Space Dimension Increases Research Articles

Two-stage many-objective evolutionary algorithm: enhanced dominance relations and control mechanisms for separated balance

Although the multiobjective evolutionary algorithms (MOEAs) have been proved to bring promising prospects for solving multiobjective optimization problems (MOPs), the performance of the algorithm deteriorates sharply in high-dimensional objective space due to the weak selection pressure and the unregulated balance, which is caused by the increase of objective space dimension. Some current MOEAs with two-stage strategy (TS) strive to address above issues by dividing the evolutionary process into two independent stages, in which convergence and diversity are handled separately within successive generations of different stages. However, TS-MOEAs have some weaknesses, such as sensitivity to stage division, and incomplete separation of convergence and diversity. In this paper, TS/KW-MaOEA is proposed for solving many-objective optimization problems (MaOPs), which keeps TS as the central and equips a perfect control mechanism for separated balance. More specifically, TS/KW-MaOEA can automatically adjust the balance trend and provide appropriate selection pressure for MaOPs according to the Kondratiev wave (KW) search model and the objective space dimension. To verify the effectiveness of the proposed algorithm, a series of experiments are carried out against seven state-of-the-art many-objective optimization algorithms on 15 benchmark problems with up to 30 objectives. Experimental results indicate that the proposed algorithm is highly competitive against peer competitors.

Distilling dynamical knowledge from stochastic reaction networks

Stochastic reaction networks are widely used in the modeling of stochastic systems across diverse domains such as biology, chemistry, physics, and ecology. However, the comprehension of the dynamic behaviors inherent in stochastic reaction networks is a formidable undertaking, primarily due to the exponential growth in the number of possible states or trajectories as the state space dimension increases. In this study, we introduce a knowledge distillation method based on reinforcement learning principles, aimed at compressing the dynamical knowledge encoded in stochastic reaction networks into a singular neural network construct. The trained neural network possesses the capability to accurately predict the state conditional joint probability distribution that corresponds to the given query contexts, when prompted with rate parameters, initial conditions, and time values. This obviates the need to track the dynamical process, enabling the direct estimation of normalized state and trajectory probabilities, without necessitating the integration over the complete state space. By applying our method to representative examples, we have observed a high degree of accuracy in both multimodal and high-dimensional systems. Additionally, the trained neural network can serve as a foundational model for developing efficient algorithms for parameter inference and trajectory ensemble generation. These results collectively underscore the efficacy of our approach as a universal means of distilling knowledge from stochastic reaction networks. Importantly, our methodology also spotlights the potential utility in harnessing a singular, pretrained, large-scale model to encapsulate the solution space underpinning a wide spectrum of stochastic dynamical systems.

Theory of (1 + 1) ES on the RIDGE

Previous research proposed the uniform mutation inside the sphere as a new mutation operator for evolution strategies (continuous evolutionary algorithms), with a case study of the elitist algorithm on the SPHERE. For that landscape, one-step success probability and expected progress were estimated analytically, and further proved to converge, as space dimension increases, to the corresponding asymptotics of the algorithm with normal mutation. This article takes the analysis further by considering the RIDGE, an asymmetric landscape almost uncovered in the literature. For the elitist algorithm, estimates of expected progress along the radial and longitudinal axes are derived, then tested numerically against the real behavior of the algorithm on several functions from this class. The global behavior of the algorithm is predicted correctly by iterating the one-step analytical formulas. Moreover, experiments show identical mean value dynamics for the algorithms with uniform and normal mutation, which implies that the derived formulas apply also to the normal case. Essential to the whole analysis is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula> , the inclination angle of the RIDGE. The behavior of the algorithm on the SPHERE and HYPERPLANE is also obtained, at the limits of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\theta $ </tex-math></inline-formula> interval (0°, 90°].

Suboptimal Kalman Filtering in Triplet Markov Models Using Model Order Reduction

When the state space dimension increases, the computational burden can become a major challenge for optimal Kalman filtering in Gaussian triplet Markov models (TMMs). In this paper, we introduce a new model order reduction technique applicable to linear time-homogeneous Gaussian TMMs. Taking advantage of the lower state dimension of the resulting approximate model, a low-complexity suboptimal Kalman filter is obtained. The proposed estimator provides complexity reduction without significant accuracy loss and is shown to outperform two classical methods in the case of Markovian process noise.

