一种新的局部空间排列算法

局部切空间排列算法(local tangent space alignment,LTSA)是一种经典的非线性流形学习方法,能够有效地对非线性分布数据进行降维,但它无法学习局部高曲率数据集.针对此问题,给出了描述数据集局部曲率的参数,并提出一种局部最小偏差空间排列(locally minimal deviation spacealignment,LMDSA)算法.该算法考虑到局部切空间低鲁棒性的缺陷,在计算局部最小偏差空间的同时,能够发现数据的局部高曲率现象,通过参数控制及邻域间的连接信息,减少计算局部高曲率空间的可能,进而利用空间排列技术进行降维,手工流形及真实数据集的实验证实了该算法学习局部高曲率数据集的有效性.     

局部切空间对齐算法的核主成分分析解释

基于核方法的降维技术和流形学习是两类有效而广泛应用的非线性降维技术,它们有着各自不同的出发点和理论基础,在以往的研究中很少有研究关注两者的联系。LTSA算法利用数据的局部结构构造一种特殊的核矩阵,然后利用该核矩阵进行核主成分分析。本文针对局部切空间对齐这种流形学习算法,重点研究了LTSA算法与核PCA的内在联系。研究表明,LTSA在本质上是一种基于核方法的主成分分析技术。     

一种改进的局部切空间排列算法

局部切空间排列(LTSA)算法是一种有效的流形学习算法,能较好地学习出高维数据的低维嵌入坐标。数据点的切空间在LTSA算法中起着重要的作用,其局部几何特征多是在样本点的切空间内表示。但是在实际中,LTSA算法是把数据点邻域的样本协方差矩阵的主元所张成的空间当做数据点的切空间,导致了在非均匀采样或样本邻域均值点与样本自身偏离程度较大时,原算法的误差增大,甚至失效。为此,提出一种更严谨的数据点切空间的计算方法,即数据点的邻域矩阵按照数据点本身进行中心化。通过数学推导,证明了在一阶泰勒展开的近似下,提出的计算方法所得到的空间即为数据点自身的切空间。在此基础上,提出了一种改进的局部切空间排列算法,并通过实验结果体现了该方法的有效性和稳定性。与已有经典算法相比,提出的计算方法没有增加任何计算复杂度。     

基于几何距离摄动的局部切空间排列算法

局部切空间排列算法(Local Tangent Space Alignment)是一种具有严格数学推理的流形学习算法,能有效地学习出高维数据的低维嵌入坐标,但也存在一些不足,如对近邻点的选取依赖性较强、不适应处理高曲率分布、稀疏分布数据源。针对这些缺点,提出了一种基于几何距离摄动的局部切空间排列算法。利用几何摄动条件把样本空间划分为一组线性分块的组合,在每一个线性块上应用LTSA算法完成降维。实验结果表明了该算法的有效性。     

改进的线性局部切空间排列算法

线性局部切空间排列算法（LLTSA）是一种能很好的适用于识别问题的非线性降维方法,但LLTSA仅仅关注了数据的局部几何结构,而没有体现数据的整体信息。本文提出了一种基于主成分分析（PCA）改进的线性局部切空间排列算法（P-LLTSA）,该算法在Linear-LTSA的基础上,考虑了样本的全局结构,进而得到更好的降维效果。在经典的三维流形和在MNIST图像库手写体识别的实验中,识别率较PCA、LPP,LLTSA有明显提高,证实了该算法在识别问题中的有效性。     

融合Gabor与局部切空间排列法的人脸识别算法

针对Gabor小波提取人脸特征存在维数高,计算复杂的问题,引入基于划分的局部切空间排列算法(Partitional Local Tangent Space Alignment)对得到的Gabor幅度特征(Gabor Magnitude Feature,GMF)进行降维,同时将主成分分析(PCA)和线性判别分析(LDA)引入到算法中,确定用最近邻分类器进行分类识别的最优投影子空间。通过在ORL人脸数据库上的实验证明了该算法的有效性,用Gabor小波提取特征对光照和表情变化等有良好的鲁棒性。     

基于局部切空间偏离度的自适应邻域选取算法

基于对局部切空间的几何性质的理论研究结果,提出一种基于局部切空间偏离度的自适应邻域选取算法.该算法基于局部切空间的正交投影计算局部中心化样本点与其切空间的夹角,更好地刻画出局部切空间的性质,能够区分不属于该邻域的样本点,同时具有较好的抗噪音能力.该算法是对该领域研究中的局部切空间排列算法的一个有效改进,具有局部高曲率的流形学习功能.实验证实该算法的有效性.     

