Geometric Lower Bounds for Parametric Matroid Optimization

We relate the sequence of minimum bases of a matroid with linearly varying weights to three problems from combinatorial geometry: k -sets, lower envelopes of line segments, and convex polygons in line arrangements. Using these relations we show new lower bounds on the number of base changes in such sequences: Ω(nr ^1/3 ) for a general n -element matroid with rank r , and Ω(mα(n)) for the special case of parametric graph minimum spanning trees. The only previous lower bound was Ω(n log r) for uniform matroids; upper bounds of O(mn ^1/2 ) for arbitrary matroids and O(mn ^1/2 / log ^* n) for uniform matroids were also known. Received November 12, 1996, and in revised form February 19, 1997.     

学查字典

本文利用层次分析法对查字典的三种方式进行了综合排序 ,此外还从查找时间的长短进行了定量的比较 .     

Lower Bounds on the State Complexity of Geometric Goppa Codes

We reinterpret the state space dimension equations for geometric Goppa codes. An easy consequence is that if deg then the state complexity of is equal to the Wolf bound. For deg , we use Clifford's theorem to give a simple lower bound on the state complexity of . We then derive two further lower bounds on the state space dimensions of in terms of the gonality sequence of . (The gonality sequence is known for many of the function fields of interest for defining geometric Goppa codes.) One of the gonality bounds uses previous results on the generalised weight hierarchy of and one follows in a straightforward way from first principles; often they are equal. For Hermitian codes both gonality bounds are equal to the DLP lower bound on state space dimensions. We conclude by using these results to calculate the DLP lower bound on state complexity for Hermitian codes.     

Analysis of Bounds for Multilinear Functions

We analyze four bounding schemes for multilinear functions and theoretically compare their tightness. We prove that one of the four schemes provides the convex envelope and that two schemes provide the concave envelope for the product of p variables over .     

高斯核正则化学习算法的泛化误差

对广义凸损失函数和变高斯核情形下正则化学习算法的泛化性能展开研究.其目标是给出学习算法泛化误差的一个较为满意上界.泛化误差可以利用正则误差和样本误差来测定.基于高斯核的特性,通过构构建一个径向基函数(简记为RBF)神经网络,给出了正则误差的上界估计,通过投影算子和再生高斯核希尔伯特空间的覆盖数给出样本误差的上界估计.所获结果表明,通过适当选取参数σ和λ,可以提高学习算法的泛化性能.     

Seiffert平均关于几何、算术和二次平均特殊组合的确界

应用实分析的方法,研究第一和第二类Seiffert平均P和T关于算术平均A与几何平均G和算术平均A与二次平均Q特殊组合的序关系,得到了四个关于第一和第二类Seiffert平均与算术平均,几何平均或二次平均特殊组合的精确双向不等式.     

基于字典学习的地震数据随机噪声压制算法

针对传统变换基函数难以获得地震数据最优的稀疏表示,提出基于字典学习的随机噪声压制算法,将地震数据分块,每一块包含多个地震记录道在一定采样时间段内波形的信息,利用自适应字典学习技术,以地震数据块为训练样本,根据地震数据邻近块中记录道相似的特点,构造超完备字典,稀疏编码地震数据,从而恢复数据的主要特征,压制随机噪声.实验表明算法具有较高的PSNR值,并且能较好的保持地震数据纹理复杂区域的局部特征.     

Sharp Bounds on Geometric Permutations of Pairwise Disjoint Balls in R d

We prove that the maximum number of geometric permutations, induced by line transversals to a collection of n pairwise disjoint balls in \R ^d , is Θ (n ^d-1 ) . This improves substantially the upper bound of O(n ^2d-2 ) known for general convex sets [9]. We show that the maximum number of geometric permutations of a sufficiently large collection of pairwise disjoint unit disks in the plane is two, improving the previous upper bound of three given in [5]. Received September 21, 1998, and in revised form March 14, 1999.     

基于字典学习和TV的能谱CT重建算法

能谱CT可以将较宽的能谱数据划分为几个单独的窄谱数据,从而同时获得多个能量通道下的投影.但由于窄谱通道内接收到的光子数较少,投影通常包含较大的噪声.针对这一问题,基于压缩感知理论提出了一种基于字典学习和全变分TV(total-variation)的迭代重建算法用于能谱CT重建,应用交替最小化方法优化相关目标函数,并采用Split-Bregman算法求解.同时,采用有序子集方法加速迭代收敛过程,提高运算速率.为了验证和评估所提出的方法,使用简单模型和实际临床小鼠模型进行了仿真实验,实验结果表明,所提出的算法有较好的去噪及细节保存能力.     

