Geometric Decoupling: Diagnosing the Structural Instability of Latent

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.

Latent diffusion models offer an attractive alternative to discrete diffusion for non-autoregressive text generation by operating on continuous text representations and denoising entire sequences in parallel. The major challenge in latent diffusion modeling is constructing a suitable latent space. In this work, we present the Latent Diffusion Language Model (LDLM), in which the latent encoder, diffusion model, and decoder are trained jointly. LDLM builds its latent space by reshaping the representations of a pre-trained language model with a trainable encoder, yielding latents that are easy to both denoise and decode into tokens. We show that naive joint training produces a low-quality diffusion model, and propose a simple training recipe consisting of an MSE decoder loss, diffusion-to-encoder warmup, adaptive timestep sampling, and decoder-input noise. Ablations show that each component substantially impacts generation performance. On OpenWebText and LM1B, LDLM achieves better generation performance than existing discrete and continuous diffusion language models while being $2{\text -}13\times$ faster, indicating that jointly learning the latent space is a key step toward making latent diffusion competitive for text generation.

Despite remarkable progress, image generation is far from solved. The dominant metric, FID, conflates sample fidelity with mode coverage and is close to being saturated. Yet a model can still exhibit mode collapse while achieving a low FID, since a handful of sharp, near-duplicate images can outscore a model that faithfully covers the full data distribution. We argue that precision and recall are essential complements to FID, and that because FID is already saturated, the more meaningful goal is to improve diversity and coverage. Achieving high recall requires a model that explicitly prioritizes mode coverage, unlike most generative models, which optimize sample fidelity. We introduce RTM, which replaces the single-pass latent mapping in style-based generators with an iterative refinement process, and show that this consistently improves both quality and diversity. Integrated with Implicit Maximum Likelihood Estimation (IMLE), which optimizes mode coverage by design, RTM achieves the highest precision and recall among current state-of-the-art approaches while maintaining competitive FID, with improvements across CIFAR-10, CelebA-HQ at 256x256, and nine few-shot benchmarks. RTM also improves StyleGAN2 and StyleGAN2-ADA on CIFAR-10 and AFHQ-v1 at 512x512, demonstrating that the benefit is not specific to IMLE. Unlike flow-matching baselines that achieve competitive FID at the expense of coverage, recursive refinement improves both quality and diversity simultaneously.