AI 中文总结
本文以经典信息几何为基础，通过梯度流探究对偶平坦几何的延伸，构建正则与奇异信息几何理论，关联多类数学结构并经数值实验验证，为奇异信息几何提供基础。
AI 中文摘要
本文以经典信息几何为出发点，通过梯度流研究对偶平坦几何如何在正则凸性、非退化性与光滑性之外延伸。正则理论基于正定Gram矩阵上的对数行列式势函数构建，确立了其Legendre对偶、Fisher–Rao度量、Bregman散度及广义勾股定理。我们将该框架与Craig–Sakamoto形变、Wolfe对偶关联，还通过Yoshizawa嵌入与Brockett–Bloch–Ratiu双括号流建立联系，连接了等谱动力学、Stiefel优化及分量学习。Bures–Wasserstein几何提供了互补的梯度流结构。奇异理论源于边界行为：凸差形变产生不定或退化Hessian，同时保留伪Hessian、对偶平坦、Legendre自对偶结构。牛顿流在非Morse临界集附近呈现有限时间坍缩或由Łojasiewicz控制的收敛。通过显式爆破解决了Birkhoff多面体与椭圆曲线模上的Fisher度量退化，得到双有理不变指数衰减律。我们进一步推导了闭式Kirillov雅可比，并引入交叉曲率作为局部逃逸率的谱诊断，包括新的Box–Cox插值。可复现的数值实验验证了闭式结果。本文未声称理论已完成，而是为以退化 pencil、不定对偶平坦性、爆破几何及Łojasiewicz型收敛为核心的奇异信息几何提供基础。
英文摘要
Taking classical information geometry as its point of departure, this paper investigates, through gradient flows, how dually flat geometry extends beyond regular convexity, non-degeneracy, and smoothness. The regular theory is developed from the log-determinant potential on positive definite Gram matrices, establishing its Legendre dual, Fisher--Rao metric, Bregman divergence, and generalized Pythagorean theorem. We connect this framework to Craig--Sakamoto deformation, Wolfe duality, and, via Yoshizawa's embedding, Brockett--Bloch--Ratiu double-bracket flows, linking isospectral dynamics, Stiefel optimization, and component learning. The Bures--Wasserstein geometry provides a complementary gradient-flow structure. The singular theory emerges from boundary behavior: difference-of-convex deformations produce indefinite or degenerate Hessians while retaining pseudo-Hessian, dually flat, Legendre-self-dual structures. Newton flows exhibit finite-time collapse or Łojasiewicz-controlled convergence near non-Morse critical sets. Fisher-metric degeneracies on the Birkhoff polytope and elliptic-curve moduli are resolved by explicit blow-ups, yielding a birationally invariant exponential decay law. We further derive a closed-form Kirillov Jacobian and introduce cross curvature as a spectral diagnostic of local escape rates, including a new Box--Cox interpolation. Reproducible numerical experiments support the closed-form results. Rather than claiming a completed theory, the paper provides foundations for singular information geometry centered on degenerate pencils, indefinite dual flatness, blow-up geometry, and Łojasiewicz-type convergence.
Comments226 pages, 6 figures
