Number Of Extreme Points Research Articles

On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaces

AbstractIn this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carathéodory.

Single-pulse-like and double-pulse-like characteristics of near-fault ground motions

An approach is proposed for quantitatively identifying the number of one-cycle pulses of ground motions. The one-cycle pulse signals are derived from the original velocity time series using the low-pass acausal fourth-order Butterworth filter and the zero-point method, and the adaptive cut-off frequency of the filter is determined by the number of extreme points in the extracted pulse signals. Then, according to the relative energy and the comprehensive evaluation index of the extracted pulse signals, ground motions are classified into four classes: nonpulse-like, single-pulse-like, double-pulse-like, and ambiguous. The statistical relationships between pulse characteristics and source parameters, including moment magnitudes, site conditions, and rupture distances, are discussed. In addition, the relationship between the first-arriving-pulse signal and the last-arriving-pulse signal in double-pulse-like ground motion is investigated. Finally, this approach analyzes the distribution of double-pulse-like and single-pulse-like ground motions recorded during the 1999 Chi-Chi earthquake.

Generalization Bounds in the Predict-Then-Optimize Framework

The predict-then-optimize framework is fundamental in many practical settings: predict the unknown parameters of an optimization problem and then solve the problem using the predicted values of the parameters. A natural loss function in this environment is to consider the cost of the decisions induced by the predicted parameters in contrast to the prediction error of the parameters. This loss function is referred to as the smart predict-then-optimize (SPO) loss. In this work, we seek to provide bounds on how well the performance of a prediction model fit on training data generalizes out of sample in the context of the SPO loss. Because the SPO loss is nonconvex and non-Lipschitz, standard results for deriving generalization bounds do not apply. We first derive bounds based on the Natarajan dimension that, in the case of a polyhedral feasible region, scale at most logarithmically in the number of extreme points but, in the case of a general convex feasible region, have linear dependence on the decision dimension. By exploiting the structure of the SPO loss function and a key property of the feasible region, which we denote as the strength property, we can dramatically improve the dependence on the decision and feature dimensions. Our approach and analysis rely on placing a margin around problematic predictions that do not yield unique optimal solutions and then providing generalization bounds in the context of a modified margin SPO loss function that is Lipschitz continuous. Finally, we characterize the strength property and show that the modified SPO loss can be computed efficiently for both strongly convex bodies and polytopes with an explicit extreme point representation. Funding: O. El Balghiti thanks Rayens Capital for their support. A. N. Elmachtoub acknowledges the support of the National Science Foundation (NSF) [Grant CMMI-1763000]. P. Grigas acknowledges the support of NSF [Grants CCF-1755705 and CMMI-1762744]. A. Tewari acknowledges the support of the NSF [CAREER grant IIS-1452099] and a Sloan Research Fellowship.

Comonotone approximation by hybrid polynomials

Let f(x):=g(x)+ax, where g∈C˜, the space of continuous 2π-periodic functions and a∈R. Denote ‖g‖:=maxx∈R⁡|g(x)|, and let ω(f,t), denote the modulus of continuity of f. Let Tn be the set of trigonometric polynomials Tn of degree <n. We call Qn(x):=Tn(x)+ax, a∈R, a hybrid polynomial.If f is monotone, then g was called, by Salem and Zygmund, of monotone type.If f has an even number of extremal points in (−π,π], then we estimateinf⁡{‖f−Qn‖:Qns.t.f′(x)Qn′(x)≥0, a.e. in R}, the error of its best comonotone approximation, in the uniform norm, by hybrid polynomials.We obtain Jackson-type estimates for the approximation of f by hybrid polynomials for a wide class of functions f. We also show cases where such estimates are invalid.

A Method of Infrared Small Target Detection in Strong Wind Wave Backlight Conditions

How to accurately detect small targets from the complex maritime environment has been a bottleneck problem. The strong wind-wave backlight conditions (SWWBC) is the most common situation in the process of distress target detection. In order to solve this problem, the main contribution of this paper is to propose a small target detection method suitable for SWWBC. First of all, for the purpose of suppressing the gray value of the background, it is analyzed that some minimum points with the lowest gray value tend to gather in the interior of the small target. As the distance from the extreme point increases, the gray value of the pixel in all directions also increases by the same extent. Therefore, an inverse Gaussian difference (IGD) preprocessing method similar to the distribution of the target pixel value is proposed to suppress the uniform sea wave and intensity of the sky background. So as to achieve the purpose of background suppression. Secondly, according to the feature that the small target tends to “ellipse shape” in both horizontal and vertical directions, a multi-scale and multi-directional Gabor filter is applied to filter out interference without “ellipse shape”. Combined with the inter-scale difference (IsD) operation and iterative normalization operator to process the results of the same direction under different scales, it can further suppress the noise interference, highlight the significance of the target, and fuse the processing results to enrich the target information. Then, according to different texture feature distributions of the target and noise in the multi-scale feature fusion results, a cross-correlation (CC) algorithm is proposed to eliminate noise. Finally, according to the dispersion of the number of extreme points and the significance of the intensity of the small target compared with the sea wave and sky noise, a new peak significance remeasurement method is proposed to highlight the intensity of the target and combined with a binary method to achieve accurate target segmentation. In order to better evaluate the performance index of the proposed method, compared with current state-of-art maritime target detection technologies. The experimental results of multiple image sequence sets confirm that the proposed method has higher accuracy, lower false alarm rate, lower complexity, and higher stability.

