Showing papers on "Maxima and minima published in 2004"

Minima hopping: An efficient search method for the global minimum of the potential energy surface of complex molecular systems

Abstract: A method is presented that can find the global minimum of very complex condensed matter systems. It is based on the simple principle of exploring the configurational space as fast as possible and of avoiding revisiting known parts of this space. Even though it is not a genetic algorithm, it is not based on thermodynamics. The efficiency of the method depends strongly on the type of moves that are used to hop into new local minima. Moves that find low-barrier escape-paths out of the present minimum generally lead into low energy minima.

Maximum likelihood estimation of cascade point-process neural encoding models

Abstract: Recent work has examined the estimation of models of stimulus-driven neural activity in which some linear filtering process is followed by a nonlinear, probabilistic spiking stage. We analyze the estimation of one such model for which this nonlinear step is implemented by a known parametric function; the assumption that this function is known speeds the estimation process considerably. We investigate the shape of the likelihood function for this type of model, give a simple condition on the nonlinearity ensuring that no non-global local maxima exist in the likelihood—leading, in turn, to efficient algorithms for the computation of the maximum likelihood estimator—and discuss the implications for the form of the allowed nonlinearities. Finally, we note some interesting connections between the likelihood-based estimators and the classical spike-triggered average estimator, discuss some useful extensions of the basic model structure, and provide two novel applications to physiological data.

Metastability in Reversible Diffusion Processes I: Sharp Asymptotics for Capacities and Exit Times

Abstract: We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form 1 +rF(·)r on R d or subsets of R d , whereF is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of F can be related, up to multiplicative errors that tend to one as # 0, to the capacities of suitably constructed sets. We show that these capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of F at the starting minimum and the relevant saddle points. As a result, we are able to give the first rigorous proof of the classical Eyring-Kramers formulain dimension larger than 1. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes.

A general variational principle and some of its applications

Abstract: In this paper, given a reflexive real Banach space X and two sequentially weakly lower semicontinuous functionals Phi, Psi on X with Psi strongly continuous and coercive, we are mainly interested in the existence of infinitely many local minima of the functional capital Phi + r Psi for each sufficiently real r.

Racetrack Inflation

Abstract: We develop a model of eternal topological inflation using a racetrack potential within the context of type IIB string theory with KKLT volume stabilization The inflaton field is the imaginary part of the Kahler structure modulus, which is an axion-like field in the 4D effective field theory This model does not require moving branes, and in this sense it is simpler than other models of string theory inflation Contrary to single-exponential models, the structure of the potential in this example allows for the existence of saddle points between two degenerate local minima for which the slow-roll conditions can be satisfied in a particular range of parameter space We conjecture that this type of inflation should be present in more general realizations of the modular landscape We also consider `irrational' models having a dense set of minima, and discuss their possible relevance for the cosmological constant problem

A Gibbs Point Process for Road Extraction from Remotely Sensed Images

Abstract: In this paper we propose a new method for the extraction of roads from remotely sensed images. Under the assumption that roads form a thin network in the image, we approximate such a network by connected line segments. To perform this task, we construct a point process able to simulate and detect thin networks. The segments have to be connected, in order to form a line-network. Aligned segments are favored whereas superposition is penalized. These constraints are enforced by the interaction model (called the Candy model). The specific properties of the road network in the image are described by the data term. This term is based on statistical hypothesis tests. The proposed probabilistic model can be written within a Gibbs point process framework. The estimate for the network is found by minimizing an energy function. In order to avoid local minima, we use a simulated annealing algorithm, based on a Monte Carlo dynamics (RJMCMC) for finite point processes. Results are shown on SPOT, ERS and aerial images.

Matching shapes with self-intersections:application to leaf classification

Abstract: We address the problem of two-dimensional (2-D) shape representation and matching in presence of self-intersection for large image databases. This may occur when part of an object is hidden behind another part and results in a darker section in the gray level image of the object. The boundary contour of the object must include the boundary of this part which is entirely inside the outline of the object. The curvature scale space (CSS) image of a shape is a multiscale organization of its inflection points as it is smoothed. The CSS-based shape representation method has been selected for MPEG-7 standardization. We study the effects of contour self-intersection on the curvature scale space image. When there is no self-intersection, the CSS image contains several arch shape contours, each related to a concavity or a convexity of the shape. Self intersections create contours with minima as well as maxima in the CSS image. An efficient shape representation method has been introduced in this paper which describes a shape using the maxima as well as the minima of its CSS contours. This is a natural generalization of the conventional method which only includes the maxima of the CSS image contours. The conventional matching algorithm has also been modified to accommodate the new information about the minima. The method has been successfully used in a real world application to find, for an unknown leaf, similar classes from a database of classified leaf images representing different varieties of chrysanthemum. For many classes of leaves, self-intersection is inevitable during the scanning of the image. Therefore the original contributions of this paper is the generalization of the curvature scale space representation to the class of 2-D contours with self-intersection, and its application to the classification of Chrysanthemum leaves.

