Tensor eigenvectors for projection pursuit

Abstract

Tensor eigenvectors naturally generalize matrix eigenvectors to multi-way arrays: eigenvectors of symmetric tensors of order k and dimension p are stationary points of polynomials of degree k in p variables on the unit sphere. Dominant eigenvectors of symmetric tensors maximize polynomials in several variables on the unit sphere, while base eigenvectors are roots of polynomials in several variables. In this paper, we focus on skewness-based projection pursuit and on third-order tensor eigenvectors, which provide the simplest, yet relevant connections between tensor eigenvectors and projection pursuit. Skewness-based projection pursuit finds interesting data projections using the dominant eigenvector of the sample third standardized cumulant to maximize skewness. Skewness-based projection pursuit also uses base eigenvectors of the sample third cumulant to remove skewness and facilitate the search for interesting data features other than skewness. Our contribution to the literature on tensor eigenvectors and on projection pursuit is twofold. Firstly, we show how skewness-based projection pursuit might be helpful in sequential cluster detection. Secondly, we show some asymptotic results regarding both dominant and base tensor eigenvectors of sample third cumulants. The practical relevance of the theoretical results is assessed with six well-known data sets.