Geometric Fourier Analysis for Computational Vision

Projective Fourier analysis — geometric Fourier analysis of the group SL(2,), the group identified in the conformal camera that provides image perspective transformations—is discussed in the framework of representation theory of semisimple Lie groups. The compact model of projective Fourier analysis is constructed, complementing the noncompact model proposed before. Detailed mathematical formulation of both models is presented. It is demonstrated that the projective Fourier analysis provides the data model for efficient perspectively covariant digital image representation well adapted to the retino-cortical mapping of biological visual system, and therefore, explicitly designed for the foveated sensors of a silicon retina, the use of which in active vision systems is presently limited due to the lack of such a model.     

伪Smarandache函数的上下界

对于正整数n,设Z(n)=min{m|m∈N,1/2m(m+1)≡0(modn)},称为n的伪Smarandache函数.设r是正整数.根据广义Ramanujan-Nagell方程的结果,运用初等数论方法证明了下列结果:i)1/2(-1+(8n+1)≤Z(n)≤2n-1.ii)当r≠1,2,3或5时,Z(2~r+1)≥1/2(-1+(2~(r+3)·5+41)).iii)当r≠1,2,3,4或12时,Z(2~r-1)≥1/2(-1+(2~(r+3)·3-23).     

Analysis and Optimization of Stability Robustness Bounds for Discretized Systems

In this paper, robustness bounds for the perturbations of continuous-time systems to ensure the stability of their discretized counterparts are developed. Both zero-order hold and P-step matrix integrators are considered. The effect of the sampling time on the robustness bounds is studied via examples. To determine how well a simulated system will retain the robustness properties of the continuous-time system being simulated, a new criterion for the selection of the simulation method and time step is introduced. Both implicit and explicit robustness measures for sampled-data systems are obtained.     

A Geometric Framework for Stochastic Shape Analysis

We introduce a stochastic model of diffeomorphisms, whose action on a variety of data types descends to stochastic evolution of shapes, images and landmarks. The stochasticity is introduced in the vector field which transports the data in the large deformation diffeomorphic metric mapping framework for shape analysis and image registration. The stochasticity thereby models errors or uncertainties of the flow in following the prescribed deformation velocity. The approach is illustrated in the example of finite-dimensional landmark manifolds, whose stochastic evolution is studied both via the Fokker–Planck equation and by numerical simulations. We derive two approaches for inferring parameters of the stochastic model from landmark configurations observed at discrete time points. The first of the two approaches matches moments of the Fokker–Planck equation to sample moments of the data, while the second approach employs an expectation-maximization based algorithm using a Monte Carlo bridge sampling scheme to optimise the data likelihood. We derive and numerically test the ability of the two approaches to infer the spatial correlation length of the underlying noise.

     

Bounds for multiplicities

Let and be a homogeneous -algebra. We establish bounds for the multiplicity of certain homogeneous -algebras in terms of the shifts in a free resolution of over . Huneke and we conjectured these bounds as they generalize the formula of Huneke and Miller for the algebras with pure resolution, the simplest case. We prove these conjectured bounds for various algebras including algebras with quasi-pure resolutions. Our proof for this case gives a new and simple proof of the Huneke-Miller formula. We also settle these conjectures for stable and square free strongly stable monomial ideals . As a consequence, we get a bound for the regularity of . Further, when is not Cohen-Macaulay, we show that the conjectured lower bound fails and prove the upper bound for almost Cohen-Macaulay algebras as well as algebras with a -linear resolution.

     

Bounds for permanents and determinants

An inequality of Johnson and Newman for determinants of real matrices is extended to complex matrices. A related inequality for permanents of real matrices is improved by means of a new rearrangement theorem.     

用于几何非线性分析的增量形式的虚功方程

本文导出了适用于几何非线性有限元分析的增量形式的虚功方程。在这个增量形式的虚功方程中,考虑了积累误差的影响。     

基于分类学习字典全局稀疏表示模型的图像修复算法研究

基于稀疏重构的图像修复依赖于图像全局自相似性信息的利用和稀疏分解字典的选择,为此提出了基于分类学习字典全局稀疏表示模型的图像修复思路.该算法首先将图像未丢失信息聚类为具有相似几何结构的多个子区域,并分别对各个子区域用K-SVD字典学习方法得到与各子区域结构特征相适应的学习字典.然后根据图像自相似性特点构建能够描述图像块空间组织结构关系的全局稀疏最大期望值表示模型,迭代地使用该模型交替更新图像块的组织结构关系和损坏图像的估计直到修复结果趋于稳定.实验结果表明,方法对于图像的纹理细节、结构信息都能起到好的修复作用.