Nonlinear dimensionality reduction is formulated here as the problem of trying to find a Euclidean feature-space embedding of a set of observations that preserves as closely as possible their intrinsic metric structure – the distances between points on the observation manifold as measured along geodesic paths. Our isometric feature mapping procedure, or isomap, is able to reliably recover low-dimensional nonlinear structure in realistic perceptual data sets, such as a manifold of face images, where conventional global mapping methods find only local minima. The recovered map provides a canonical set of globally meaningful features, which allows perceptual transformations such as interpolation, extrapolation, and analogy – highly nonlinear transformations in the original observation space – to be computed with simple linear operations in feature space. 1

. Several measures have been proposed for comparing nonlinear projection methods but so far no comparisons have taken into account one of their most important properties, the trustworthiness of the resulting neighborhood or proximity relationships. One of the main uses of nonlinear mapping methods is to visualize multivariate data, and in such visualizations it is crucial that the visualized proximities can be trusted upon: If two data samples are close to each other on the display they should be close-by in the original space as well. A local measure of trustworthiness is proposed and it is shown for three data sets that neighborhood relationships visualized by the Self-Organizing Map and its variant, the Generative Topographic Mapping, are more trustworthy than visualizations produced by traditional multidimensional scalingbased nonlinear projection methods. 1

Many settings of unsupervised learning can be viewed as quantization problems - the minimization

We propose an incremental method to find principal curves. Line segments are fitted and connected to form polygonal lines. New segments are inserted until a performance criterion is met. Experimental results illustrate the performance of the method compared to other existing approaches.

INTRODUCTION Consider a multivariate random variable X in R p with density function f and a random sample from X, namely X 1 , ..., X n . The first principal component can be viewed as the straight line which best fits the cloud of data (see, e.g., [17, pp. 386#387]). When the distribution of X is ellipsoidal the population first principal component is the main axis of the ellipsoids of equal concentration. In the past 40 years many works have appeared proposing extensions of principal components to distributions with nonlinear structure. We cite Shepard and Carroll [24], Gnanadesikan and Wilk [13], Srivastava [27], Etezadi-Amoli and McDonald [10], Yohai, Ackermann and Haigh [33], Koyak [19] and Gifi [12], among others. Some of them look for nonlinear transformations of the observable variables into spaces admitting a doi:10

In this paper we apply the latest techniques in sparse Gaussian process regression (GPR) to the Gaussian process latent variable model (GP-LVM). We review three techniques and discuss how they may be implemented in the context of the GP-LVM. Each approach is then implemented on a well known benchmark data set and compared with earlier attempts to sparsify the model. 1

We study the application of Self-Organizing Maps for the analyses of remote sensing spectral images. Advanced airborne and satellite-based imaging spectrometers produce very high-dimensional spectral signatures that provide key information to many scientific inves- tigations about the surface and atmosphere of Earth and other planets. These new, so- phisticated data demand new and advanced approaches to cluster detection, visualization, and supervised classification. In this article we concentrate on the issue of faithful topo- logical mapping in order to avoid false interpretations of cluster maps created by an SaM. We describe several new extensions of the standard SaM, developed in the past few years: the Growing Self-Organizing Map, magnification control, and Generalized Relevance Learn- ing Vector Quantization, and demonstrate their effect on both low-dimensional traditional multi-spectral imagery and 200-dimensional hyperspectral imagery.

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In this paper, we develop methods for outlier removal and noise reduction based on weighted local linear smoothing for a set of noisy points sampled from a nonlinear manifold. The methods can be used by manifold learning methods such as Isomap, LLE and LTSA as a preprocessing procedure so as to obtain a more accurate reconstruction of the underlying nonlinear manifolds. Weighted principal component analysis is used as a building block of our methods and we develop an iterative weight selection scheme that leads to robust local linear fitting. We also develop an e#cient and e#ective bias-reduction method to deal with the trim the peak and fill the valley phenomenon in local linear smoothing. Several illustrative examples are presented to show that nonlinear manifold learning methods combined with weighted local linear smoothing give more accurate reconstruction of the underlying nonlinear manifolds.

The recent explosion of publicly available biology gene sequences and chemical compounds offers an unprecedented opportunity for data mining. To make data analysis feasible for such vast volume and high-dimensional scientific data, we apply high performance dimension reduction algorithms. It facilitates the investigation of unknown structures in a three dimensional visualization. Among the known dimension reduction algorithms, we utilize the multidimensional scaling and generative topographic mapping algorithms to configure the given high-dimensional data into the target dimension. However, both algorithms require large physical memory as well as computational resources. Thus, the authors propose an interpolated approach to utilizing the mapping of only a subset of the given data. This approach effectively reduces