Distribution of Distances between Elements in a Compact Set

In this article, we propose a review of studies evaluating the distribution of distances between elements of a random set independently and uniformly distributed over a region of space in a normed R -vector space (for example, point events generated by a homogeneous Poisson process in a compact set). The distribution of distances between individuals is present in many situations when interaction depends on distance and concerns many disciplines, such as statistical physics, biology, ecology, geography, networking, etc. After reviewing the solutions proposed in the literature, we present a modern, general and unified resolution method using convolution of random vectors. We apply this method to typical compact sets: segments, rectangles, disks, spheres and hyperspheres. We show, for example, that in a hypersphere the distribution of distances has a typical shape and is polynomial for odd dimensions. We also present various applications of these results and we show, for example, that variance of distances in a hypersphere tends to zero when space dimension increases.

Extreme lattices: symmetries and decorrelation

We study statistical and structural properties of extreme lattices, which are the local minima in the density landscape of lattice sphere packings in d-dimensional Euclidean space . Specifically, we ascertain statistics of the densities and kissing numbers as well as the numbers of distinct symmetries of the packings for dimensions 8 through 13 using the stochastic Voronoi algorithm. The extreme lattices in a fixed dimension of space d () are dominated by typical lattices that have similar packing properties, such as packing densities and kissing numbers, while the best and the worst packers are in the long tails of the distribution of the extreme lattices. We also study the validity of the recently proposed decorrelation principle, which has important implications for sphere packings in general. The degree to which extreme-lattice packings decorrelate as well as how decorrelation is related to the packing density and symmetry of the lattices as the space dimension increases is also investigated. We find that the extreme lattices decorrelate with increasing dimension, while the least symmetric lattices decorrelate faster.

Proper generalized decomposition for solving coupled heat and moisture transfer

This paper proposes a reduced order model to simulate heat and moisture behaviour of material based on proper general decomposition (PGD). This innovative method is an a priori model reduction method. It proposes an alternative way for computing solutions of the problem by considering a separated representation of the solution. PGD offers an interesting reduction of numerical cost. In this paper, the PGD solution is first compared with a finite element solution and the commercial validated model Delphin in an 1D case. The results show that the PGD resolution techniques enable the field of interest to be represented with accuracy, with a relative error rate of less than 0.1%. The study remains in the hygroscopic range of the material. As the numerical gain of the method becomes interesting when the space dimension increases, this resolution strategy was then used on a 2D multi-layered test case. The dynamics and amplitude of hygrothermal fields are perfectly represented by the PGD solution. Temperature and vapour pressure modelled with PGD can be used for post-processing and analysing the behaviour of an assembly.

Variability-Aware, Discrete Optimization for Analog Circuits

This paper explores the use of discrete optimization techniques for variability-aware analog circuit synthesis, with an observation that the continuous design space can be effectively covered by a finite number of discrete points when parameter variation is present. Three algorithms are described that can leverage a discretized design space yet mitigate its dimensionality scaling problem: an isotropic discretization scheme, which can fill the neighborhood of any given point with only quadratically-increasing number of nearest neighbors placed at equal distances as the space dimension increases, a stochastic hill-climbing algorithm, which evaluates only a partial set of nearest neighbors yet finds the local optimum at the reduced cost, and an incremental Monte Carlo sampling algorithm, which draws the minimal number of Monte Carlo samples just enough to determine the superior design point during the local search process. These algorithms help in finding the optimal design of analog circuits efficiently. For instance, for a digitally-controlled oscillator example, its discretized design space consists of 5 565 907 points but the optimal point was found by evaluating only 40 design points and running only 21 Monte Carlo simulations per point in average (824 in total).

ψ-Epistemic models are exponentially bad at explaining the distinguishability of quantum states.

The status of the quantum state is perhaps the most controversial issue in the foundations of quantum theory. Is it an epistemic state (state of knowledge) or an ontic state (state of reality)? In realist models of quantum theory, the epistemic view asserts that nonorthogonal quantum states correspond to overlapping probability measures over the true ontic states. This naturally accounts for a large number of otherwise puzzling quantum phenomena. For example, the indistinguishability of nonorthogonal states is explained by the fact that the ontic state sometimes lies in the overlap region, in which case there is nothing in reality that could distinguish the two states. For this to work, the amount of overlap of the probability measures should be comparable to the indistinguishability of the quantum states. In this Letter, I exhibit a family of states for which the ratio of these two quantities must be ≤2de-cd in Hilbert spaces of dimension d that are divisible by 4. This implies that, for large Hilbert space dimension, the epistemic explanation of indistinguishability becomes implausible at an exponential rate as the Hilbert space dimension increases.