Design of photonic crystal fibers with flat dispersion and three zero dispersion wavelengths for coherent supercontinuum generation in both normal and anomalous regions

A photonic crystal fiber (PCF) structure with three-zero dispersion wavelengths (ZDWs) and a flat dispersion curve was designed by adjusting the structural parameters (the diameters of small air holes inserted between standard lattices in the first and third rings) and fiber core size to obtain the required shape, number of extreme points and the dynamic range of waveguide dispersion (Dw) curve. A quantitative method which was realized by fixing the positions of the ZDWs and tailoring different dispersion slopes was proposed to design another PCF with the same ZDWs and a slightly larger dispersion slope. Both of the designed PCFs were used for contrast experiments of supercontinuum (SC) generation. The simulation results show that the flat dispersion can produce the coherent SC in both normal and anomalous dispersion regions for the designed two PCFs. The degree of flatness and the wavelength range of coherent SC are determined by the great time-domain compression due to the direct soliton spectrum tunneling (DSST) resulting from the mismatch between self-phase modulation and gradually decreasing dispersion. A wide, flat, and coherent SC with a wavelength range of 1281–2200 nm and power range of −14.8 ~ −9.4 dB was obtained for the PCF with flatter dispersion by pumping in the anomalous dispersion region with 50 fs incident pulse. This study is instructive for PCF design in different application scenarios and finds a new way for the generation of wide flat coherent spectrum.

The Restless Hidden Markov Bandit With Linear Rewards and Side Information

In this paper we present a model for the hidden Markovian bandit problem with linear rewards. As opposed to current work on Markovian bandits, we do not assume that the state is known to the decision maker before making the decision. Furthermore, we assume structural side information where the decision maker knows in advance that there are two types of hidden states; one is common to all arms and evolves according to a Markovian distribution, and the other is unique to each arm and is distributed according to an i.i.d. process that is unique to each arm. We present an algorithm and regret analysis to this problem. Surprisingly, we can recover the hidden states and maintain logarithmic regret in the case of a convex polytope action set. Furthermore, we show that the structural side information leads to expected regret that does not depend on the number of extreme points in the action space. Therefore, we obtain practical solutions even in high dimensional problems.

MMW compressed sensing target reconstruction based on AMPSO search algorithm

ABSTRACT Introducing compressed sensing theory into the millimeter-wave near-field holographic imaging algorithm, it can break the Nyquist sampling limit, reconstruct the compressed echo signal, and invert the target image. In the reconstruction process, there are defects such as missing target key information, excessive invalid search volume and so on. Aiming at this problem, an adaptive multi-extreme particle swarm optimization (AMPSO) algorithm is proposed. Its advantages are that it can retain more target information, search for more extreme values, and improve the convergence speed. At the same time, the search probability in the strong scattering area is also increased, the search time is avoided in the noise area, and the number of extreme points is adjusted on a global scale. The effectiveness of the algorithm is verified by simulation and actual measurement of multiple types of targets under different experimental conditions.

Unifying neighbourhood and distortion models: part II – new models and synthesis

Neighbourhoods of precise probabilities are instrumental to perform robustness analysis, as they rely on very few parameters. In the first part of this study, we introduced a general, unified view encompassing such neighbourhoods, and revisited some well-known models (pari mutuel, linear vacuous, constant odds-ratio). In this second part, we study models that have received little to no attention, but are induced by classical distances between probabilities, such as the total variation, the Kolmogorov and the distances. We finish by comparing those models in terms of a number of properties: precision, number of extreme points, n-monotonicity, …thus providing possible guidelines to select a neighbourhood rather than another.

