用于复杂神经网络下降的凯勒流形及保证，包括对卡拉比-丘流形的搜索与破坏
Kähler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold
- Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究将复几何与深度学习结合，以凯勒流形为框架分析复杂神经网络下降的损失景观，揭示卡拉比-丘流形相关特性及负曲率对损失景观的破坏，为深度学习理论提供几何分析视角。
AI中文摘要：
我们研究复参数化网络的损失景观。我们的方法受参数的信息论流形视角以及经典优化保证的启发，尽管涉及复几何变体，例如通过多贝阿尔特渐近。下降路径在交叉熵下通过对数似然势上的Wirtinger海森矩阵获得凯勒信息度量。我们关注的下降更新规则是通过逆度量缩放的微分损失的自然梯度下降，因此下降路径保持在全纯切丛中。我们强调卡拉比-丘信息流形，其因曲率条件不佳的景观而破坏理论保证。在卡拉比-丘度量下，特别是在具有全局势的非紧设置中，该势是几何定义的而非调用卡拉比猜想的拓扑要求，无处为零的全纯形式的楔积是凯勒形式的最高外积（相差常数），产生常数行列式条件。在固定行列式下，度量在特征值容差下几乎低秩会导致爆破效应。此外，已发现负曲率会破坏损失景观，特别是截面曲率，因此我们扩展此内容并建立与负定里奇曲率的关联。我们的论证主要存在于几何分析模式中，尽管我们在深度学习理论中确立了根基，例如通过初始化时的渐近分析，以及在里奇曲率消失和为负时神经网络保证失效模式的关联。
英文摘要：
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. 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.
