Basic Triangle Inequality Approach Versus Metric VP-Tree and Projection in Determining Euclidean and Cosine Neighbors

Abstract

The Euclidean distance and the cosine similarity are often applied for clustering or classifying objects or simply for determining most similar objects or nearest neighbors. In fact, the determination of nearest neighbors is typically a subtask of both clustering and classification. In this chapter, we discuss three principal approaches to efficient determination of nearest neighbors: namely, using the triangle inequality when vectors are ordered with respect to their distances to one reference vector, using a metric VP-tree and using a projection onto a dimension. Also, we discuss a combined application of a number of reference vectors and/or projections onto dimensions and compare two variants of VP-tree. The techniques are well suited to any distance metrics such as the Euclidean distance, but they cannot be directly used for searching nearest neighbors with respect to the cosine similarity. However, we have shown recently that the problem of determining a cosine similarity neighborhood can be transformed to the problem of determining a Euclidean neighborhood among normalized forms of original vectors. In this chapter, we provide an experimental comparison of the discussed techniques for determining nearest neighbors with regard to the Euclidean distance and the cosine similarity.