Precise algorithm to generate random sequential addition of hard hyperspheres at saturation

The study of the packing of hard hyperspheres in d-dimensional Euclidean space R^{d} has been a topic of great interest in statistical mechanics and condensed matter theory. While the densest known packings are ordered in sufficiently low dimensions, it has been suggested that in sufficiently large dimensions, the densest packings might be disordered. The random sequential addition (RSA) time-dependent packing process, in which congruent hard hyperspheres are randomly and sequentially placed into a system without interparticle overlap, is a useful packing model to study disorder in high dimensions. Of particular interest is the infinite-time saturation limit in which the available space for another sphere tends to zero. However, the associated saturation density has been determined in all previous investigations by extrapolating the density results for nearly saturated configurations to the saturation limit, which necessarily introduces numerical uncertainties. We have refined an algorithm devised by us [S. Torquato, O. U. Uche, and F. H. Stillinger, Phys. Rev. E 74, 061308 (2006)] to generate RSA packings of identical hyperspheres. The improved algorithm produce such packings that are guaranteed to contain no available space in a large simulation box using finite computational time with heretofore unattained precision and across the widest range of dimensions (2≤d≤8). We have also calculated the packing and covering densities, pair correlation function g(2)(r), and structure factor S(k) of the saturated RSA configurations. As the space dimension increases, we find that pair correlations markedly diminish, consistent with a recently proposed "decorrelation" principle, and the degree of "hyperuniformity" (suppression of infinite-wavelength density fluctuations) increases. We have also calculated the void exclusion probability in order to compute the so-called quantizer error of the RSA packings, which is related to the second moment of inertia of the average Voronoi cell. Our algorithm is easily generalizable to generate saturated RSA packings of nonspherical particles.

Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses

In the first paper of this series [S. Torquato, J. Chem. Phys. 136, 054106 (2012)], analytical results concerning the continuum percolation of overlapping hyperparticles in d-dimensional Euclidean space R(d) were obtained, including lower bounds on the percolation threshold. In the present investigation, we provide additional analytical results for certain cluster statistics, such as the concentration of k-mers and related quantities, and obtain an upper bound on the percolation threshold η(c). We utilize the tightest lower bound obtained in the first paper to formulate an efficient simulation method, called the rescaled-particle algorithm, to estimate continuum percolation properties across many space dimensions with heretofore unattained accuracy. This simulation procedure is applied to compute the threshold η(c) and associated mean number of overlaps per particle N(c) for both overlapping hyperspheres and oriented hypercubes for 3 ≤ d ≤ 11. These simulations results are compared to corresponding upper and lower bounds on these percolation properties. We find that the bounds converge to one another as the space dimension increases, but the lower bound provides an excellent estimate of η(c) and N(c), even for relatively low dimensions. We confirm a prediction of the first paper in this series that low-dimensional percolation properties encode high-dimensional information. We also show that the concentration of monomers dominate over concentration values for higher order clusters (dimers, trimers, etc.) as the space dimension becomes large. Finally, we provide accurate analytical estimates of the pair connectedness function and blocking function at their contact values for any d as a function of density.

Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes

We show analytically that the [0, 1], [1, 1], and [2, 1] Padé approximants of the mean cluster number S for both overlapping hyperspheres and overlapping oriented hypercubes are upper bounds on this quantity in any Euclidean dimension d. These results lead to lower bounds on the percolation threshold density η(c), which become progressively tighter as d increases and exact asymptotically as d → ∞, i.e., η(c) → 2(-d). Our analysis is aided by a certain remarkable duality between the equilibrium hard-hypersphere (hypercube) fluid system and the continuum percolation model of overlapping hyperspheres (hypercubes). Analogies between these two seemingly different problems are described. We also obtain Percus-Yevick-like approximations for the mean cluster number S in any dimension d that also become asymptotically exact as d → ∞. We infer that as the space dimension increases, finite-sized clusters become more ramified or "branch-like." These analytical estimates are used to assess simulation results for η(c) up to 20 dimensions in the case of hyperspheres and up to 15 dimensions in the case of hypercubes. Our analysis sheds light on the radius of convergence of the density expansion for S and naturally leads to an analytical approximation for η(c) that applies across all dimensions for both hyperspheres and oriented hypercubes. Finally, we describe the extension of our results to the case of overlapping particles of general anisotropic shape in d dimensions with a specified orientational probability distribution.

