空间结构化欧拉核及其应用  

Spatial Euler kernel and its applications

作　　者：刘爽[1] 陈松灿[1] 

机构地区：[1]南京航空航天大学计算机科学与技术学院,南京210016

出　　处：《中国科学：信息科学》2016年第2期179-192,共14页Scientia Sinica(Informationis)

基　　金：国家自然科学基金(批准号:61170151);高等学校博士点基金(批准号:20133218110032)资助项目

摘　　要：自Yang等将向量主成分分析(1D-PCA)推广至面向图像的2D-PCA以来,众多基于向量的1D形式算法被相继推广至对应的2D形式.因利用了图像/矩阵空间特定的结构先验知识,在处理矩阵型数据时2D算法自然导致了较传统1D算法更好的学习性能,这与"没有免费午餐定理"相符.但2D算法仍有其不足,主要表现在:(1)2D算法几乎都是线性的,因此对非线性数据处理的能力有限;(2)2D算法的空间结构信息利用仍不够充分.针对第一个不足,本文利用核方法进行改进,但相对于1D算法,2D算法因难以利用表示定理而导致核化困难,因此本文绕过表示定理,通过改变度量获得一个简洁的核化方法;针对第二个不足,本文采用在核空间直接对空间结构信息进行补偿的方法.但这需要在核空间中描述矩阵数据的空间结构,如果使用隐式核进行核化可能会导致矩阵数据空间结构扭曲,从而使对空间结构信息的描述和利用变得困难;如果使用显式核进行核化,会导致维数灾难而失去隐式核的优势.若核映射是一个显式、等维且各分量非耦合的映射,就能自然地描述出矩阵数据在核空间中的结构.幸运的是,存在众多符合以上要求的显式核(如Hellinger核和欧拉核)和隐式加性核(如Intersection核、JS核和χ2核)的近似显式形式.因欧拉核形式上的简洁性及其良好的行为,本文以欧拉核作为样例,首次尝试进行矩阵的核化及其在核空间的空间结构信息补偿.尽管存在若干空间结构信息的补偿方法,如空间结构信息约束、图像距离度量等,本文围绕现有的图像欧氏距离加以阐述,从而为矩阵或图像数据构建出对应的空间结构化欧拉核.最后将其应用于典型2D算法并通过实验验证了其有效性.Since the Principal Component Analysis（1D-PCA） extended to the image oriented 2D-PCA by Yang et al, a number of 1D methods have been extended to their corresponding 2D variants. Because of the use of natural spatial information of matrix-form data（e.g., an image）, the 2D methods usually yield much better performance than their 1D versions, which is consistent with the theorem of ＂No Free Lunch＂. However, the 2D methods still suffer from two main drawbacks：（1） they are almost all linear, which might not match the nonlinear structure of actual data; and（2） the spatial information of data is not fully used in existing 2D methods. To address the first drawback, although the kernel trick is theoretically feasible, it is practically difficult since the representation theorem cannot be straightforwardly extended for 2D-form data. To this end, we in this work propose to obtain a simple kernelizing method by changing the measurement without the representation theorem.In view of the second shortcoming, we aim to use the spatial information of data in the nonlinear mapping feature space（or say kernel space）. Unfortunately, it usually requires to describe the data in the nonlinear feature space（i.e., kernel space）, which is generally implemented through data dimension-increasing as well as implicit kernel mapping. To fulfill this goal, it usually refers to the implicit or explicit kernel mapping, the former, however,might distort the spatial structure of data while the latter leads to dimension risk. As a result, we can preferably preserve the spatial structure of data naturally if we employ the form of explicit kernels in which the dimension is identical and each component is uncoupled. Fortunately, many explicit kernel（e.g., Hellinger and Euler kernels）and approximate explicit mathematical formulations of some implicit additive kernels（e.g., Intersection, JS andχ2kernels） meet the requirements. Considering the conciseness and good generalization of the Euler kernel, we in this work at

关 键 词：2D算法 核方法 空间结构信息 欧拉核 图像欧氏距离 

分 类 号：TP391.41[自动化与计算机技术—计算机应用技术]