Contractive Embedding Methods for Similarity Searching in Metric Spaces

Complex data types (e.g., images, documents, DNA sequences, etc) are becoming increasingly important in database applications. The term multimedia database is often used to characterize such databases. A typical query for such data seeks to nd objects that are similar to some target object, where (dis)similarity is de ned by some distance function. Often, the cost of evaluating the distance of 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, such 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, our focus is on embedding methods that allow returning the same query result as if the actual distances of the objects are consulted, thus ensuring that no relevant objects are left out (i.e., there are no false dismissals). Particular attention was paid to SparseMap, a variant of Lipschitz embeddings, and FastMap, which is designed to be a heuristic alternative to the KLT (and the equivalent PCA and SVD) method for dimensionality reduction. We show that neither SparseMap nor FastMap guarantee that queries on the embedded objects have no false dismissals. However, we describe a variant of SparseMap allows queries with no false dismissals. Moreover, we show that with FastMap, 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 to guarantee no false dismissals. This work was supported in part by the National Science Foundation under Grant IRI-97-12715.