Unique Global Minimum Research Articles

Correlation at Low Temperature: II. Asymptotics

The present paper is a continuation of ref. 4, where the truncated two-point correlation function for a class of lattice spin systems was proved to have exponential decay at low temperature, under a weak coupling assumption. In this paper we compute the asymptotics of the correlation function as the temperature goes to zero. This paper thus extends ref. 3 in two directions: The Hamiltonian function is allowed to have several local minima other than a unique global minimum, and we do not require translation invariance of the Hamiltonian function. We are in particular able to handle spin systems on a general lattice.

Relevance of structural segregation and chain compaction for the thermodynamics of folding of a hydrophobic protein model.

The relevance of inside-outside segregation and chain compaction for the thermodynamics of folding of a hydrophobic protein model is probed by complete enumeration of two-dimensional chains of up to 18 monomers in the square lattice. The exact computation of Z scores for uniquely designed sequences confirms that Z tends to decrease linearly with sigma square root of N, as previously suggested by theoretical analysis and Monte Carlo simulations, where sigma, the standard deviation of the number of contacts made by different monomers in the target structure, is a measure of structural segregation and N is the chain length. The probability that the target conformation is indeed the unique global energy minimum of the designed sequence is found to increase dramatically with sigma, approaching unity at maximal segregation. However, due to the huge number of conformations with sub-maximal values of sigma, which correspond to intermediate, only mildly discriminative, values of Z, in addition to significant oscillations of Z around its estimated value, the probability that a correctly designed sequence corresponds to a maximally segregated conformation is small. This behavior of Z also explains the observed relation between sigma and different measures of folding cooperativity of correctly designed sequences.

Minimizers of the Lawrence–Doniach energy in the small-coupling limit: finite width samples in a parallel field

In this paper we study the Lawrence–Doniach model for layered superconductors, for a sample with finite width subjected to a magnetic field parallel to the superconducting layers. We provide a rigorous analysis of the energy minimizers in the limit as the coupling between adjacent superconducting layers tends to zero. We identify a unique global minimizer of the Gibbs free energy in this regime (“vortex planes”), and reveal a sequence of first-order phase transitions by which Josephson vortices are nucleated via the boundary. The small coupling limit is studied via degenerate perturbation theory based on a Lyapunov–Schmidt decomposition which reduces the Lawrence–Doniach system to a finite-dimensional variational problem. Finally, a lower bound on the radius of validity of the perturbation expansion (in terms of various parameters appearing in the model) is obtained.

The Recovery of an Anisotropic Conductivity in Groundwater Modelling

In order to model groundwater flow effectively, one is faced inevitably with the problem of suitably estimating the various subsurface parameters in the differential equations governing the flow from measurements, over time, of the piezometric head at various well sites, together with certain ancillary data. A new method is presented that allows for the estimation of an anisotropic hydraulic conductivity as the unique global minimum of a convex functional.

A finite newton method for classification

A fundamental classification problem of data mining and machine learning is that of minimizing a strongly convex, piecewise quadratic function on the n -dimensional real space R n . We show finite termination of a Newton method to the unique global solution starting from any point in R n . If the function is well conditioned, then no stepsize is required from the start and if not, an Armijo stepsize is used. In either case, the algorithm finds the unique global minimum solution in a finite number of iterations.

On a finite horizon production lot size inventory model for deteriorating items: An optimal solution

A generalized production lot size inventory model for deteriorating items over a finite planning horizon is considered. The demand, production, and deteriorating rates are assumed to be known and continuous functions of time. Shortages are allowed and completely backlogged. The conditions under which the system total cost attains its (unique) global minimum are derived. An example which illustrate the applicability of theoretical results is also introduced.

Adaptive minor component extraction with modular structure

An information criterion for adaptively estimating multiple minor eigencomponents of a covariance matrix is proposed. It is proved that the proposed criterion has a unique global minimum at the minor subspace and that all other equilibrium points are saddle points. Based on the gradient search approach of the proposed information criterion, an adaptive algorithm called adaptive minor component extraction (AMEX) is developed. The proposed algorithm automatically performs the multiple minor component extraction in parallel without the inflation procedure. Similar to the adaptive lattice filter structure, the AMEX algorithm also has the flexibility wherein increasing the number of the desired minor component does not affect the previously extracted minor components. The AMEX algorithm has a highly modular structure and the various modules operate completely in parallel without any delay. Simulation results are given to demonstrate the effectiveness of the AMEX algorithm for both the minor component analysis (MCA) and the minor subspace analysis (MSA).

Neural network learning dynamics in a path integral framework

A path-integral formalism is proposed for studying the dynamical evolution in time of patterns in an artificial neural network in the presence of noise. An effective cost function is constructed which determines the unique global minimum of the neural network system. The perturbative method discussed also provides a way for determining the storage capacity of the network.

