Geodesic Regression and the Theory of Least Squares on Riemannian Manifolds

This paper develops the theory of geodesic regression and least-squares estimation on Riemannian manifolds. Geodesic regression is a method for finding the relationship between a real-valued independent variable and a manifold-valued dependent random variable, where this relationship is modeled as a geodesic curve on the manifold. Least-squares estimation is formulated intrinsically as a minimization of the sum-of-squared geodesic distances of the data to the estimated model. Geodesic regression is a direct generalization of linear regression to the manifold setting, and it provides a simple parameterization of the estimated relationship as an initial point and velocity, analogous to the intercept and slope. A nonparametric permutation test for determining the significance of the trend is also given. For the case of symmetric spaces, two main theoretical results are established. First, conditions for existence and uniqueness of the least-squares problem are provided. Second, a maximum likelihood criteria is developed for a suitable definition of Gaussian errors on the manifold. While the method can be generally applied to data on any manifold, specific examples are given for a set of synthetically generated rotation data and an application to analyzing shape changes in the corpus callosum due to age.     

最小二乘隐空间支持向量机

在隐空间中采用最小二乘损失函数,提出了最小二乘隐空间支持向量机（LSHSSVMs）．同隐空间支持向量机（HSSVMs）一样,最小二乘隐空间支持向量机不需要核函数满足正定条件,从而扩展了支持向量机核函数的选择范围．由于采用了最小二乘损失函数,最小二乘隐空问支持向量机产生的优化问题为无约束凸二次规划,这比隐空间支持向量机产生的约束凸二次规划更易求解．仿真实验结果表明所提算法在计算时间和推广能力上较隐空间支持向量机存在一定的优势．     

GSOR Method for the Equality Constrained Least Squares Problems and the Generalized Least Squares Problems

In a recent paper [4], Li et al . gave a generalized successive overrelaxation (GSOR) method for the least squares problems. In this paper, we show that the GSOR method can be applied to the equality constrained least squares (LSE) problems and the generalized least squares (GLS) problems.     

基于最小二乘的多雷达观察精度分析

多传感器数据融合是目前指控系统获取目标真实信息的重要途径.雷达作为一种重要的传感器,其对目标的观察精度分析是数据融合的重中之重,而对于观察同一目标的多雷达精度分析尤为重要.论文通过建立外推估算法精度估计模型和基于最小二乘的离散优化多雷达精度排序模型给出了多雷达对目标观察精度的排序分析.     

Analysis of a Discontinuous Least Squares Spectral Element Method

This paper addresses the development of a Discontinuous Spectral Least-Squares method. Based on pre-multiplication with a mesh-dependent function a discontinuous functional can be set up. Coercivity of this functional will be established. An example of the approximation to a continuous solution and a solution in which a jump is prescribed will be presented. The discontinuous least-squares method preserves symmetry and positive definiteness of the discrete system.     

On the Least Trimmed Squares Estimator

The linear least trimmed squares (LTS) estimator is a statistical technique for fitting a linear model to a set of points. Given a set of n points in ? ^d and given an integer trimming parameter h≤n, LTS involves computing the (d?1)-dimensional hyperplane that minimizes the sum of the smallest h squared residuals. LTS is a robust estimator with a 50 %-breakdown point, which means that the estimator is insensitive to corruption due to outliers, provided that the outliers constitute less than 50 % of the set. LTS is closely related to the well known LMS estimator, in which the objective is to minimize the median squared residual, and LTA, in which the objective is to minimize the sum of the smallest 50 % absolute residuals. LTS has the advantage of being statistically more efficient than LMS. Unfortunately, the computational complexity of LTS is less understood than LMS. In this paper we present new algorithms, both exact and approximate, for computing the LTS estimator. We also present hardness results for exact and approximate LTS. A number of our results apply to the LTA estimator as well.     