A parallel updating scheme for approximating and optimizing high fidelity computer simulations

Abstract: Approximation methods are often used to construct surrogate models, which can replace expensive computer simulations for the purposes of optimization. One of the most important aspects of such optimization techniques is the choice of model updating strategy. In this paper we employ parallel updates by searching an expected improvement surface generated from a radial basis function model. We look at optimization based on standard and gradient-enhanced models. Given Np processors, the best Np local maxima of the expected improvement surface are highlighted and further runs are performed on these designs. To test these ideas, simple analytic functions and a finite element model of a simple structure are analysed and various approaches compared.

An Efficient Method Based on Lattice Construction and the Genetic Algorithm for Optimization of Large Lennard-Jones Clusters.

Abstract: Geometric methods for the construction of three structural motifs, the icosahedron, Ino's decahedron, and the complete octahedron, are proposed. On the basis of the constructed lattices and the genetic algorithm, a method for optimization of large size Lennard-Jones (LJ) clusters is presented. Initially, the proposed method is validated by optimization of LJ13-309 clusters with the above structural motifs. Results show that the proposed method successfully located all the lowest known minima with an excellent performance; for example, based on Ino's decahedron with 147 lattice sites, the mean time consumed for successful optimization of LJ75 is only 0.61 s (Pentium III, 1 GHz), and the percentage success is 100%. Then, putative global minima of LJ310-561 clusters are predicted with the method. By theoretical analysis, these global minima are reasonable, although further verification or proof is still needed.

Recovering Volatility from Option Prices by Evolutionary Optimization

Abstract: We propose a probabilistic approach for estimating parameters of an option pricing model from a set of observed option prices. Our approach is based on a stochastic optimization algorithm which generates a random sample from the set of global minima of the in-sample pricing error and allows for the existence of multiple global minima. Starting from an IID population of candidate solutions drawn from a prior distribution of the set of model parameters, the population of parameters is updated through cycles of independent random moves followed by selection according to pricing performance. We examine conditions under which such an evolving population converges to a sample of calibrated models. The heterogeneity of the obtained sample can then be used to quantify the degree of ill-posedness of the inverse problem: it provides a natural example of a coherent measure of risk, which is compatible with observed prices of benchmark (vanilla) options and takes into account the model uncertainty resulting from incomplete identification of the model. We describe in detail the algorithm in the case of a diffusion model, where one aims at retrieving the unknown local volatility surface from a finite set of option prices, and illustrate its performance on simulated and empirical data sets of index options.

Nonlinear estimation of the fundamental matrix with minimal parameters

Abstract: The purpose of this paper is to give a very simple method for nonlinearly estimating the fundamental matrix using the minimum number of seven parameters. Instead of minimally parameterizing it, we rather update what we call its orthonormal representation, which is based on its singular value decomposition. We show how this method can be used for efficient bundle adjustment of point features seen in two views. Experiments on simulated and real data show that this implementation performs better than others in terms of computational cost, i.e., convergence is faster, although methods based on minimal parameters are more likely to fall into local minima than methods based on redundant parameters.

An Introduction to Nonlinear Analysis

Abstract: 1. Extrema 2. Critical points 3. Boundary value problems 4. Saddle points 5. Calculus of variations 6. Degree theory 7. Conditional extrema 8. Minimax methods 9. Jumping nonlinearities 10. Higher dimensions.

A stochastic optimization approach for parameter tuning of support vector machines

Abstract: Support vector machines (SVMs) are both mathematically well-funded and efficient in a large number of real-world applications. However, the classification results highly depend on the parameters of the model: the scale of the kernel and the regularization parameter. Estimating these parameters is referred to as tuning. Tuning requires to estimate the generalization error and to find its minimum over the parameter space. Classical methods use a local minimization approach. After empirically showing that the tuning of parameters presents local minima, we investigate in this paper the use of global minimization techniques, namely genetic algorithms and simulated annealing. This latter approach is compared to the standard tuning frameworks and provides a more reliable tuning method.