Constrained minimum-BER multiuser detection

A new linear multiuser detector that directly minimizes the bit-error rate (BER) subject to a set of reasonable constraints is proposed. It is shown that the constrained BER cost function has a unique global minimum. This allows us to develop an efficient Newton barrier method for finding the coefficients of the proposed detector using information about timing, amplitudes, channels, and the signature signals of all users. Although the new detector cannot be shown to be optimal among linear multiuser detectors without the constraints imposed, extensive simulations demonstrate that it achieves the lowest BER. Furthermore, in some cases, the BER of the proposed detector can be significantly lower than that of the decorrelating and MMSE detectors.

Application of extreme value theory to level estimation in nonlinearly distorted hidden Markov models

This paper is concerned with the application of extreme value theory (EVT) to the state level estimation problem for discrete-time, finite-state Markov chains hidden in additive colored noise and subjected to unknown nonlinear distortion. If the nonlinear distortion affects only those observations with small magnitudes or those that lie outside a finite interval, we show that the level estimation problem can be reduced to a curve fitting problem with a unique global minimum. Compared with optimum maximum likelihood estimation algorithms, the developed level estimation algorithms are computationally inexpensive and are not affected by the unknown nonlinearity as long as the extreme values of observations are not distorted. This work has been motivated by unknown deadzone and saturation nonlinearities introduced by sensors in data measurement systems. We illustrate the effectiveness of the new EVT-based level estimation algorithms with computer simulations.

Optimizing Nonadaptive Group Tests for Objects with Heterogeneous Priors

We investigate nonadaptive group testing designs for heterogeneous mixtures of objects, independently positive with individual prior probabilities. In our model of the prior probabilities, the objects occur in one of several disjoint subsets and the number of positives in each subset is known. Furthermore, the positives are uniformly distributed within the subsets. The expected number of unresolved negative objects is minimized, and a unique global minimum is found for a family of stochastic, random incidence designs: all v group tests are constructed independently. The optimum incidence probabilities for the objects are well approximated by an asymptotic power series in v-1. We find the three leading coefficients of this series. The dependence of the optimum incidence probability upon the prior probability is, to leading order, logarithmic. Objects with larger prior probability of being positive have smaller optimum incidence probability. Furthermore, this logarithmic dependence can be nonnegligible fo...

Product of second moments in time and frequency for discrete-time signals and the uncertainty limit

The product of ordinary second moments in time and frequency for discrete-time signals has a unique global minimum for the unit sample sequence. When the ordinary-second-moment product is used as the measure of localization in the continuous-time case, Gaussian signals are the best localized simultaneously in time and frequency. But sampling of the Gaussian signals provides discrete-time signals which are far from being the best localized. In general, it turns out that the product of ordinary second moments in time and frequency is not a most reasonable measure of simultaneous time–frequency localization for discrete-time signals. Motivated by this discrepancy, in this paper an uncertainty relation for the product of generalized second signal moments in time and frequency is derived, and optimal signals that reach the uncertainty limit are found. The requirement for convolution invariance between the optimal signals offers a new uncertainty relation for the discrete-time case. The set of optimal discrete-time signals, beside the unit sample sequence, binomial sequences, the sampled band-limited minimum-effective-duration signal, contains also, as a limit case, the Gaussian signals. The second moments involved in the uncertainty relation are described in more details.

Gauge-fixing approach to lattice chiral gauge theories

We report on recent progress with the definition of lattice chiral gauge theories, using a lattice action that includes a discretized Lorentz gauge-fixing term. This gauge-fixing term has a unique global minimum, and allows us to use perturbation theory in order to study the influence of the gauge degrees of freedom on the fermions. For the abelian case, we find, both in perturbation theory and numerically, that the fermions remain chiral, and that there are no doublers.

Global Convexity in a Three-Dimensional Inverse Acoustic Problem

We consider an inverse scattering problem (ISP) for the acoustic equation $% u_{tt}=c_{}^2(x)\Delta u,u|_{t=0}=0,u_t|_{t=0}=\delta (x),x\in {\Bbb R}^3.$ The ISP consists of the determination of the speed of sound $c(x)$ inside a bounded domain $\Omega \subset {\Bbb R}^3$ given $c(x)$ outside $\Omega $ and measurements of the amplitude $u(x,t)$ of the sound at the boundary $% \partial \Omega ,\;u|_{_{\partial \Omega }}=\varphi (x,t).$ This problem is nonoverdetermined since only a single source location at $\left\{ 0\right\}$ is counted. Assuming regularity of the rays generated by $c(x)$ and using the Carleman's weight functions, we construct a cost functional $J_\lambda $. The main result is Theorem 3.1, which claims global strict convexity of $J_\lambda $ on "reasonable" compact sets of solutions. Therefore, global convergence on such a set of a number of standard minimization algorithms to the unique global minimum of $J_\lambda $ (i.e., solution of the ISP) is guaranteed. This in turn shows a possibility of constructions of numerical methods for this ISP which would not be affected by the problem of local minima.