Robust algorithm to generate a diverse class of dense disordered and ordered sphere packings via linear programming

We have formulated the problem of generating dense packings of nonoverlapping, nontiling nonspherical particles within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem called the adaptive-shrinking cell (ASC) formulation [S. Torquato and Y. Jiao, Phys. Rev. E 80, 041104 (2009)]. Because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in d-dimensional Euclidean space R(d), it is most suitable and natural to solve the corresponding ASC optimization problem using sequential-linear-programming (SLP) techniques. We implement an SLP solution to produce robustly a wide spectrum of jammed sphere packings in R(d) for d=2, 3, 4, 5, and 6 with a diversity of disorder and densities up to the respective maximal densities. A novel feature of this deterministic algorithm is that it can produce a broad range of inherent structures (locally maximally dense and mechanically stable packings), besides the usual disordered ones (such as the maximally random jammed state), with very small computational cost compared to that of the best known packing algorithms by tuning the radius of the influence sphere. For example, in three dimensions, we show that it can produce with high probability a variety of strictly jammed packings with a packing density anywhere in the wide range [0.6, 0.7408...], where π/√18 = 0.7408... corresponds to the density of the densest packing. We also apply the algorithm to generate various disordered packings as well as the maximally dense packings for d=2, 4, 5, and 6. Our jammed sphere packings are characterized and compared to the corresponding packings generated by the well-known Lubachevsky-Stillinger (LS) molecular-dynamics packing algorithm. Compared to the LS procedure, our SLP protocol is able to ensure that the final packings are truly jammed, produces disordered jammed packings with anomalously low densities, and is appreciably more robust and computationally faster at generating maximally dense packings, especially as the space dimension increases.

Multidimensional electrostatic energy and classical renormalization

Recent interest in problems in higher space di mensions is becoming increasingly important and attracted the attention of many investigators in variety of fields in physics. In this paper, the electrostatic energy of two geometries (a charged spherical shell and a nonconducting sphere) is calculated in higher space dimension, N. It is shown that as the space dimension increases, up to N = 9, the electrostatic energy of the two geometries decreases and beyond N = 9 it increases. Furthermore, we discuss a simple example which illustrates classical renormalization in electrostatics in higher dimensions.

Topological and statistical properties of quantum control transition landscapes

A puzzle arising in the control of quantum dynamics is to explain the relative ease with which high-quality control solutions can be found in the laboratory and in simulations. The emerging explanation appears to lie in the nature of the quantum control landscape, which is an observable as a function of the control variables. This work considers the common case of the observable being the transition probability between an initial and a target state. For any controllable quantum system, this landscape contains only global maxima and minima, and no local extrema traps. The probability distribution function for the landscape value is used to calculate the relative volume of the region of the landscape corresponding to good control solutions. The topology of the global optima of the landscape is analysed and the optima are shown to have inherent robustness to variations in the controls. Although the relative landscape volume of good control solutions is found to shrink rapidly as the system Hilbert space dimension increases, the highly favourable landscape topology at and away from the global optima provides a rationale for understanding the relative ease of finding high-quality, stable quantum optimal control solutions.

Random sequential addition of hard spheres in high Euclidean dimensions

Sphere packings in high dimensions have been the subject of recent theoretical interest. Employing numerical and theoretical methods, we investigate the structural characteristics of random sequential addition (RSA) of congruent spheres in d -dimensional Euclidean space R{d} in the infinite-time or saturation limit for the first six space dimensions (1< or =d < or =6) . Specifically, we determine the saturation density, pair correlation function, cumulative coordination number and the structure factor in each of these dimensions. We find that for 2< or =d <or =6 , the saturation density phi{s} scales with dimension as phi{s}=c{1}/2{d}+c{2}d/2{d} , where c{1}=0.202048 and c{2}=0.973872 . We also show analytically that the same density scaling is expected to persist in the high-dimensional limit, albeit with different coefficients. A byproduct of this high-dimensional analysis is a relatively sharp lower bound on the saturation density for any d given by phi{s}> or =(d+2)(1-S{0})2;{d+1} , where S{0}[0,1] is the structure factor at k=0 (i.e., infinite-wavelength number variance) in the high-dimensional limit. We demonstrate that a Palàsti-type conjecture (the saturation density in R{d} is equal to that of the one-dimensional problem raised to the d th power) cannot be true for RSA hyperspheres. We show that the structure factor S(k) must be analytic at k=0 and that RSA packings for 1< or =d< or =6 are nearly "hyperuniform." Consistent with the recent "decorrelation principle," we find that pair correlations markedly diminish as the space dimension increases up to six. We also obtain kissing (contact) number statistics for saturated RSA configurations on the surface of a d -dimensional sphere for dimensions 2< or =d< or =5 and compare to the maximal kissing numbers in these dimensions. We determine the structure factor exactly for the related "ghost" RSA packing in R{d} and demonstrate that its distance from "hyperuniformity" increases as the space dimension increases, approaching a constant asymptotic value of 12 . Our work has implications for the possible existence of disordered classical ground states for some continuous potentials in sufficiently high dimensions.

