Complex data types—such as images, documents, DNA sequences, etc.—are becoming increasingly important in modern database applications. A typical query in many of these applications seeks to find objects that are similar to some target object, where (dis)similarity is defined by some distance function. Often, the cost of evaluating the distance between two objects is very high. Thus, the number of distance evaluations should be kept at a minimum, while (ideally) maintaining the quality of the result. One way to approach this goal is to embed the data objects in a vector space so that the distances of the embedded objects approximates the actual distances. Thus, queries can be performed (for the most part) on the embedded objects. In this paper, we are especially interested in examining the issue of whether or not the embedding methods will ensure that no relevant objects are left out (i.e., there are no false dismissals and, hence, the correct result is reported). Particular attention is paid to the SparseMap, FastMap, and MetricMap embedding methods. SparseMap is a variant of Lipschitz embeddings, while FastMap and MetricMap are inspired by dimension reduction methods for Euclidean spaces (using KLT or the related PCA and SVD). We show that, in general, none of these embedding methods guarantee that queries on the embedded objects have no false dismissals, while also demonstrating the limited cases in which the guarantee does hold. Moreover, we describe a variant of SparseMap that allows queries with no false dismissals. In addition, we show that with FastMap and MetricMap, the distances of the embedded objects can be much greater than the actual distances. This makes it impossible (or at least impractical) to modify FastMap and MetricMap to guarantee no false dismissals.

The bandwidth problem is the problem of enumerating the vertices of a given graph G such that the maximum difference between the numbers of adjacent vertices is minimal. The problem has a long history and a number of applications. There was not much known though on approximation hardness of this problem, till recently. Karpinski and Wirtgen [KW 97] showed that there are no polynomial time approximation algorithms with an absolute error guarantee of n 1\Gammaffl for any ffl ? 0 unless P = NP . In this paper we show, that there is no PTAS for the bandwidth problem unless P = NP , even for trees. More precisely we show that there are no polynomial time approximation algorithms for general graphs with an approximation ratio better than 1:5, and for the trees with an approximation ratio better than 4=3 ß 1:332. Dept. of Computer Science, University of Bonn, 53117 Bonn. Email: blache@cs.bonn.edu. y Dept. of Computer Science, University of Bonn, 53117 Bonn, and International Compute...

In this paper, we propose the concept of Manifold of Facial Expression based on the observation that images of a subject's facial expressions define a smooth manifold in the high dimensional image space. Such a manifold representation can provide a unified framework for facial expression analysis. We first apply Active Wavelet Networks (AWN) on the image sequences for facial feature localization. To learn the structure of the manifold in the feature space derived by AWN, we investigated two types of embeddings from a high dimensional space to a low dimensional space: locally linear embedding (LLE) and Lipschitz embedding. Our experiments show that LLE is suitable for visualizing expression manifolds. After applying Lipschitz embedding, the expression manifold can be approximately considered as a super-spherical surface in the embedding space. For manifolds derived from different subjects, we propose a nonlinear alignment algorithm that keeps the semantic similarity of facial expression from different subjects on one generalized manifold. We also show that nonlinear alignment outperforms linear alignment in expression classification.