3173726
Approximate GCDs of polynomials and sparse SOS relaxations
The problem of computing approximate GCDs of several polynomials with real or complex coefficients can be formulated as computing the minimal perturbation such that the perturbed polynomials have an exact GCD of given degree. We present algorithms based on SOS (Sums Of Squares) relaxations for solving the involved polynomial or rational function optimization problems with or without constraints.
10784212
Approximate GCDs of polynomials and SOS relaxation
min p,u1,...,us ‖f1−p·u1‖2+‖f2−p·u2‖2+· · ·+‖fs−p·us‖2 (1) where p, u1, . . . , us are polynomials such that deg(p) ≤ k, deg(p · ui) ≤ di = deg fi for 1 ≤ i ≤ s. Here ‖f‖2 denotes the norm of the coefficient vector of polynomial f . The minimization problem has many different formulations and various numeric optimization strategies have been proposed, see [1, 3] and references therein. In particular, Karmarkar and Lakshman [4] proposed an algorithm based on the global minimization of a rational function to compute the approximate GCD of univariate polynomials. The most expensive part of their algorithm is to find all the real solutions of two bivariate polynomials with high degrees. It has been shown in [8] that sum of squares (SOS) relaxation[5, 9] can be used to find the global minimum of the rational function arising from the approximate GCD computation. The SOS relaxation can be solved by reformulating them as semidefinite programs (SDPs), which in turn are solved efficiently by using interior point methods [7]. Motivated by the interesting results obtained in [8], we show how to apply SOS relaxation to solve different optimization problems formulated in [1, 3, 4]. Let fi,ui,p be coefficient vectors corresponding to polynomials fi, ui, p respectively, and Ai = Ai(p) be convolution ∗ This research was partially supported by NKBRPC (2004CB318000) and the Chinese National Natural Science Foundation under Grant 10401035 (Li and Zhi).
14862024
Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars
We generalize the technique by Peyrl and Parillo [Proc. SNC 2007] to computing lower bound certificates for several well-known factorization problems in hybrid symbolic-numeric computation. The idea is to transform a numerical sum-of-squares (SOS) representation of a positive polynomial into an exact rational identity. Our algorithms successfully certify accurate rational lower bounds near the irrational global optima for benchmark approximate polynomial greatest common divisors and multivariate polynomial irreducibility radii from the literature, and factor coefficient bounds in the setting of a model problem by Rump (up to n = 14, factor degree = 13.
 The numeric SOSes produced by the current fixed precision semi-definite programming (SDP) packages (SeDuMi, SOSTOOLS, YALMIP) are usually too coarse to allow successful projection to exact SOSes via Maple 11's exact linear algebra. Therefore, before projection we refine the SOSes by rank-preserving Newton iteration. For smaller problems the starting SOSes for Newton can be guessed without SDP ("SDP-free SOS"), but for larger inputs we additionally appeal to sparsity techniques in our SDP formulation.
15051068
Study on Epipolar Geometry Restoration of Parallel Binocular Vision
The epipolar geometry restoration offline and online for the parallel binocular vision system were discussed in this paper. For the former, a method that uses a common calibration board to realize automatic and accurate matching of the image corner pairs was developed; for the latter, a connotative constraint was proved by geometry theory and the corresponding cross-correlation sparse matching algorithm was introduced. The constraint was that for any space point projected into two images, the horizontal coordinate in the left image of this point is bigger than that in the right image. Standard and real images experiments were made for each kind of problem and the results show that the algorithms for each problem were valid.
3439449
Classification of functional data: A segmentation approach
We suggest a classification and feature extraction method on functional data where the predictor variables are curves. The method, called functional segment discriminant analysis (FSDA), combines the classical linear discriminant analysis and support vector machine. FSDA is particularly useful for irregular functional data, characterized by spatial heterogeneity and local patterns like spikes. FSDA not only reduces the computation and storage burden by using a fraction of the spectrum, but also identifies important predictors and extracts features. FSDA is highly flexible, easy to incorporate information from other data sources and/or prior knowledge from the investigators. We apply FSDA to two public domain data sets and discuss the understanding developed from the study.
11047335
Dynamic modeling for locomotion-manipulation of a snake-like robot by using geometric methods
A snake-like robot can locomote in various environments; and it can manipulate objects when one end is fixed. A method of dynamic modeling for locomotion-manipulation of the snake-like robot is developed in order to unify the dynamic equations of two states. A virtual structure for orientation and position and the product-of-exponentials formula describe the mechanism and the kinematics of the robot. The dynamics of the robot are established in a Riemannian manifold. Furthermore, the dynamics of manipulation can be directly degenerated from those of locomotion. This method unifies the dynamics of locomotion and manipulation of the snake-like robot in the differential geometry formulation. Finally, simulation results of the method are presented.
16012768
Continuous Optimization based-on Boosting Gaussian Mixture Model
A new estimation of distribution algorithm (EDA) based on Gaussian mixture model (GMM) is proposed, in which boosting, an efficient ensemble learning method, is adopted to estimate GMM. By boosting simple GMM with two components, it has the ability of learning the model structure and parameters automatically without any requirement for prior knowledge. Moreover, since boosting can be viewed as a gradient search for a good fit of some objective in function space, the new EDA is time efficient. A set of experiments is implemented to evaluate the efficiency and performance of the new algorithm. The results show that, with a relatively smaller population and less number of generations, the new algorithm can perform as well as compared EDAs in optimizing multimodal functions
1881312
Design of a mobile mechanism possessing driving ability and detecting function for in-pipe inspection
In this paper, a mobile mechanism with driving capability and detecting function is proposed for in-pipe inspection task. Based on this mechanism, a robot is designed and fabricated. The advantage of this robot is that it has mobile ability in the pipe and detecting function for inspection, while only one DC motor is installed. This results in low energy consumption and low cost to make. The robot propels itself in the pipe under a driving mode, and it is used for finding the defect of the pipe under a detecting mode. By switching these two working modes, the robot performs the inspection task without other extra DC motors. Moreover, a velocity change mechanism is introduced to adapt the change of the payload through adjusting the incline angle of the roller. The characteristics of this mechanism are analyzed by comparison with a classical screw drive robot and a direct drive robot. Finally, basic experiments are conducted to testify the mobility and efficiency of this robot.