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

arXiv: https://arxiv.org/abs/2608.19584 · HTML: https://mlsanity.com/paper/2608.19584/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 76 references that are in the map.
- First posted on arXiv 20 Aug 2026.

## Builds on (most cited first, 15 of 15)

- [Meta-Principled Family of Hyperparameter Scaling Strategies](https://mlsanity.com/paper/2210.04909/) (2022)
- [On the asymptotics of wide networks with polynomial activations](https://mlsanity.com/paper/2006.06687/) (2020)
- [Asymptotics of Wide Convolutional Neural Networks](https://mlsanity.com/paper/2008.08675/) (2020)
- [Natural Gradient Methods: Perspectives, Efficient-Scalable Approximations, and Analysis](https://mlsanity.com/paper/2303.05473/) (2023)
- [On the Convergence Analysis of DCA](https://mlsanity.com/paper/2211.10942/) (2022)
- [Genus expansion for non-linear random matrix ensembles with applications to neural networks](https://mlsanity.com/paper/2407.08459/) (2024)
- [Mean-Field Analysis of Two-Layer Neural Networks: Global Optimality with Linear Convergence Rates](https://mlsanity.com/paper/2205.09860/) (2022)
- [A Riemannian Optimization Perspective of the Gauss-Newton Method for Feedforward Neural Networks](https://mlsanity.com/paper/2412.14031/) (2024)
- [Complex-valued Neural Networks -- Theory and Analysis](https://mlsanity.com/paper/2312.06087/) (2023)
- [The lazy (NTK) and rich (μP) regimes: a gentle tutorial](https://mlsanity.com/paper/2404.19719/) (2024)
- [Continuous-Time Dynamics of the Difference-of-Convex Algorithm](https://mlsanity.com/paper/2604.06926/) (2026)
- [A note on Linear Bottleneck networks and their Transition to Multilinearity](https://mlsanity.com/paper/2206.15058/) (2022)
- [Quotient Geometry, Effective Curvature, and Implicit Bias in Simple Shallow Neural Networks](https://mlsanity.com/paper/2603.21502/) (2026)
- [Navigating Potholes with Geometry-Aware Sharpness Minimization](https://mlsanity.com/paper/2605.16134/) (2026)
- [Local Information Operators for Spatial Identifiability in Distributed-Parameter Inverse Problems in Computational Mechanics](https://mlsanity.com/paper/2605.28601/) (2026)

## Abstract (arXiv)

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.