The Triangle Inequality versus Projection onto a Dimension in Determining Cosine Similarity Neighborhoods of Non-negative Vectors

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In many applications, objects are represented by non-negative vectors and cosine similarity is used to measure their similarity. It was shown recently that the determination of the cosine similarity of two vectors can be transformed to the problem of determining the Euclidean distance of normalized forms of these vectors. This equivalence allows applying the triangle inequality to determine cosine similarity neighborhoods efficiently. Alternatively, one may apply the projection onto a dimension to this end. In this paper, we prove that the triangle inequality is guaranteed to be a pruning tool, which is not less efficient than the projection in determining neighborhoods of non-negative vectors.

This work was supported by the National Centre for Research and Development (NCBiR) under Grant No. SP/I/1/77065/10 devoted to the Strategic scientific research and experimental development program: ‘Interdisciplinary System for Interactive Scientific and Scientific-Technical Information’.