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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.15421 |
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| _version_ | 1866912831394283520 |
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| author | Hassan, Ishrak Alhajj |
| author_facet | Hassan, Ishrak Alhajj |
| contents | We study the discrete dynamical system obtained by repeatedly applying the Pearson correlation operator to a real matrix. Each step centers every row, normalizes each centered row to unit Euclidean norm, and forms the Gram matrix of the resulting rows. This produces a nonlinear map that underlies the classical CONCOR and GAP procedures. Despite its simple formulation and long history, the global behavior of this iteration has remained analytically unresolved.
We present a geometric formulation that separates directions associated with changes in row means and row norms from directions that preserve them. This formulation clarifies why local analysis does not extend to a global convergence theorem: the iteration is nonlinear, the structure of its fixed-point set is not fully characterized, and standard uniform contractive or Fejer-type techniques do not directly apply.
Empirically, the iteration stabilizes at a block plus or minus one pattern, exhibits finite total variation, and displays rapid decay once trajectories enter a neighborhood of a fixed pattern. We develop a dimension-uniform experimental framework and perform a large-scale numerical study over dimensions from 3 to 2000 with thousands of random initializations. Using the Frobenius step size, the entrywise step size, and the one-step ratio, we identify four universal empirical laws that persist uniformly across all tested dimensions. These observations provide a quantitative, dimension-uniform description of the iteration and formulate a precise target for future global analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15421 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Empirical Laws for Iterated Correlation Matrices Hassan, Ishrak Alhajj Dynamical Systems 37M10 We study the discrete dynamical system obtained by repeatedly applying the Pearson correlation operator to a real matrix. Each step centers every row, normalizes each centered row to unit Euclidean norm, and forms the Gram matrix of the resulting rows. This produces a nonlinear map that underlies the classical CONCOR and GAP procedures. Despite its simple formulation and long history, the global behavior of this iteration has remained analytically unresolved. We present a geometric formulation that separates directions associated with changes in row means and row norms from directions that preserve them. This formulation clarifies why local analysis does not extend to a global convergence theorem: the iteration is nonlinear, the structure of its fixed-point set is not fully characterized, and standard uniform contractive or Fejer-type techniques do not directly apply. Empirically, the iteration stabilizes at a block plus or minus one pattern, exhibits finite total variation, and displays rapid decay once trajectories enter a neighborhood of a fixed pattern. We develop a dimension-uniform experimental framework and perform a large-scale numerical study over dimensions from 3 to 2000 with thousands of random initializations. Using the Frobenius step size, the entrywise step size, and the one-step ratio, we identify four universal empirical laws that persist uniformly across all tested dimensions. These observations provide a quantitative, dimension-uniform description of the iteration and formulate a precise target for future global analysis. |
| title | Empirical Laws for Iterated Correlation Matrices |
| topic | Dynamical Systems 37M10 |
| url | https://arxiv.org/abs/2512.15421 |