Intrinsic-Hybrid Latent Diffusion Models for Generative Modeling on Unknown Manifolds

2026-08-05

AI Summary

This work addresses the limitation of existing diffusion models in high-dimensional generation, which often ignore the intrinsic manifold geometry of data, while conventional latent diffusion models impose an Euclidean structure that struggles to capture complex geometries under data sparsity. To overcome this, we propose the Intrinsic Latent Diffusion Model (ILDM), the first framework to integrate Riemannian manifold geometry into the diffusion process. ILDM treats the latent space as a coordinate chart of an unknown manifold and jointly models geometric structure and uncertainty via a probabilistic decoder. We introduce a Riemannian–Euclidean hybrid forward diffusion mechanism, supported by a local uncertainty–driven diffusion strategy, a probabilistic metric tensor, and a tailored approximate denoising score matching objective. Experiments on COIL-100, MNIST, and cardiac MRI demonstrate that ILDM significantly outperforms existing methods, achieving lower FID and LPIPS scores and superior generation quality.

Abstract

We introduce the Intrinsic Hybrid Latent Diffusion Model (ILDM), a generative framework that integrates probabilistic dimensionality reduction with geometry-aware diffusion on unknown manifolds. While diffusion models (DMs) have achieved state-of-the-art results in high-dimensional data synthesis, they rely on large training datasets and ignore intrinsic geometric structure. Latent diffusion models (LDMs) address the high dimensionality by learning a latent space, but they typically impose a Euclidean structure, failing to capture the underlying manifold geometry, especially problematic in data-sparse regimes. ILDM addresses these limitations by interpreting the latent space as a chart of an unknown Riemannian manifold, with geometry and uncertainty quantified through a probabilistic decoder. The forward process is a hybrid diffusion that switches between Riemannian and Euclidean dynamics based on local uncertainty, where the Riemannian component is governed by a probabilistic metric tensor derived from the decoder. To learn the generative dynamics, we introduce an approximate denoising score matching method tailored to the hybrid diffusion setting, enabling a backward process defined by hybrid Langevin dynamics. Experiments on COIL-100, MNIST, and cardiac MRI datasets demonstrate that ILDM significantly improves generation quality, achieving lower FID and LPIPS scores compared to standard diffusion and latent diffusion models.

Problem

• diffusion models
• latent space
• manifold geometry
• probabilistic dimensionality reduction
• Riemannian manifold

Innovation

• Latent Diffusion Models
• Riemannian Manifold
• Hybrid Diffusion
• Probabilistic Metric Tensor
• Denoising Score Matching