Transcriptomic phase transitions along the Alzheimer's disease continuum using riemmaninan tensor model: a bifurcation-intermediate variance framework on the ROSMAP cohort

Alzheimers disease (AD) progression involves systemic network transitions. To capture these using ROSMAP bulk RNA-seq (n=624), we focused on the geometry of the covariance structure, performing a Riemannian (Log-Euclidean) analysis of stage-wise covariance matrices as points on the manifold of symmetric positive-definite (SPD) matrices. On the SPD manifold the three stages were non-collinear: geodesic distances were non-uniform and MCI was displaced from the NCI-AD chord, while the von Neumann entropy of the covariance structure dipped at MCI (S = 2.760, 2.639, 2.647 for normal cognitive intact NCI, mild cognitive impairment MCI, AD) and the path-curvature profile reached a minimum there -- together identifying MCI as a saddle/bifurcation state. The differential covariance spectrum (CAD - CNCI) separated AD-amplified ("structural collapse") from AD-suppressed ("protective loss") modes. Ultimately, second-order statistics analyzed through Riemannian geometry, rather than Euclidean summaries, reveal AD progression structure invisible to mean-level analysis.