An introduction to schoenberg's approximation

For a given function B and a non-zero real number h, Schoenberg's approximation defines from some data the function . For people not used to this kind of approximation, this paper intends to do a summary of the main definitions, properties and utilizations of Schoenberg's approximation: we show that the main tool to handle Schoenberg's approximation is the Fourier transform of B and even more its modified version, the transfer function of B; we give conditions for convergence of when h tends to zero, and we give various ways to define various B as combinations of translates of some function (usually is either some radial function, or obtained by a tensor product of some radial function), depending on the properties we want for the associated Schoenberg's approximation. Last, we show how multi-resolution analysis, subdivision techniques, and wavelets techniques, are nicely connected to Schoenberg's approximation.     

An efficient evolutionary algorithm for accurate polygonal approximation

An optimization problem for polygonal approximation of 2-D shapes is investigated in this paper. The optimization problem for a digital contour of N points with the approximating polygon of K vertices has a search space of C(N, K) instances, i.e., the number of ways of choosing K vertices out of N points. A genetic-algorithm-based method has been proposed for determining the optimal polygons of digital curves, and its performance is better than that of several existing methods for the polygonal approximation problems. This paper proposes an efficient evolutionary algorithm (EEA) with a novel orthogonal array crossover for obtaining the optimal solution to the polygonal approximation problem. It is shown empirically that the proposed EEA outperforms the existing genetic-algorithm-based method under the same cost conditions in terms of the quality of the best solution, average solution, variance of solutions, and the convergence speed, especially in solving large polygonal approximation problems.     

Some remarks on Fitzpatrick and Flynn’s Gröbner basis technique for Padé approximation

In Fitzpatrick and Flynn (J. Symbolic Comput. 13 (1992) 133), a Gröbner basis technique for multivariable Padéapproximation problems was developed under a rather restrictive hypothesis on the shape of the numerator and denominator in relation to the approximation conditions desired. In this article, we show that their hypotheses can be replaced by other less stringent conditions, and we show how to compute some standard forms of multivariable approximants through several examples.     

M/G/c/K blocking probability models and system performance

An exact solution for the M/G/c/K model is only possible for special cases, such as exponential service, a single server, or no waiting room at all. Instead of basing the approximation on an infinite capacity queue as is often the case, an approximation based on a closed-form expression derivable from the finite capacity exponential queue is presented. Properties of the closed-form expression along with its use in approximating the blocking probability of M/G/c/K systems are discussed. Extensive experiments are provided to test and verify the efficacy of our approximate results.     

Efficient high-dimensional indexing by sorting principal component

The vector approximation file (VA-file) approach is an efficient high-dimensional indexing method for image retrieval in large database. Some extensions of VA-file have been proposed towards better query performance. However, all of these methods applying sequential scan need read the whole vector approximation file. In this paper, we present a new indexing structure based on vector approximation method, in which only a small part of approximation file need be accessed. First, principal component analysis is used to map multidimensional points to a 1D line. Then a B⁺-tree is built to index the approximate vector according to principal component. When performing k-nearest neighbor search, the partial distortion searching algorithm is used to reject the improper approximate vectors. Only a small set of approximate vectors need to be sequentially scanned during the search, which can reduce the CPU cost and I/O cost dramatically. Experiment results on large image databases show that the new approach provides a faster search speed than the other VA-file approaches.     

An improved lower bound on approximation algorithms for the Closest Substring problem

The Closest Substring problem (the CSP problem) is a basic NP-hard problem in the study of computational biology. It is known that the problem has polynomial time approximation schemes. In this paper, we prove that unless the Exponential Time Hypothesis fails, the CSP problem has no polynomial time approximation schemes of running time f(1/ε)n^o(1/ε) for any function f. This essentially excludes the possibility that the CSP problem has a practical polynomial time approximation scheme even for moderate values of the error bound ε. As a consequence, it is unlikely that the study of approximation schemes for the CSP problem in the literature would lead to practical approximation algorithms for the problem for small error bound ε.     