Optimizing blind source separation with guided genetic algorithms

This paper proposes a novel method for blindly separating unobservable independent component (IC) signals based on the use of a genetic algorithm. It is intended for its application to the problem of blind source separation (BSS) on post-nonlinear mixtures. The paper also includes a formal proof on the convergence of the proposed algorithm using guiding operators, a new concept in the GA scenario. This approach is very useful in many fields such as forecasting indexes in financial stock markets, where the search for independent components is the major task to include exogenous information into the learning machine; or biomedical applications which usually use a high number of input signals. The guiding GA (GGA) presented in this work, is able to extract IC with faster rate than the previous ICA algorithms, as input space dimension increases. It shows significant accuracy and robustness than the previous approaches in any case. In addition, we present a simple though effective contrast function which evaluates individuals of each population (candidate solutions) based (a) on estimating the probability densities of the outputs through histogram approximation and (b) evaluating higher-order statistics of the outputs.

Solitary waves in media with dispersion and dissipation (a review)

The latest results relating to the theory of nonlinear waves in dispersive and dissipative media are reviewed. Attention is concentrated on small-amplitude solitary waves and, in particular, on the classification of types of solitary waves, their conditions of existence, the evolution of local perturbations associated with the presence of solitary waves of various types, and problems of the existence of nonlinear waves localized with respect to a particular direction as the space dimension increases (spontaneous dimension breaking). As examples of dispersive and dissipative media admitting plane solitary waves of various types, we consider a cold collisionless plasma, an ideal incompressible fluid of finite depth beneath an elastic plate and with surface tension, and a fluid in a rapidly oscillating rectangular vessel (Faraday resonance). Examples of spontaneous dimension breaking are considered for the generalized Kadomtsev-Petviashvili equation.

Discriminant Analysis When the Sample is Finite and the Feature Space Dimension Increases

Previous article Next article Discriminant Analysis When the Sample is Finite and the Feature Space Dimension IncreasesS. V. SemivskiiS. V. Semivskiihttps://doi.org/10.1137/1132076PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. D. Deev, Representation of statistics of discriminant analysis and asymptotic expansions when space dimensions are comparable to sample size, Soviet Math. Dokl., 11 (1970), 1547–1552 0236.62044 Google Scholar[2] D. D. Meshalkin and , V. I. Serdobol'skii, Errors in the classification of multi-variate observations, Theory Probab. Appl., 23 (1978), 741–750 10.1137/1123090 0451.62045 LinkGoogle Scholar[3] V. S. Korolyuk, Handbook on Probability Theory and Mathematical Statistics, Naukova Dumka, Kiev, 1978, (In Russian.) Google Scholar[4] B. Chandrasekaran and , A. Jain, Independence, measurement complexity, and classification performance, IEEE Trans. Systems Man Cybernet., SMC-5 (1975), 240–244 58:32051a 0298.68066 CrossrefGoogle Scholar[5] John W. Van Ness, Dimensionality and classification performance with independent coordinates, IEEE Trans. Systems Man Cybernet., SMC-7 (1977), 560–564 58:8573 0359.62047 Google Scholar[6] Willem Schaafsma and , Ton Steerneman, Discriminant analysis when the number of features is unbounded, IEEE Trans. Systems Man Cybernet., 11 (1981), 144–151 82k:92075 0477.62044 CrossrefGoogle Scholar[7] L. N. Bol'shev and , N. V. Smirnov, Tables of Mathematical Statistics, Nauka, Moscow, 1968, (In Russian.) Google Scholar[8] T. W. Anderson, An introduction to multivariate statistical analysis, Wiley Publications in Statistics, John Wiley & Sons Inc., New York, 1958xii+374 19,992a 0083.14601 Google Scholar[9] Peter Lancaster, Theory of matrices, Academic Press, New York, 1969xii+316 39:6885 0186.05301 Google Scholar[10] A. Ludwig, On the asymptotic distribution of the eigenvalues of random matrices, J. Math. Anal. Appl., 20 (1967), 262–268 10.1016/0022-247X(67)90089-3 36:922 0246.60029 CrossrefGoogle Scholar[11] A. Bikyalis, Estimate of the remainder term in the central limit theorem, Lit. Mat. Sb., 6 (1966), 323–346, (In Russian.) Google Scholar[12] S. V. Semovskii, Two problems in discriminant analysis, II Vilnius Conference on Probability Theory and Mathematical Statistics, Abstracts, Vol. 2, Vol. 2, Lith. Acad. Sci. Inst. Math. and Physics, 2, 1977, 153–156, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 32, Issue 3| 1988Theory of Probability & Its Applications History Submitted:02 April 1985Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1132076Article page range:pp. 521-525ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics