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

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. We focus on Calabi-Yau metrics specifically in a non-compact setting with a global potential, so defined geometrically rather than invoking the topological requirements of the Calabi conjecture. In non-compact settings, we can write the metric determinant with respect to a background in terms of a pluriharmonic or real-valued function. Under bounded, nonuniform, and almost low-rank assumptions, we get a partial eigenvalue blow-up effect. In an empirical setting, a Ricci-flat metric will not form, but the blow-up effect is a local condition and can partially hold empirically on open sets. We isolate the Calabi-Yau case in a theoretical setting, and we counteract the corrupted geometries under regularization. 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 via geometric analysis, 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.