Fixed Points, Stability, Basin Geometry, and Global Convergence of the $3\times3$ Correlation Map
arXiv:2608.03404
2026
Dynamics
1 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper provides an exact dynamical mechanism for repeated row-wise Pearson correlation: centering induces rank reduction, every nondegenerate orbit remains defined, and in the 3x3 case all trajectories converge to one of seven explicitly classified fixed points. The most transferable asset is a recurrent correlation-normalization layer whose state is constrained to the correlation elliptope and whose rank collapse provides a measurable certificate of convergence or representation collapse. A practical neural use is an inference-time or training-time relational module operating on groups of three feature vectors, with the paper's kernel-coordinate dynamics used to predict convergence and detect exceptional unstable basins.
Ideas from this paper
Unverified
2026
Insert a recurrent layer that repeatedly replaces a three-by-three feature affinity matrix by the Pearson correlations of its rows. Unlike an unconstrained recurrent affinity update, the state remains a valid correlation matrix, becomes rank at most two after one step, and in dimension three converges globally to one of seven fixed points. Use the converged patterned fixed point as a differentiable or stop-gradient clustering/relational embedding, while monitoring rank and kernel-coordinate…
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