We interpret several well-known algorithms for dimensionality reduction of manifolds as kernel methods. Isomap, graph Laplacian eigenmap, and locally linear embedding (LLE) all utilize local neighborhood information to construct a global embedding of the manifold. We show how all three algorithms can be described as kernel PCA on specially constructed Gram matrices, and illustrate the similarities and differences between the algorithms with representative examples.

In this paper, we show a direct relation between spectral embedding methods and kernel PCA, and how both are special cases of a more general learning problem, that of learning the principal eigenfunctions of an operator defined from a kernel and the unknown data generating density. Whereas

We describe an algorithm for nonlinear dimensionality reduction based on semidefinite programming and kernel matrix factorization. The algorithm learns a kernel matrix for high dimensional data that lies on or near a low dimensional manifold. In earlier work, the kernel matrix was learned by maximizing the variance in feature space while preserving the distances and angles between nearest neighbors. In this paper, adapting recent ideas from semi-supervised learning on graphs, we show that the full kernel matrix can be very well approximated by a product of smaller matrices. Representing the kernel matrix in this way, we can reformulate the semidefinite program in terms of a much smaller submatrix of inner products between randomly chosen landmarks. The new framework leads to order-of-magnitude reductions in computation time and makes it possible to study much larger problems in manifold learning. 1

Abstract We give a tutorial overview of several geometric methods for feature extraction and dimensional reduction. We divide the methods into projective methods and methods that model the manifold on which the data lies. For projective methods, we review projection pursuit, principal component analysis (PCA), kernel PCA, probabilistic PCA, and oriented PCA; and for the manifold methods, we review multidimensional scaling (MDS), landmark MDS, Isomap, locally linear embedding, Laplacian eigenmaps and spectral clustering. The Nyström method, which links several of the algorithms, is also reviewed. The goal is to provide a self-contained review of the concepts and mathematics underlying these algorithms.

We show how nonlinear embedding algorithms popular for use with shallow semisupervised learning techniques such as kernel methods can be applied to deep multilayer architectures, either as a regularizer at the output layer, or on each layer of the architecture. This provides a simple alternative to existing approaches to deep learning whilst yielding competitive error rates compared to those methods, and existing shallow semi-supervised techniques. 1.

Recently, the Isomap algorithm has been proposed for learning a nonlinear manifold from a set of unorganized high-dimensional data points. It is based on extending the classical multidimensional scaling method for dimension reduction. In this paper, we present a continuous version of Isomap which we call continuum isomap and show that manifold learning in the continuous framework is reduced to an eigenvalue problem of an integral operator. We also show that the continuum isomap can perfectly recover the underlying natural parametrization if the nonlinear manifold can be isometrically embedded onto an Euclidean space. Several numerical examples are given to illustrate the algorithm.

Relevant Component Analysis (RCA) has been proposed for learning distance metrics with contextual constraints for image retrieval. However, RCA has two important disadvantages. One is the lack of exploiting negative constraints which can also be informative, and the other is its incapability of capturing complex nonlinear relationships between data instances with the contextual information. In this paper, we propose two algorithms to overcome these two disadvantages, i.e., Discriminative Component Analysis (DCA) and Kernel DCA. Compared with other complicated methods for distance metric learning, our algorithms are rather simple to understand and very easy to solve. We evaluate the performance of our algorithms on image retrieval in which experimental results show that our algorithms are effective and promising in learning good quality distance metrics for image retrieval. 1

We introduce a new perspective on spectral dimensionality reduction which views these methods as Gaussian random fields (GRFs). Our unifying perspective is based on the maximum entropy principle which is in turn inspired by maximum variance unfolding. The resulting probabilistic models are based on GRFs. The resulting model is a nonlinear generalization of principal component analysis. We show that parameter fitting in the locally linear embedding is approximate maximum likelihood in these models. We directly maximize the likelihood and show results that are competitive with the leading spectral approaches on a robot navigation visualization and a human motion capture data set. 1

Abstract. This paper gives an overview of the eigenvalue problems encountered in areas of data mining that are related to dimension reduction. Given some input high-dimensional data, the goal of dimension reduction is to map them to a lowdimensional space such that certain properties of the initial data are preserved. Optimizing the above properties among the reduced data can be typically posed as a trace optimization problem that leads to an eigenvalue problem. There is a rich variety of such problems and the goal of this paper is to unravel relations between them as well as to discuss effective solution techniques. First, we make a distinction between projective methods that determine an explicit linear projection from the high-dimensional space to the low-dimensional space, and nonlinear methods where the mapping between the two is nonlinear and implicit. Then, we show that all of the eigenvalue problems solved in the context of explicit projections can be viewed as the projected analogues of the so-called nonlinear or implicit projections. We also discuss kernels as a means of unifying both types of methods and revisit some of the equivalences between methods established in this way. Finally, we provide some illustrative examples to showcase the behavior and the particular characteristics of the various dimension reduction methods on real world data sets.

Classical multidimensional scaling (MDS) is a method for visualizing high-dimensional point clouds by mapping to low-dimensional Euclidean space. This mapping is defined in terms of eigenfunctions of a matrix of interpoint dissimilarities. In this paper we analyze in detail multidimensional scaling applied to a specific dataset: the 2005 United States House of Representatives roll call votes. Certain MDS and kernel projections output “horseshoes” that are characteristic of dimensionality reduction techniques. We show that, in general, a latent ordering of the data gives rise to these patterns when one only has local information. That is, when only the interpoint distances for nearby points are known accurately. Our results provide a rigorous set of results and insight into manifold learning in the special case where the manifold is a curve. 1. Introduction. Classical