空间光滑且完整的子空间学习算法

提出一种空间光滑且完整的子空间学习算法.它融合了主成分分析、空间光滑的子空间学习算法和局部敏感判别投影的技术特点.不但保持了数据流形的全局和局部几何结构,而且保持了它的判别信息和空间关系.从原始样本提取全局和局部特征经线性变换组成新样本,再从新样本中提取最佳分类特征,最后由分类器完成分类识别.同一般的子空间算法相比,该算法提高了识别率.实验结果验证了该算法的有效性.     

一种基于稀疏嵌入分析的降维方法

近几年局部流形学习算法研究得到了广泛的关注, 如局部线性嵌入以及局部切空间排列算法等.这些算法都是基于局部可线性化的假设而提出的, 但局部是否可线性化的问题没有得到很好有效的解决, 使得目前的降维算法对自然数据效果不佳. 自然数据中有很多是稀疏的,对稀疏数据的降维是局部线性嵌入算法所面临的一个问题. 基于对数据自然属性的考虑,利用数据的统计信息动态确定局部线性化范围, 依据数据的分布提出一种排列的稀疏局部线性嵌入算法(Sparse local linear embedding algorithm, SLLEA). 在数据集稀疏的情况下,该算法能够很好地把握数据的局部和整体信息. 将该算法应用于手工流形及图像检索等试验中,验证了该算法的有效性.     

一种邻域线性竞争的排列降维方法

局部线性嵌入算法以及局部切空间排列算法是目前对降维研究有着重要影响的算法, 但对于稀疏数据及噪声数据, 在使用这些经典算法降维时效果欠佳。一个重要问题就是这些算法在处理局部邻域时存在信息涵盖量不足。对经典算法中全局信息和局部信息的提取机制进行分析后, 提出一种邻域线性竞争的排列方法（neighborhood linear rival alignment algorithm, NLRA）。通过对数据点的近邻作局部结构提取, 有效挖掘稀疏数据内部信息, 使得数据整体降维效果更加稳定。通过手工流形和真实数据集的实验, 验证了算法的有效性和稳定性。     

Robust local tangent space alignment via iterative weighted PCA

Recently manifold learning has attracted extensive interest in machine learning and related communities. This paper investigates the noise manifold learning problem, which is a key issue in applying manifold learning algorithm to practical problems. We propose a robust version of LTSA algorithm called RLTSA. The proposed RLTSA algorithm makes LTSA more robust from three aspects: firstly robust PCA algorithm based on iterative weighted PCA is employed instead of the standard SVD to reduce the influence of noise on local tangent space coordinates; secondly RLTSA chooses neighborhoods that are well approximated by the local coordinates to align with the global coordinates; thirdly in the alignment step, the influence of noise on embedding result is further reduced by endowing clean data points and noise data points with different weights into the local alignment errors. Experiments on both synthetic data sets and real data sets demonstrate the effectiveness of our RLTSA when dealing with noise manifold.     

Regression reformulations of LLE and LTSA with locally linear transformation

Locally linear embedding (LLE) and local tangent space alignment (LTSA) are two fundamental algorithms in manifold learning. Both LLE and LTSA employ linear methods to achieve their goals but with different motivations and formulations. LLE is developed by locally linear reconstructions in both high- and low-dimensional spaces, while LTSA is developed with the combinations of tangent space projections and locally linear alignments. This paper gives the regression reformulations of the LLE and LTSA algorithms in terms of locally linear transformations. The reformulations can help us to bridge them together, with which both of them can be addressed into a unified framework. Under this framework, the connections and differences between LLE and LTSA are explained. Illuminated by the connections and differences, an improved LLE algorithm is presented in this paper. Our algorithm learns the manifold in way of LLE but can significantly improve the performance. Experiments are conducted to illustrate this fact.     

