Kähler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold

2026-08-19

AI Summary

研究复参数化网络的优化问题，通过Kähler信息度量和自然梯度下降法解决，并探讨了Calabi-Yau流形对优化过程的影响。

Abstract

We study landscapes for complex-parameterized networks. Our approach is motivated with an information-theoretic manifold perspective of the parameter and via classical optimization guarantees although of complex geometric variety such as through Dolbeault asymptotics. The descent path admits a Kähler information metric under a cross-entropy via the Wirtinger Hessian on the log-likelihood potential. We restrict attention to a descent update rule with natural gradient descent via a differentiated loss scaled by the inverse metric, so the descent path remains in the holomorphic tangent bundle. We emphasize Calabi-Yau information manifolds which profane theoretical guarantees via an ill-curvature-conditioned landscape. Under a Calabi-Yau metric, specifically in a non-compact setting with a global potential so defined geometrically rather than invoking the topological requirements of the Calabi conjecture, a wedged nowhere-vanishing holomorphic form is the top exterior product of the Kähler form up to constants, yielding a constant determinant condition. Under a fixed determinant, a metric almost low rank up to an eigenvalue tolerance implies a blow-up effect. Moreover, it has been discovered that negative curvature subverts the loss landscape, specifically sectional curvature, so we expand on this and draw interconnections to negative-definite Ricci curvature. Our arguments primarily exist in a geometric analytic modality, although we establish roots in deep learning theory such as through asymptotics at initialization and connections through failure modes of neural network guarantees under vanishing and negative Ricci curvature.

Problem

• Kähler information metric
• Calabi-Yau manifold
• complex-parameterized networks
• negative curvature
• Ricci curvature

Innovation

• Kähler information metric
• natural gradient descent
• Calabi-Yau manifold
• negative curvature
• Ricci curvature