Block-Quantized Support Vector Ordinal Regression

Support vector ordinal regression (SVOR) is a recently proposed ordinal regression (OR) algorithm. Despite its theoretical and empirical success, the method has one major bottleneck, which is the high computational complexity. In this brief, we propose a both practical and theoretical guaranteed algorithm, block-quantized support vector ordinal regression (BQSVOR), where we approximate the kernel matrix K with [(K)tilde] that is composed of k ² constant blocks. We provide detailed theoretical justification on the approximation accuracy of BQSVOR. Moreover, we prove theoretically that the OR problem with the block-quantized kernel matrix [(K)tilde] could be solved by first separating the data samples in the training set into k clusters with kernel k-means and then performing SVOR on the k cluster representatives. Hence, the algorithm leads to an optimization problem that scales only with the number of clusters, instead of the data set size. Finally, experiments on several real-world data sets support the previous analysis and demonstrate that BQSVOR improves the speed of SVOR significantly with guaranteed accuracy.     

Finding nondegenerate Hopf bifurcation points for four-dimensional two-parametric systems

We establish a numerically feasible algorithm to find a simplicial approximation A to a certain part of the boundary of the set of stable (or Hurwitz) polynomials of degree 4. Moreover, we have that . Using this, we build an algorithm to find a piecewise-linear approximation to the intersection curve of a given surface contained in ⁴ with . We have also devised an efficient computer program to perform all these operations. The main motivation is to find the curve of nondegenerate bifurcation points in parameter space for a given 2-parametric Hopf bifurcation problem of dimension 4.     

Alphabet indexing for approximating features of symbols

We consider two maximization problems to find a mapping from a large alphabet forming given two sets of strings to a set of a very few symbols specifying a symbol wise transformation of strings. First we show that the problem to find a mapping that transforms the most of the strings as to form disjoint sets cannot be approximated within a ratio n^1/16 in polynomial time, unless P = NP. Next we consider a mapping that retains the difference of the maximum number of pairs of strings over the given sets. We present a polynomial-time approximation algorithm that guarantees a ratio k/(k − 1) for mappings to k symbols, as well as proving that the problem is hard to approximate within an arbitrary small ratio in polynomial time. Furthermore, we extend this algorithm as to deal with not only pairs but also tuples of strings and show that it achieves a constant approximation ratio.     

Stabilization of a class of multi-inputs nonlinear systems

In this paper, we investigate the local stabilization of affine analytic systems with n − 1 inputs. We state a sufficient condition on the first approximation of f for which the local asymptotic stabilizability can be achieved; furthermore, we give explicitly the stabilizing feedback.     

t-Spanners for metric space searching

The problem of Proximity Searching in Metric Spaces consists in finding the elements of a set which are close to a given query under some similarity criterion. In this paper we present a new methodology to solve this problem, which uses a t-spanner G′(V, E) as the representation of the metric database. A t-spanner is a subgraph G′(V, E) of a graph G(V, A), such that E  A and G′ approximates the shortest path costs over G within a precision factor t.

Our key idea is to regard the t-spanner as an approximation to the complete graph of distances among the objects, and to use it as a compact device to simulate the large matrix of distances required by successful search algorithms such as AESA. The t-spanner properties imply that we can use shortest paths over G′ to estimate any distance with bounded-error factor t.

For this sake, several t-spanner construction, updating, and search algorithms are proposed and experimentally evaluated. We show that our technique is competitive against current approaches. For example, in a metric space of documents our search time is only 9% over AESA, yet we need just 4% of its space requirement. Similar results are obtained in other metric spaces.

Finally, we conjecture that the essential metric space property to obtain good t-spanner performance is the existence of clusters of elements, and enough empirical evidence is given to support this claim. This property holds in most real-world metric spaces, so we expect that t-spanners will display good behavior in most practical applications. Furthermore, we show that t-spanners have a great potential for improvements.     

Guarding galleries and terrains

Let P be a polygon with n vertices. We say that two points of P see each other if the line segment connecting them lies inside (the closure of) P. In this paper we present efficient approximation algorithms for finding the smallest set G of points of P so that each point of P is seen by at least one point of G, and the points of G are constrained to be belong to the set of vertices of an arbitrarily dense grind. We also present similar algorithms for terrains and polygons with holes.     

Multivariable systems design using the inverse Nyquist array

This paper describes some recent developments in the design of multivariable control systems using frequency-response methods.

Rosenbrock's inverse Nyquist array design technique, which makes equal use of the power of modern algebra and the well established analytical techniques of classical control theory, is introduced. Essentially, a multivariable controller is determined which reduces the design of an interacting system with m inputs and m outputs to the design of m single-loop controllers. Also, engineering constraints are more easily satisfied with this approach than with state-feedback design methods.