Pattern search methods for user-provided points: application to molecular geometry problems ∗

Abstract: This paper deals with the application of pattern search methods to the numerical solution of a class of molecular geometry problems with important applications in molecular physics and chemistry. The goal is to find a configuration of a cluster or a molecule with minimum total energy. The minimization problems in this class of molecular geometry problems have no constraints, and the objective function is smooth. The difficulties arise from the existence of several local minima and, especially, from the expensive function evaluation (total energy) and the possible nonavailability of first-order derivatives. We introduce a pattern search approach that attempts to exploit the physical nature of the problem by using energy lowering geometrical transformations and to take advantage of parallelism without the use of derivatives. Numerical results for a particular instance of this new class of pattern search methods are presented, showing the promise of our approach. The new pattern search methods can be used in any other context where there is a user-provided scheme to generate points leading to a potential objective function decrease.

Speed and convergence properties of gradient algorithms for optimization of IMRT.

Abstract: Gradient algorithms are the most commonly employed search methods in the routine optimization of IMRT plans. It is well known that local minima can exist for dose-volume-based and biology-based objective functions. The purpose of this paper is to compare the relative speed of different gradient algorithms, to investigate the strategies for accelerating the optimization process, to assess the validity of these strategies, and to study the convergence properties of these algorithms for dose-volume and biological objective functions. With these aims in mind, we implemented Newton's, conjugate gradient (CG), and the steepest decent (SD) algorithms for dose-volume- and EUD-based objective functions. Our implementation of Newton's algorithm approximates the second derivative matrix (Hessian) by its diagonal. The standard SD algorithm and the CG algorithm with "line minimization" were also implemented. In addition, we investigated the use of a variation of the CG algorithm, called the "scaled conjugate gradient" (SCG) algorithm. To accelerate the optimization process, we investigated the validity of the use of a "hybrid optimization" strategy, in which approximations to calculated dose distributions are used during most of the iterations. Published studies have indicated that getting trapped in local minima is not a significant problem. To investigate this issue further, we first obtained, by trial and error, and starting with uniform intensity distributions, the parameters of the dose-volume- or EUD-based objective functions which produced IMRT plans that satisfied the clinical requirements. Using the resulting optimized intensity distributions as the initial guess, we investigated the possibility of getting trapped in a local minimum. For most of the results presented, we used a lung cancer case. To illustrate the generality of our methods, the results for a prostate case are also presented. For both dose-volume and EUD based objective functions, Newton's method far outperforms other algorithms in terms of speed. The SCG algorithm, which avoids expensive "line minimization," can speed up the standard CG algorithm by at least a factor of 2. For the same initial conditions, all algorithms converge essentially to the same plan. However, we demonstrate that for any of the algorithms studied, starting with previously optimized intensity distributions as the initial guess but for different objective function parameters, the solution frequently gets trapped in local minima. We found that the initial intensity distribution obtained from IMRT optimization utilizing objective function parameters, which favor a specific anatomic structure, would lead to a local minimum corresponding to that structure. Our results indicate that from among the gradient algorithms tested, Newton's method appears to be the fastest by far. Different gradient algorithms have the same convergence properties for dose-volume- and EUD-based objective functions. The hybrid dose calculation strategy is valid and can significantly accelerate the optimization process. The degree of acceleration achieved depends on the type of optimization problem being addressed (e.g., IMRT optimization, intensity modulated beam configuration optimization, or objective function parameter optimization). Under special conditions, gradient algorithms will get trapped in local minima, and reoptimization, starting with the results of previous optimization, will lead to solutions that are generally not significantly different from the local minimum.

Energy minimization and flux domain structure in the intermediate state of a type-I superconductor

Abstract: The intermediate state of a type-I superconductor involves a fine-scale mixture of normal and superconducting domains. We take the viewpoint, due to Landau, that the realizable domain patterns are (local) minima of a nonconvex variational problem. We examine the scaling law of the minimum energy and the qualitative properties of domain patterns achieving that law. Our analysis is restricted to the simplest possible case: a superconducting plate in a transverse magnetic field. Our methods include explicit geometric constructions leading to upper bounds and ansatz-free inequalities leading to lower bounds. The problem is unexpectedly rich when the applied field is near-zero or near-critical. In these regimes there are two small parameters, and the ground state patterns depend on the relation between them.

