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Clustering, multicollinearity, and singular vectors

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  • Usefi, Hamid

Abstract

Let A be a matrix with its Moore-Penrose pseudo-inverse A†. It is proved that, after re-ordering the columns of A, the projector P=I−A†A has a block-diagonal form, that is there is a permutation matrix Π such that ΠPΠT=diag(S1,S2,…,Sk). It is further proved that each block Si corresponds to a cluster of columns of A that are linearly dependent with each other. A clustering algorithm is provided that allows to partition the columns of A into clusters where columns in a cluster correlate only with columns within the same cluster. Some applications in supervised and unsupervised learning, specially feature selection, clustering, and sensitivity of solutions of least squares solutions are discussed.

Suggested Citation

  • Usefi, Hamid, 2022. "Clustering, multicollinearity, and singular vectors," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    • Handle: RePEc:eee:csdana:v:173:y:2022:i:c:s0167947322001037
    DOI: 10.1016/j.csda.2022.107523
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