Manifold learning methods for unsupervised nonlinear dimensionality reduction have proven effective in the visualization of high dimensional data sets. When dealing with classification tasks, supervised extensions of manifold learning techniques, in which class labels are used to improve the embedding of the training points, require an appropriate method for out-of-sample mapping. In this paper we propose multi-output kernel ridge regression (KRR) for out-of-sample mapping in supervised manifold learning, in place of general regression neural networks (GRNN) that have been adopted by previous studies on the subject. Specifically, we consider a supervised agglomerative variant of Isomap and compare the performance of classification methods when the out-of-sample embedding is based on KRR and GRNN, respectively. Extensive computational experiments, using support vector machines and k-nearest neighbors as base classifiers, provide statistical evidence that out-of-sample mapping based on KRR consistently dominates its GRNN counterpart, and that supervised agglomerative Isomap with KRR achieves a higher accuracy than direct classification methods on most data sets.

Kernel ridge regression for out-of-sample mapping in supervised manifold learning

Extracting low-dimensional structures from high-dimensional space through spectral analysis has been prevalent in the fields of machine learning and computer vision. However, most manifold learning methods assume that there is a dominant low-dimensional manifold, while other variations are usually considered as noise or even ignored. This paper proposes a novel submanifold decomposition (SMD) algorithm, which simultaneously considers multiple submanifolds intertwined in the same high-dimensional space for decomposition. It makes full use of multi-category labels of a dataset to improve the modeling of manifolds of each label. In order to applied the proposed method to practical applications, the linear version of SMD is developed subsequently. Comparative experiments demonstrate that the proposed method not only effectively extracts submanifolds by subspace learning, but also outperforms traditional manifold and subspace learning methods for visual recognition tasks.

https://ieeexplore.ieee.org/iel5/6425799/6460043/06460490.pdf

Submanifold decomposition

Determining, or selecting a distance measure over the input feature space is a fundamental problem in pattern recognition. A notable metric, called the image euclidean distance (IMED) was proposed by Wang et al. [5], which is demonstrated consistent performance improvements in many real-world problems. In this paper, we present a fast implementation of IMED, which is referred as the convolution standardizing transform (CST). It can reduce the space complexity from O(n1 2n2 2 ) to O(1) , and the time complexity from O(n1 2n2 2 ) to O(n1n2), for n1 X n2 images. Both theoretical analysis and experimental results show the efficiency of our algorithm.

A Fast Algorithm for Image Euclidean Distance

The IMage Euclidean Distance (IMED) is a class of image metrics, in which the spatial relationship between pixels is taken into consideration. It was shown that calculating the IMED of two images is equivalent to performing a linear transformation called Standardizing Transform (ST) and then followed by the traditional Euclidean distance. However, while the IMED is invariant to image shift, the ST is not a Shift-Invariant (SI) filter. This left as an open problem whether IMED is equivalent to SI transformation plus traditional Euclidean distance. In this paper, we give a positive answer to this open problem. Specifically, for a wider class of metrics, including IMED, we construct closed-form SI transforms. Based on the SI metric-transform connection, we next develop an image metric learning algorithm by learning a metric filter in the transform domain. This is different from all previous metric approaches. Experimental results on benchmark datasets demonstrate that the learned image metric has promising performances.

http://vigir.missouri.edu/~gdesouza/Research/Conference_CDs/IEEE_CVPR_2009/data/papers/0743.pdf

Learning IMED via shift-invariant transformation

确定、或选择某种距离度量是模式识别、机器学习和聚类分析的基本问题.图像欧氏距离和广义欧氏距离是重要的两种距离度量,它们可以嵌入到大多数识别算法中去,并且在大量应用中都提升了识别效率.文中揭示了包括IMED和GED在内的一类距离度量与线性平移不变系统的关系.证明了以下结论:输入空间上的平移不变度量等价于对输入信号进行线性滤波再计算传统欧氏度量.这一结果扩展了对距离度量的认识,将在距离度量的选择上起到一定的作用.

度量滤波器: 距离度量的滤波视角

In recent years, some nonlinear dimension reduction methods, named manifold learning, have been proposed and widely used in data visualization and pattern recognition. Of them, Isomap is a representative, which can project data from high-dimensional space into low-dimensional space with local structure preserved perfectly. However, Isomap suffers from the topological stability and is sensitive to noise. Moreover, it can only run in a batch mode, so cannot be directly used in pattern classification. In this paper, firstly, an improved Isomap based on image distance, namely IMD-Isomap, is proposed. Because spatial information of images is considered in image distance, as our experiments will show, IMD-Isomap outperforms Isomap for data visualization especially when noise is added. Then, combining IMD-Isomap and generalized regression neural network, which has a good ability for approximation, a classification method is proposed. Experimental results showed that our methods are robust to noise for data visualization or image classification when compared with KNN, Isomap or eigenface.

IMD-Isomap for Data Visualization and Classification

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IMD-Isomap for Data Visualization and Classification