arXiv subjects

Ishrak Alhajj Hassan

Publications and source records attributed to Ishrak Alhajj Hassan.

3 recordsLinked to original sources

arXiv2026

A Two-Channel F-Transform Representation for Early Trajectory Characterization in Iterated Correlation Dynamics

Many nonlinear iterative systems generate high-dimensional trajectories whose early behavior is informative but difficult to compare directly. This paper derives a fixed-dimensional F-transform coordinate representation for early trajectories of iterated Pearson correlation matrices. The construction is defined on the first five-point post-transient signal window, which is the shortest sampled window that simultaneously places the three symmetric fuzzy nodes at observed positions and supports a nondegenerate centered first-degree F-transform coefficient, thereby providing the earliest feasible local level--trend characterization within this sampled geometry. The representation combines two logarithmic observables of the post-transient dynamics: step size and contraction ratio. Applying this same four-coordinate construction to the step-size and contraction-ratio signals yields the eight-dimensional descriptor $Ψ=(v_1,v_2,v_3,s_2,u_1,u_2,u_3,r_2)$, with a common coordinate form across matrix sizes. For the fixed construction, $Ψ=M(q_2,\ldots,q_7)^{\top}$, $q_k=\logδ_k$, with $\operatorname{rank}M=6$. Thus, the descriptor is an injective, overcomplete representation of the six logged step sizes underlying the two channels. The representation is Lipschitz stable, and the centered first-degree coefficient recovers affine trends exactly. Convergence-length approximation is used as a downstream test of retained dynamical information. Across 22 matrix dimensions and 22,000 trajectories, repeated train--test evaluation shows predictive performance comparable to raw two-channel and statistical-summary representations. PCA shows that the first two principal components explain on average $84.47\%$ of the descriptor variance. Clustering reveals reproducible coarse organization, with the strongest mean silhouette at $k=2$ and high stability for smaller numbers of clusters.

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arXiv2026

Fixed Points, Stability, Basin Geometry, and Global Convergence of the $3\times3$ Correlation Map

For more than half a century, iterated Pearson correlation has underpinned methods for network blockmodeling, clustering, seriation, and information visualization. Despite its continued use and the longstanding assumption of convergence, the global convergence problem remained open even in dimension three. We resolve this problem completely for the $3\times3$ correlation map: every admissible orbit is well defined for all forward iterates and converges to one of exactly seven fixed points. In arbitrary dimension, we establish an exact row-wise Gram factorization and the identity $\operatorname{rank}C(A)=\operatorname{rank}(AH_n)$, which gives the precise rank-reduction mechanism and forces every fixed point to be singular. We identify the all-ones matrix as the unique Pearson-degenerate correlation matrix, prove forward invariance of the Pearson-nondegenerate elliptope, and classify the $2^{n-1}-1$ nondegenerate sign-valued fixed points. For $n=3$, we prove a complete analytical fixed-point classification: three rank-one patterned points, three rank-two mixed points, and one rank-two equicorrelation point. We also prove the complete relative Lyapunov stability classification: precisely the patterned points are locally asymptotically stable, while the mixed and equicorrelation points are unstable. The global proof reduces the rank-two dynamics to a one-dimensional projective kernel coordinate; an exact order identity produces monotone projective ratios, excludes nontrivial periodic and recurrent limit sets, and forces convergence to a fixed point. Finally, the initial conditions converging to the four unstable fixed points form a Lebesgue-null set. Hence almost every admissible initial condition converges to a patterned fixed point; the three patterned basins are relatively open and permutation-equivalent, and each has Lebesgue measure exactly one third of that of the elliptope.

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arXiv2026

Empirical Laws for Iterated Correlation Matrices

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.

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