Adaptive Laguerre-lattice filters

Adaptive Laguerre-based filters provide an attractive alternative to adaptive FIR filters in the sense that they require fewer parameters to model a linear time-invariant system with a long impulse response. We present an adaptive Laguerre-lattice structure that combines the desirable features of the Laguerre structure (i.e., guaranteed stability, unique global minimum, and small number of parameters M for a prescribed level of modeling error) with the numerical robustness and low computational complexity of adaptive FIR lattice structures. The proposed configuration is based on an extension to the IIR case of the FIR lattice filter; it is a cascade of identical sections but with a single-pole all-pass filter replacing the delay element used in the conventional (FIR) lattice filter. We utilize this structure to obtain computationally efficient adaptive algorithms (O(M) computations per time instant). Our adaptive Laguerre-lattice filter is an extension of the gradient adaptive lattice (GAL) technique, and it demonstrates the same desirable properties, namely, (1) excellent steady-state behavior, (2) relatively fast initial convergence (comparable with that of an RLS algorithm for Laguerre structure), and good numerical stability. Simulation results indicate that for systems with poles close to the unit circle, where an (adaptive) FIR model of very high order would be required to meet a prescribed modeling error, an adaptive Laguerre-lattice model of relatively low order achieves the prescribed bound after just a few updates of the recursions in the adaptive algorithm.

Parameter Estimation for Optimal Object Recognition: Theory andApplication

Object recognition systems involve parameters such as thresholds, bounds and weights. These parameters have to be tuned before the system can perform successfully. A common practice is to choose such parameters manually on an {\it add\ hoc} basis, which is a disadvantage. This paper presents a novel theory of parameter estimation for optimization-based object recognition where the optimal solution is defined as the global minimum of an energy function. The theory is based on supervised learning from examples. {\it Correctness} and {\it instability} are established as criteria for evaluating the estimated parameters. A correct estimate enables the labeling implied in each exemplary configuration to be encoded in a unique global energy minimum. The instability is the ease with which the minimum is replaced by a non-exemplary configuration after a perturbation. The optimal estimate minimizes the instability. Algorithms are presented for computing correct and minimal-instability estimates. The theory is applied to the parameter estimation for MRF-based recognition and promising results are obtained.

Behavioral analysis of anisotropic diffusion in image processing

In this paper, we analyze the behavior of the anisotropic diffusion model of Perona and Malik (1990). The main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem, so its behavior can be analyzed based on the shape of the corresponding energy surface. We show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem. It is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space. We give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition for it to be ill posed due to the dense global minima. The mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operator into two parts: one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator, and the other that flows normally to the edges and thus acts as an enhancement operator.

Folding kinetics of proteinlike heteropolymers

Using a simple three-dimensional lattice copolymer model and Monte Carlo dynamics, we study the collapse and folding of proteinlike heteropolymers. The polymers are 27 monomers long and consist of two monomer types. Although these chains are too long for exhaustive enumeration of all conformations, it is possible to enumerate all the maximally compact conformations, which are 3 ×3×3 cubes. This allows us to select sequences that have a unique global minimum. We then explore the kinetics of collapse and folding and examine what features determine the various rates. The folding time has a plateau over a broad range of temperatures and diverges at both high and low temperatures. The folding time depends on sequence and is related to the amount of energetic frustration in the native state. The collapse times of the chains are sequence independent and are a few orders of magnitude faster than the folding times, indicating a two-phase folding process. Below a certain temperature the chains exhibit glasslike behavior, characterized by a slowing down of time scales and loss of self-averaging behavior. We explicitly define the glass transition temperature (Tg), and by comparing it to the folding temperature (Tf), we find two classes of sequences: good folders with Tf≳Tg and non-folders with Tf<Tg.

A generalized Gaussian image model for edge-preserving MAP estimation

The authors present a Markov random field model which allows realistic edge modeling while providing stable maximum a posterior (MAP) solutions. The model, referred to as a generalized Gaussian Markov random field (GGMRF), is named for its similarity to the generalized Gaussian distribution used in robust detection and estimation. The model satisfies several desirable analytical and computational properties for map estimation, including continuous dependence of the estimate on the data, invariance of the character of solutions to scaling of data, and a solution which lies at the unique global minimum of the a posteriori log-likelihood function. The GGMRF is demonstrated to be useful for image reconstruction in low-dosage transmission tomography.

A new approach to the optimization of robust antenna array processors

The authors present a method for solving a class of optimization problems with nonsmooth constraints. In particular, they apply the method to the design of narrowband minimum-power antenna array processors which are robust in the presence of errors such as array element placement, look direction misalignment, and frequency offset. They first show that the constrained minimum power problem has a unique global minimum provided that the constraint set is nonempty. Then it is shown how the design problem derived directly from considerations of the sensitivity of the antenna array processor to errors can be transformed into a quadratic programming problem with linear inequality constraints which can be solved efficiently by the standard active set strategy. They also present numerical results for two types of nonsmooth constraints developed to provided robustness. These results confirm the effectiveness of the method. >