Lower bounds for local search by quantum arguments

Abstract: The problem of finding a local minimum of a black-box function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube 0,1n, we show a lower bound of Ω(2n/4/n) on the number of queries needed by a quantum computer to solve this problem. More surprisingly, our approach, based on Ambainis's quantum adversary method, also yields a lower bound of Ω(2n/2/n2) on the problem's classical randomized query complexity. This improves and simplifies a 1983 result of Aldous. Finally, in both the randomized and quantum cases, we give the first nontrivial lower bounds for finding local minima on grids of constant dimension d≥3.

Indexing of Compressed Time Series

Abstract: We describe a procedure for identifying major minima and maxima of a time series, and present two applications of this procedure. The first application is fast compression of a series, by selecting major extrema and discarding the other points. The compression algorithm runs in linear time and takes constant memory. The second application is indexing of compressed series by their major extrema, and retrieval of series similar to a given pattern. The retrieval procedure searches for the series whose compressed representation is similar to the compressed pattern. It allows the user to control the trade-off between the speed and accuracy of retrieval. We show the effectiveness of the compression and retrieval for stock charts, meteorological data, and electroencephalograms.

Genetic algorithm optimization of a convolutional neural network for autonomous crack detection

Abstract: Detecting cracks is an important function in building, tunnel and bridge structural analysis. Successful automation of crack detection can provide a uniform and timely means for preventing further damage to structures. This laboratory has successfully applied convolutional neural networks (CNNs) to online crack detection. CNNs represent an interesting method for adaptive image processing and form a link between artificial neural networks, and finite impulse response filters. As with most artificial neural networks, the CNN is susceptible to multiple local minima, thus complexity and time must be applied in order to avoid becoming trapped within the local minima. This paper employs a standard genetic algorithm (GA) to train the weights of a 4-5x5 filter CNN in order to pass through the local minima. This technique resulted in a 92.3/spl plusmn/1.4% average success rate using 25 GA-trained CNNs presented with 100 crack (320x240 pixel) images.

Image Sharpening by Flows Based on Triple Well Potentials

Abstract: Image sharpening in the presence of noise is formulated as a non-convex variational problem. The energy functional incorporates a gradient-dependent potential, a convex fidelity criterion and a high order convex regularizing term. The first term attains local minima at zero and some high gradient magnitude, thus forming a triple well-shaped potential (in the one-dimensional case). The energy minimization flow results in sharpening of the dominant edges, while most noisy fluctuations are filtered out.

Iterative phase retrieval without support

Abstract: An iterative phase retrieval method for nonperiodic objects has been developed from the charge-flipping algorithm proposed in crystallography. A combination of the hybrid input-output (HIO) algorithm and the flipping algorithm has greatly improved performance. In this combined algorithm the flipping algorithm serves to find the support (object boundary) dynamically, and the HIO part improves convergence and moves the algorithm out of local minima. It starts with a single intensity measurement in the Fourier domain and does not require a priori knowledge of the support in the image domain. This method is suitable for general image recovery from oversampled diffuse elastic x-ray and electron-diffraction intensities. The relationship between this algorithm and the output-output algorithm is elucidated.

ℓâ‚€-norm Minimization for Basis Selection

Abstract: Finding the sparsest, or minimum l0-norm, representation of a signal given an overcomplete dictionary of basis vectors is an important problem in many application domains. Unfortunately, the required optimization problem is often intractable because there is a combinatorial increase in the number of local minima as the number of candidate basis vectors increases. This deficiency has prompted most researchers to instead minimize surrogate measures, such as the l1-norm, that lead to more tractable computational methods. The downside of this procedure is that we have now introduced a mismatch between our ultimate goal and our objective function. In this paper, we demonstrate a sparse Bayesian learning-based method of minimizing the l0-norm while reducing the number of troublesome local minima. Moreover, we derive necessary conditions for local minima to occur via this approach and empirically demonstrate that there are typically many fewer for general problems of interest.

Preservation of stochastic orders for random minima and maxima, with applications

Abstract: It is shown, in this note, that the right spread order and the increasing convex order are both preserved under the taking of random maxima, and the total time on test transform order and the increasing concave order are preserved under the taking of random minima. Some inequalities and preservation properties in reliability and economics are given as applications. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.