Geometric Decoupling: Diagnosing the Structural Instability of Latent

Latent Diffusion Models (LDMs) achieve high-fidelity synthesis but suffer from latent space brittleness, causing discontinuous semantic jumps during editing. We introduce a Riemannian framework to diagnose this instability by analyzing the generative Jacobian, decomposing geometry into Local Scaling (capacity) and Local Complexity (curvature). Our study uncovers a "Geometric Decoupling": while curvature in normal generation functionally encodes image detail, Out-of-Distribution (OOD) generation exhibits a functional decoupling where extreme curvature is wasted on unstable semantic boundaries rather than perceptible details. This geometric misallocation identifies "Geometric Hotspots" as the structural root of instability, providing a robust intrinsic metric for diagnosing generative reliability. Our code is at https://github.com/Byronliang8/Diffusion-Geometry.

Latent Diffusion Models (LDMs) achieve high-fidelity text-to-image synthesis, but the latent space used can involve complicated local geometry, leading to quick semantic jumps even with small changes of latent during editing. In this paper, we introduce a mathematical framework to diagnose this instability. Specifically, we analyze the local geometry of the mapping from the latent to the generative output, and decompose the geometry into local scaling (which measures local volume expansion related to representation capacity), and local complexity (which measures local curvature). Our study uncovers a Geometric Decoupling phenomenon. For normal generation for regions where the model is well trained, local curvature corresponds to encoding image detail, but for generation involving abnormal cases (e.g., hallucinated generation), extreme curvature is wasted on unstable semantic boundaries and does not correlate with fine image details. Our research leads to robust metrics for diagnosing generative reliability. Our code is at https://github.com/Byronliang8/Diffusion-Geometry.