A method to recover the Hessian/Fisher metric of generative model latent spaces is validated on Ising and TASEP and applied to claim fractal phase transitions in diffusion models.
Hessianizability of surface metrics
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abstract
A symmetric quadratic form $g$ on a surface~$M$ is said to be locally Hessianizable if each $p\in M$ has an open neighborhood~$U$ on which there exists a local coordinate chart $(x^1,x^2):U\to\mathbb{R}^2$ and a function $f:U\to\mathbb{R}$ such that, on $U$, we have $$ g = \frac{\partial^2 f}{\partial x^i\partial x^j}\,\mathrm{d} x^i\circ\mathrm{d} x^j. $$ In this article, I show that, when $g$ is nondegenerate and smooth, it is always smoothly locally Hessianizable.
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Hessian Geometry of Latent Space in Generative Models
A method to recover the Hessian/Fisher metric of generative model latent spaces is validated on Ising and TASEP and applied to claim fractal phase transitions in diffusion models.