate variables and labels. By applying appropriate techniques to encode the multivariate labels in face recognition as the vertices of the regular simplex which can separate points with highest degree of symmetry, RR maps the face images into a face subspace where the images from each individual will locate near their individual targets. However, as a holistic method, RR operates directly on a whole face region represented as a vector and thus cannot effectively recognize the faces with illumination variations and partial occlusions. In this paper, we present a novel algorithm, termed as local ridge regression (LRR). Different from RR, LRR emphasizes on each local face region matching rather than the whole. As a result, LRR can not only enhance the robustness to the local variations by utilizing the spatial and geometrical information of facial components, but also avoid the dimensionality reduction in the holistic RR as a preprocessing. Furthermore, an efficient cross-validation algorithm is adopted to select the regularization parameters in each local region. Experiments on two standard face databases demonstrate that the proposed algorithm significantly outperforms RR and the two popular linear face recognition techniques (Eigenface and Fisherface). Although we concentrate on RR in this paper, following the proposed line of the research, many current multi-category classifiers can also be applied in face recognition through combining the characteristics of face images and may obtain better recognition accuracies.