Least Squares Fitting of Circles

Fitting standard shapes or curves to incomplete data (which represent only a small part of the curve) is a notoriously difficult problem. Even if the curve is quite simple, such as an ellipse or a circle, it is hard to reconstruct it from noisy data sampled along a short arc. Here we study the least squares fit (LSF) of circular arcs to incomplete scattered data. We analyze theoretical aspects of the problem and reveal the cause of unstable behavior of conventional algorithms. We also find a remedy that allows us to build another algorithm that accurately fits circles to data sampled along arbitrarily short arcs.Nikolai Chernov PhD in mathematics from Moscow University in 1984, scientist in Joint Institute for Nuclear Research (Dubna, Russia) 1983–1991, professor of mathematics in UCLA 1991–92, Georgia Tech 1992–93, Princeton University 1993–94, University of Alabama at Birmingham since 1994.Claire Lesort MS in mathematics from University of Limoges in 1994, MS in mathematics from University of Alabama at Birmingham 2000, PhD in Statistics from University of Alabama at Birmingham 2004. Statistician at BellSouth Telecommunication Inc. since 2003.     

Relations between Bias-Eliminating Least Squares, the Frisch scheme and Extended Compensated Least Squares methods for identifying errors-in-variables systems

There are many methods for identifying errors-in-variables systems. Among them Bias-Eliminating Least Squares (BELS), the Frisch scheme and Extended Compensated Least Squares (ECLS) methods are attractive approaches because of their simplicity and good estimation accuracy. These three methods are all based on a Bias-Compensated Least-Squares (BCLS) principle. In this paper, the relationships between them are considered. In particular, the set of nonlinear equations utilized in these three methods are proved to be equivalent under different noise conditions also for finite samples. It is shown that BELS, Frisch and ECLS methods have the same asymptotic estimation accuracy providing the same extended vector is used.     

Data‐Driven Adaptive Critic Approach for Nonlinear Optimal Control via Least Squares Support Vector Machine

This paper develops an online adaptive critic algorithm based on policy iteration for partially unknown nonlinear optimal control with infinite horizon cost function. In the proposed method, only a critic network is established, which eliminates the action network, to simplify its architecture. The online least squares support vector machine (LS‐SVM) is utilized to approximate the gradient of the associated cost function in the critic network by updating the input‐output data. Additionally, a data buffer memory is added to alleviate computational load. Finally, the feasibility of the online learning algorithm is demonstrated in simulation on two example systems.     

Scales and Scale‐like Structures

We present a method for generating scales and scale‐like structures on a polygonal mesh through surface replacement. As input, we require a triangular mesh that will be covered with scales and one or more proxy‐models to be used as the scale's shape. A user begins scale generation by drawing a lateral line on the model to control the distribution and orientation of scales on the surface. We then create a vector field over the surface to control an anisotropic Voronoi tessellation, which represents the region occupied by each scale. Next we replace these regions by cutting the proxy model to match the boundary of the Voronoi region and deform the cut model onto the surface. The result is a fully connected 2‐manifold that is suitable for subsequent post‐processing applications like surface subdivision.     

Scale‐aware Structure‐Preserving Texture Filtering

This paper presents a novel method to enhance the performance of structure‐preserving image and texture filtering. With conventional edge‐aware filters, it is often challenging to handle images of high complexity where features of multiple scales coexist. In particular, it is not always easy to find the right balance between removing unimportant details and protecting important features when they come in multiple sizes, shapes, and contrasts. Unlike previous approaches, we address this issue from the perspective of adaptive kernel scales. Relying on patch‐based statistics, our method identifies texture from structure and also finds an optimal per‐pixel smoothing scale. We show that the proposed mechanism helps achieve enhanced image/texture filtering performance in terms of protecting the prominent geometric structures in the image, such as edges and corners, and keeping them sharp even after significant smoothing of the original signal.     

Scale‐adaptive Structure‐preserving Texture Filtering

This paper proposes a scale‐adaptive filtering method to improve the performance of structure‐preserving texture filtering for image smoothing. With classical texture filters, it usually is challenging to smooth texture at multiple scales while preserving salient structures in an image. We address this issue in the concept of adaptive bilateral filtering, where the scales of Gaussian range kernels are allowed to vary from pixel to pixel. Based on direction‐wise statistics, our method distinguishes texture from structure effectively, identifies appropriate scope around a pixel to be smoothed and thus infers an optimal smoothing scale for it. Filtering an image with varying‐scale kernels, the image is smoothed according to the distribution of texture adaptively. With commendable experimental results, we show that, needing less iterations, our proposed scheme boosts texture filtering performance in terms of preserving the geometric structures of multiple scales even after aggressive smoothing of the original image.     

