Abstract:We develop interconnections between the complex normalizing flow for data drawn from Borel probability measures on the twofold realification of the complex manifold and a nonlinear flow nearly Kähler-Ricci. The complex normalizing flow relates the initial and target realified densities under the complex change of variables, necessitating the log determinant of the ensemble of Wirtinger Jacobians. The Ricci curvature of a Kähler manifold is the second order mixed Wirtinger partial derivative of the log of the local density of the volume form. Therefore, we reconcile these two facts by drawing forth the connection that the log determinant used in the complex normalizing flow matches a Ricci curvature term under differentiation and conditions. The log density under the normalizing flow is kindred to a spatial Fisher information metric under an augmented Jacobian and a Bayesian perspective to the parameter, thus under the continuum limit the log likelihood matches a Fisher metric, recovering a Kähler-Ricci flow variation up to a time derivative and expectation, or an average-valued Kähler-Einstein flow. Using this framework, we establish other relevant results, attempting to bridge the statistical and ordinary behaviors of the complex normalizing flow to the geometric features of our derived Kähler flow.
| Subjects: | Differential Geometry (math.DG); Machine Learning (cs.LG) |
| Cite as: | arXiv:2604.17954 [math.DG] |
| (or arXiv:2604.17954v2 [math.DG] for this version) | |
| https://doi.org/10.48550/arXiv.2604.17954 arXiv-issued DOI via DataCite |
From: Andrew Gracyk [view email]
[v1]
Mon, 20 Apr 2026 08:36:35 UTC (4,924 KB)
[v2]
Thu, 14 May 2026 13:24:34 UTC (4,924 KB)
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