K\"ahler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold

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 K\"ahler 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 K\"ahler form up to constants, yielding a constant determinant condition with respect to a background metric and ill-conditioned eigenvalues under nonuniform and almost low-rank assumptions. 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.