基于最小二乘支持向量机的函数型数据回归分析

部分函数线性模型是用于处理输入变量包含函数型和数值型两种数据类型而输出变量为数值的一类回归机.为提高该模型的预测精度,基于函数系数在再生核Hilbert空间上的表示,得到模型的结构化表示,将函数系数的估计转化为参数向量的估计问题,并运用最小二乘支持向量机方法得到参数估计形式.实验表明,文中算法对数值型数据的向量系数的估计与其他参数估计方法性能相近,但对函数型数据的函数系数的估计更加准确稳健,有助于确保学习机的预测精度.     

用DCT进行最小二乘相位估值的求解

已知相位差进行相位估值，在干涉雷达，自适应光学，补偿式成像，图象处理等方面起着关键作用。文中以２－范数的概念，推证了最小二乘相位估值问题可以转化成泊松方程，并用离散余弦变换（ＤＣＴ）的方法进行求解，效果很好。     

A Least Squares Solution for Camera Distortion Parameters

Most applications in optical metrology need a well calibrated camera. In particular, a calibrated camera includes a distortion mapping, parameters of which are determined in a final non-linear optimization over all camera parameters. In this article we present a closed form solution for the distortion parameters provided that all other camera parameters are known. We show that for radial, tangential, and thin prism distortions the determination of the parameters form a linear least squares problem. Therefore, a part of the camera calibration error function can be minimized by linear methods in closed form: We are able to decouple the calculation of the distortion parameters from the non-linear optimization. The number of parameters in the non-linear minimization are reduced. Several experimental results confirm the benefit of the approach.     

TISO-OEAR模型的分解递推最小二乘辨识方法

针对输出误差模型参数估计过程中的计算量较大的问题,提出了基于分解的两输入单输出（TISO）输出误差自回归模型（OEAR）的分解递推最小二乘（DRLS）算法．基本的思想是分解TISO系统为3个子系统,并通过递推最小二乘分别辨识每个子系统．DRLS算法是解决大规模系统的计算量大和复杂辨识模型的辨识难题的一种有效的方法．最后通过仿真实例验证和分析了所提出算法的有效性与优越性,并对两种算法的特点进行了总结．     

总体最小二乘波达方向估计方法

在到达时间差定位技术中,当传感器位置坐标存在误差时,最小二乘法(LS)所得估计值不再具有最优无偏性,导致测向精度下降。针对该问题,提出基于总体最小二乘的测向方法。将非线性的观测方程转化为伪线性方程,构成增广矩阵并对其进行奇异值分解,从而得到目标位置。理论分析和仿真结果表明,与经典的LS法和LS-Tylar法相比,该方法的定位精度较高。     

Derivation of Error Distribution in Least Squares Steganalysis

This paper considers the least squares method (LSM) for estimation of the length of payload embedded by least-significant bit replacement in digital images. Errors in this estimate have already been investigated empirically, showing a slight negative bias and substantially heavy tails (extreme outliers). In this paper, (approximations for) the estimator distribution over cover images are derived: this requires analysis of the cover image assumption of the LSM algorithm and a new model for cover images which quantifies deviations from this assumption. The theory explains both the heavy tails and the negative bias in terms of cover-specific observable properties, and suggests improved detectors. It also allows the steganalyst to compute precisely, for the first time, a p-value for testing the hypothesis that a hidden payload is present. This is the first derivation of steganalysis estimator performance     

非均匀周期采样系统的递阶最小二乘辨识方法

针对非均匀周期采样系统,通过状态空间模型离散化方法得到其输入输出表达形式.鉴于参数化后得到的辨识模型同时包含1个参数向量和1个参数矩阵,利用递阶辨识原理,将辨识模型分解为分别含有参数向量和参数矩阵的2个虚拟子系统;考虑到系统的因果约束问题,将包含参数矩阵的子系统分解为子子系统进行辨识,从而提出这类非均匀采样系统的递阶最小二乘辨识方法.仿真例子表明该算法是有效的.