Unifying neighbourhood and distortion models: part I – new results on old models

ABSTRACT Neighbourhoods of precise probabilities are instrumental to perform robustness analysis, as they rely on very few parameters. Many such models, sometimes referred to as distortion models, have been proposed in the literature, such as the pari mutuel model, the linear vacuous mixtures or the constant odds ratio model. This paper is the first part of a two paper series where we study the sets of probabilities induced by such models, regarding them as neighbourhoods defined over specific metrics or premetrics. We also compare them in terms of a number of properties: precision, number of extreme points, n-monotonicity, behaviour under conditioning, etc. This first part tackles this study on some of the most popular distortion models in the literature, while the second part studies less known neighbourhood models and summarises our findings.

A Multi-Surface Plasticity Method for Lower Bound Shakedown Load

A new direct method for calculation of lower bound shakedown limits based on Melan’s theorem and a novel, non-smooth multi-surface plasticity model is proposed and implemented in a Finite Element environment. The load history is defined by a finite number of extreme points defining the load-envelope of a periodic load set. The shakedown problem is stated as a plasticity problem in terms of a finite number of independent yield conditions, solved for a residual stress field that satisfies a piecewise, non-smooth yield surface defined by the intersection of multiple yield surfaces. The implemented Finite Element procedure is applied to two shakedown problems and the results compared with lower and upper bound elastic shakedown solutions given by the Linear Matching Method, LMM. The example analyses show that the proposed Elastic-Shakedown Multi Surface Plasticity (EMSP) method defines robust lower bound shakedown limits between the LMM lower and upper bound limits, close to the LMM upper bound.

Non-smooth bursting analysis of a Filippov-type system with multiple-frequency excitations

The main purpose of this paper is to explore the patterns of the bursting oscillations and the non-smooth dynamical behaviours in a Filippov-type system which possesses parametric and external periodic excitations. We take a coupled system consisting of Duffing and Van der Pol oscillators as an example. Owing to the existence of an order gap between the exciting frequency and the natural one, we can regard a single periodic excitation as a slow-varying parameter, and the other periodic excitations can be transformed as functions of the slow-varying parameter when the exciting frequency is far less than the natural one. By analysing the subsystems, we derive equilibrium branches and related bifurcations with the variation of the slow-varying parameter. Even though the equilibrium branches with two different frequencies of the parametric excitation have a similar structure, the tortuousness of the equilibrium branches is diverse, and the number of extreme points is changed from 6 to 10. Overlying the equilibrium branches with the transformed phase portrait and employing the evolutionary process of the limit cycle induced by the Hopf bifurcation, the critical conditions of the homoclinic bifurcation and multisliding bifurcation are derived. Numerical simulation verifies the results well.

On the Lengths of Curves Passing Through Boundary Points of a Planar Convex Shape

We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that, for any convex shape K, there exist four points on the boundary of K such that the length of any curve passing through these points is at least half of the perimeter of K. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of K. Moreover, the factor ½ cannot be achieved with any fixed number of extreme points. We conclude the paper with a few other inequalities related to the perimeter of a convex shape.

An Acceleration-Based Gait Assessment Method for Children with Cerebral Palsy.

With the aim of providing an objective tool for motion disability assessment in clinical diagnosis and rehabilitation therapy of cerebral palsy (CP) patients, an acceleration-based gait assessment method was proposed in this paper. To capture gait information, three inertial measurement units (IMUs) were placed on the lower trunk and thigh, respectively. By comparing differences in the gait acceleration modes between children with CP and healthy subjects, an assessment method based on grey relational analysis and five gait parameters, including Pearson coefficient, variance ratio, the number of extreme points, harmonic ratio and symmetry was established. Twenty-two children with cerebral palsy (7.49 ± 2.86 years old), fourteen healthy adults (24.2 ± 1.55 years old) and ten healthy children (7.03 ± 1.49 years old) participated in the gait data acquisition experiment. The results demonstrated that, compared to healthy subjects, the symptoms and severity of motor dysfunction of CP children could result in abnormality of the gait acceleration modes, and the proposed assessment method was able to effectively evaluate the degree gait abnormality in CP children.

On FGP Approach to Multiobjective Quadratic Fractional Programming Problem

Multi objective quadratic fractional programming (MOQFP) problem involves optimization of several objective functions in the form of a ratio of numerator and denominator functions which involve both contains linear and quadratic forms with the assumption that the set of feasible solutions is a convex polyhedral with a nite number of extreme points and the denominator part of each of the objective functions is non-zero in the constraint set. In this paper, we extend the procedure as suggested by Lachhwani (Proc. Nat. Acad. Sci. India, 82(4), 317-322) based on fuzzy goal programming approach for the solution of multi objective quadratic fractional programming (MOQFP) problem. The proposed technique is simple, ecient and requires less computational work. In the proposed FGP model formulation, corresponding objectives of equivalent multi objective programming problem are transformed into fuzzy goals (membership functions) by means of assigning an aspiration level to each of them and suitable membership function is dened for each objectives. Then achievement of the highest membership value of each of fuzzy goals is formulated by minimizing the sum of negative deviational variables. The proposed methodology is illustrated with numerical example in order to support the proposed methodology.