This design technique has been successfully applied to several industrial control problems, using the computer-aided design facilities developed at the Control Systems Centre, U.M.I.S.T.     

The unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan

In this paper we consider the unbounded single machine parallel batch scheduling problem with family jobs and release dates to minimize makespan. We show that this problem is strongly NP-hard, and give an O(n(n/m+1)^m) time dynamic programming algorithm and an O(mk^k+1P^2k−1) time dynamic programming algorithm, where n is the number of jobs, m is the number of families, k is the number of distinct release dates and P is the sum of the processing times of all families. We further give a heuristic with a performance ratio 2. We also give a polynomial-time approximation scheme for the problem.     

Finite element approximation of a nonlinear diffusion problem

This paper deals with the finite element approximation of the nonlinear diffusion problem: −div (\vbgrad u\vb^p−2 grad u) = f. Glowinski and Marrocco[3] have been shown that the rate of convergence decreases as p increases. In this paper we show that the rate of convergence is optimal and independent of p. This theoretical result agrees with the numerical experiments reported in the last section.     

Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues

In many calculations, spectral discretization in space is coupled with a standard ordinary differential equation formula in time. To analyze the stability of such a combination, one would like simply to test whether the eigenvalues of the spatial discretization operator (appropriately scaled by the time step k) lie in the stability region for the o.d.e. formula, but it is well known that this kind of analysis is in general invalid. In the present paper we rehabilitate the use of stability regions by proving that a discrete linear multistep ‘method of lines’ approximation to a partial differential equation is Lax-stable, within a small algebraic factor, if and only if all of the -pseudo-eigenvalues of the spatial discretization operator lie within O() of the stability region as → 0. An -pseudo-eigenvalue of a matrix A is any number that is an eigenvalue of some matrix A + E with E ; our arguments make use of resolvents and are closely related to the Kreiss matrix theorem. As an application of our general result, we show that an explicit N-point Chebyshev collocation approximation of u_t = −xu_x on [−1, 1] is Lax-stable if and only if the time step satisfies k = O(N⁻²), although eigenvalue analysis would suggest a much weaker restriction of the form k CN⁻¹.     

Pole shifting for matrices over real algebras

Pole shifting of multivariable control system over finite dimensional real algebras is studied. The main results are expressed in terms of the minimal polynomial of A + BF, where F is a feedback.     

Delphi与Matlab接口以及脱离Matlab运行

通过动态链接库，利用Delphi面向对象编程的特点以及Matlab强大的计算功能，详细的介绍了Delphi下调用Matlab函数的方法及特点，并以一个例程的实现，阐述了Delphi下通过动态链接库调用Matlab函数，并最终脱离Matlab环境下运行。     

k-Anonymization with Minimal Loss of Information

The technique of k-anonymization allows the releasing of databases that contain personal information while ensuring some degree of individual privacy. Anonymization is usually performed by generalizing database entries. We formally study the concept of generalization, and propose three information-theoretic measures for capturing the amount of information that is lost during the anonymization process. The proposed measures are more general and more accurate than those that were proposed by Meyerson and Williams and Aggarwal et al. We study the problem of achieving k-anonymity with minimal loss of information. We prove that it is NP-hard and study polynomial approximations for the optimal solution. Our first algorithm gives an approximation guarantee of O(ln k) for two of our measures as well as for the previously studied measures. This improves the best-known O(k)-approximation in. While the previous approximation algorithms relied on the graph representation framework, our algorithm relies on a novel hypergraph representation that enables the improvement in the approximation ratio from O(k) to O(ln k). As the running time of the algorithm is O(n^2k}), we also show how to adapt the algorithm in in order to obtain an O(k)-approximation algorithm that is polynomial in both n and k.     

基于图勾勒的图链路预测方法

针对已有链路预测算法复杂度高,不适于在大规模图上进行链接预测的问题,本文基于图勾勒近似技术对已有链路预测方法进行优化,提出了基于图勾勒的链路预测方法。该方法将链路预测算法的计算复杂度由O（n³）降低至O（n²k²log²n）。为进一步提高链接预测效率,给出了基于Spark的并行化链路预测实现方法。在真实图数据集上进行测试,实验结果表明本文方法在保证链接预测精度的前提下,可有效提升算法效率。