Local Hessian Spectral Filtering for Robust Intrinsic Dimension Estimation

Diffusion models are prone to generating structural hallucinations - samples that match the statistical properties of the training data yet defy underlying structural rules, resulting in anomalies like hands with more than five fingers. Recent research studied this failure mode from several viewpoints, offering partial explanations to their occurrence, such as mode interpolation. In this work, we propose a complementary perspective that treats hallucinations as instabilities on the model-induced manifold. We begin by showing that a hallucination filter based on such instabilities matches or exceeds the performance of the recently proposed temporal one. By tracing the source of these instabilities, we identify local intrinsic dimension (LID) as their primary driver and propose Intrinsic Quenching (IQ), a direct corrective mechanism that deflates it to alleviate hallucinations. IQ consistently outperforms standard hallucination reduction baselines across a wide array of benchmarks and offers a highly promising solution for enforcing anatomical consistency in downstream medical imaging tasks.

Diffusion models generate high-dimensional data with remarkable quality, yet how their training efficiently learns the score function, bypassing the curse of dimensionality when data is supported on low-dimensional manifolds, remains theoretically unexplained. We identify a collapse-and-refine mechanism driven by the geometry of the score function itself: at small noise scales, the diverging singularity of the score drives a rapid dimensional collapse of the induced denoising map onto the data manifold projection; at moderate noise scales, training refines the intrinsic density on the learned manifold. We instantiate this principle as Score-induced Latent Diffusion (SiLD), a two-stage framework in which both manifold learning and density estimation emerge from a single denoising score matching objective, replacing the heuristic KL regularization of VAE-based latent diffusion models. We prove that the resulting sample complexity depends on the intrinsic dimension rather than the ambient dimension. Experiments on Stacked MNIST, CelebA variants, and molecular generation benchmarks show that SiLD matches or outperforms VAE-based LDMs in generation quality and consistently improves reconstruction, validating our theoretical predictions.

Diffusion models perform remarkably well on high-dimensional data such as images, often using only a modest number of reverse-time steps. Despite this practical success, existing convergence theory does not fully explain why such samplers remain efficient in high dimensions. Many prior KL guarantees bound the discretization error in terms of the ambient dimension, while other improved results replace this dependence using intrinsic-dimensional or geometric structure assumptions. In this work, we develop an alternative information-theoretic perspective on diffusion sampler convergence. We prove that, for Gaussian mixture targets, the discretization error is controlled by the Shannon entropy of the latent mixture component rather than by the ambient dimension. Consequently, the leading step complexity scales linearly with latent entropy and depends only logarithmically on the second moment of the data. Our analysis also extends to discrete target distributions, where the relevant complexity is the entropy of the target rather than the dimension of the embedding space. These results suggest that diffusion sampling can remain efficient in high-dimensional spaces when the data distribution admits a compact latent representation, as is widely believed to be the case for natural images.