基于重叠片排列的流形学习算法

针对线性数据降维算法对处理非线性结构数据的降维效果不是很好,提出一种基于重叠片排列的流形学习算法,该算法根据局部的线性贴片处在非线性流形中的特性,将流形划分为线性互相重叠的局部区域贴片,且利用主成分分析方法得到局部区域贴片的低维表示,然后排列且对齐其低维坐标,以获得整体数据的低维坐标.通过仿真结果证明,基于重叠片排列的流形学习算法在应用于人脸识别和分类问题时以及在识别准确率方面要优于其他经典的流形学习算法.     

Locally affine patch mapping and global refinement for image super-resolution

This paper deals with the super-resolution (SR) problem based on a single low-resolution (LR) image. Inspired by the local tangent space alignment algorithm in [16] for nonlinear dimensionality reduction of manifolds, we propose a novel patch-learning method using locally affine patch mapping (LAPM) to solve the SR problem. This approach maps the patch manifold of low-resolution image to the patch manifold of the corresponding high-resolution (HR) image. This patch mapping is learned by a training set of pairs of LR/HR images, utilizing the affine equivalence between the local low-dimensional coordinates of the two manifolds. The latent HR image of the input (an LR image) is estimated by the HR patches which are generated by the proposed patch mapping on the LR patches of the input. We also give a simple analysis of the reconstruction errors of the algorithm LAPM. Furthermore we propose a global refinement technique to improve the estimated HR image. Numerical results are given to show the efficiency of our proposed methods by comparing these methods with other existing algorithms.     

一种改进的局部线性嵌入算法

局部线性嵌入算法(Local Linear Embedding,简称LLE)是一种非线性流形学习算法,能有效地学习出高维采样数据的低维嵌入坐标,但也存在一些不足,如不能处理稀疏的样本数据.针对这些缺点,提出了一种基于局部映射的线性嵌入算法(Local Project Linear Embedding,简称LPLE).通过假定目标空间的整体嵌入函数,重新构造样本点的局部邻域特征向量,最后将问题归结为损失矩阵的特征向量问题从而构造出目标空间的全局坐标.LPLE算法解决了传统LLE算法在源数据稀疏情况下的不能有效进行降维的问题,这也是其他传统的流形学习算法没有解决的.通过实验说明了LPLE算法研究的有效性和意义.     

Local tangent space alignment based on Hilbert–Schmidt independence criterion regularization

Pattern Analysis and Applications - Local tangent space alignment (LTSA) is a famous manifold learning algorithm, and many other manifold learning algorithms are developed based on LTSA. However,...     

Nonlinear Dimensionality Reduction by Locally Linear Inlaying

High-dimensional data is involved in many fields of information processing. However, sometimes, the intrinsic structures of these data can be described by a few degrees of freedom. To discover these degrees of freedom or the low-dimensional nonlinear manifold underlying a high-dimensional space, many manifold learning algorithms have been proposed. Here we describe a novel algorithm, locally linear inlaying (LLI), which combines simple geometric intuitions and rigorously established optimality to compute the global embedding of a nonlinear manifold. Using a divide-and-conquer strategy, LLI gains some advantages in itself. First, its time complexity is linear in the number of data points, and hence LLI can be implemented efficiently. Second, LLI overcomes problems caused by the nonuniform sample distribution. Third, unlike existing algorithms such as isometric feature mapping (Isomap), local tangent space alignment (LTSA), and locally linear coordination (LLC), LLI is robust to noise. In addition, to evaluate the embedding results quantitatively, two criteria based on information theory and Kolmogorov complexity theory, respectively, are proposed. Furthermore, we demonstrated the efficiency and effectiveness of our proposal by synthetic and real-world data sets.      

Extended local tangent space alignment for classification

The local tangent space alignment (LTSA) has demonstrated promising results in finding meaningful low-dimensional structures hidden in high-dimensional data. However, LTSA may have a limited effectiveness on the data which are organized in multiple classes or contain noisy points. In this paper, the distances between the samples and their neighbors are rescaled by using the reconstruction weights to overcome the limitation. An extension of LTSA is proposed based on the local rescaled distance matrix. Numerical experiments on both synthetic and real-world data sets are used to show the improvement of our extension for classification and the robustness to noisy data.