Single metal catalysis: DFT and CAS modelling of species involved in the Fe cation assisted transformation of acetylene to benzene

ABSTRACTGas phase conversion of acetylene to benzene, assisted by a single metal cation such as Fe(+), Ru(+) and Rh(+), offers an attractive prospect for application of computational modelling techniques to catalytic processes. Gas phase processes are not complicated by environmental effects and the participation of a single metal atom is a significant simplification. Still the process is complex, owing to the possibility of several low-energy spin states and the abundance of alternative structures. By density functional theory modelling using recently developed models with range and dispersion corrections, we locate and characterise a number of extreme points on the FeC6H6(+) surface, some of which have not been described previously. These include eta-1, eta-2 and eta-3 complexes of Fe(+) with the C4H4 ring. We identify new FeC6H6(+) structures as well, which may be landmarks for the Fe(+)-catalysed production of benzene from acetylene. The Fe(+) benzene complex is the most stable species on the FeC6H6 cation surface. With the abundant energy of complexation available in the isolated gas phase species, detachment of the Fe(+) and production of benzene can be efficient. We address the issue raised by other investigators whether multi-configurational self-consistent field methods are essential to the proper description of these systems. We find that the relative energy of intrinsically multi-determinant doublets is strongly affected, but judge that the density functional theory (DFT) description provides more accurate estimates of energetics and a more plausible reaction path.

New Optimal Pivot Rule for the Simplex Algorithm

The purpose of this paper is to introduce a new pivot rule of the simplex algorithm. The simplex algorithm first presented by George B. Dantzig, is a widely used method for solving a linear programming problem (LP). One of the important steps of the simplex algorithm is applying an appropriate pivot rule to select the basis-entering variable corresponding to the maximum reduced cost. Unfortunately, this pivot rule not only can lead to a critical cycling (solved by Bland’s rules), but does not improve efficiently the objective function. Our new pivot rule 1) solves the cycling problem in the original Dantzig’s simplex pivot rule, and 2) leads to an optimal improvement of the objective function at each iteration. The new pivot rule can lead to the optimal solution of LP with a lower number of iterations. In a maximization problem, Dantzig’s pivot rule selects a basis-entering variable corresponding to the most positive reduced cost; in some problems, it is well-known that Dantzig’s pivot rule, before reaching the optimal solution, may visit a large number of extreme points. Our goal is to improve the simplex algorithm so that the number of extreme points to visit is reduced; we propose an optimal improvement in the objective value per unit step of the basis-entering variable. In this paper, we propose a pivot rule that can reduce the number of such iterations over the Dantzig’s pivot rule and prevent cycling in the simplex algorithm. The idea is to have the maximum improvement in the objective value function: from the set of basis-entering variables with positive reduced cost, the efficient basis-entering variable corresponds to an optimal improvement of the objective function. Using computational complexity arguments and some examples, we prove that our optimal pivot rule is very effective and solves the cycling problem in LP. We test and compare the efficiency of this new pivot rule with Dantzig’s original pivot rule and the simplex algorithm in MATLAB environment.

Alexandrov spaces with maximal number of extremal points

We show that any n-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of Euclidean n -space by an action of a crystallographic group. We describe all such actions.

변동성과 전환점에 기반한 한국어 음소 'ㅅ', 'ㅈ', 'ㅊ' 음성 인식

A phoneme is the minimal unit of speech, and it plays a very important role in speech recognition. This paper proposes a novel method that can be used to recognize 'ㅅ', 'ㅈ', and 'ㅊ' among Korean phonemes. The proposed method is based on a volatility indicator and a turning point indicator that are calculated for each constituting block of the input speech signal. The volatility indicator is the sum of the differences between the values of each two samples adjacent in a block, and the turning point indicator is the number of extremal points at which the direction of the increment or decrement of the values of the sample are inverted in a block. A phoneme recognition algorithm combines the two indicators to finally determine the positions at which the three target phonemes mentioned above are recognized by utilizing optimized thresholds related with those indicators. The experimental results show that the proposed method can markedly reduce the error rate of the existing methods both in terms of the false reject rate and the false accept rate.

Number of extremal subsets in Alexandrov spaces and rigidity

In this paper we announce the following result. We show that any $n$-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of $\mathbb{R}^n$ by an action of a crystallographic group. We describe all such actions. We start with a history, results and open questions concerning estimates on the number of extremal subsets in Alexandrov spaces. Then we give the plan of the proof of our result; the complete proof will published elsewhere.