Local quadratic convergence of iterated correlation matrices: bounds and counterexamples
Abstract
Kruskal's unpublished manuscript on CONCOR gives a local argument for convergence of successive correlations to a prescribed two-class sign pattern. We identify a failure in its stated one-step estimate and establish a replacement bound for correlation matrices in arbitrary dimension. A geometric estimate for sign-aligned centered rows yields an explicit sufficient neighbourhood of convergence and a quadratic error bound. We construct invariant families of correlation matrices with an exact scalar recurrence in every dimension at least three. These families yield counterexamples to the original estimate in every dimension at least four. The failure persists on relatively open sets of positive-definite correlation matrices. In dimensions divisible by four, an explicitly solvable subfamily provides counterexamples arbitrarily close to the prescribed fixed point and attains the asymptotic coefficient in the new bound. Quadratic order is attained by the constructed families in every dimension at least three. These counterexamples concern the quantitative estimate, not the convergence conclusion: all the exhibited trajectories converge to their prescribed sign patterns.
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BibTeX
@misc{hassan2026local,
title = {{Local quadratic convergence of iterated correlation matrices: bounds and counterexamples}},
author = {Ishrak Alhajj Hassan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24592},
eprint = {2610.24592},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24592},
}
RIS
TY - UNPB
TI - Local quadratic convergence of iterated correlation matrices: bounds and counterexamples
AU - Ishrak Alhajj Hassan
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24592
UR - https://arxiv.org/abs/2610.24592
AB - Kruskal's unpublished manuscript on CONCOR gives a local argument for convergence of successive correlations to a prescribed two-class sign pattern. We identify a failure in its stated one-step estimate and establish a replacement bound for correlation matrices in arbitrary dimension. A geometric estimate for sign-aligned centered rows yields an explicit sufficient neighbourhood of convergence and a quadratic error bound. We construct invariant families of correlation matrices with an exact scalar recurrence in every dimension at least three. These families yield counterexamples to the original estimate in every dimension at least four. The failure persists on relatively open sets of positive-definite correlation matrices. In dimensions divisible by four, an explicitly solvable subfamily provides counterexamples arbitrarily close to the prescribed fixed point and attains the asymptotic coefficient in the new bound. Quadratic order is attained by the constructed families in every dimension at least three. These counterexamples concern the quantitative estimate, not the convergence conclusion: all the exhibited trajectories converge to their prescribed sign patterns.
ER -