First non-asymptotic sample complexity bounds for structure learning of polynomial exponential families via score matching, with polynomial dependence on model dimension.
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Convergence of dif- fusion models under the manifold hypothesis in high-dimensions.arXiv preprint arXiv:2409.18804
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FHDMs achieve minimax optimal TV convergence rates for spherically supported Sobolev data distributions up to log factors, the first optimality result for random-time denoising diffusion models.
Transformers converge globally to the optimal DDPM denoiser for multi-token GMMs via self-attention mean denoising, with explicit token and iteration requirements.
Score-based generative models attain intrinsic Wasserstein-1 sample rates of order n to the power of -(beta+1)/(d+2beta) on d-dimensional smooth manifolds with beta-Holder densities.
Diffusion and flow processes forget dependencies to define valid copulas then learn to remember them for density estimation and sampling, outperforming prior copula methods on complex datasets.
For a broad class of coefficients, diffusion models achieve Õ(k/ε) iteration complexity for ε-accurate TV sampling under low-dimensional structure, independent of ambient dimension.
MIND integrates discrete patch tokenization into diffusion score functions via soft top-k and dual-branch layers, achieving FID 22.73 (no guidance) and 2.06 (with guidance) on ImageNet-256 after 80 epochs, outperforming DiT and larger LlamaGen models.
SiLD is a score-matching framework that learns both manifold projection and intrinsic density from a single objective, with proven sample complexity depending only on intrinsic dimension.
CNNs achieve dimension-dependent Sobolev approximation rates on manifolds, and a spectral boundary loss using Laplace-Beltrami eigenmodes enables stable PINN solutions for elliptic problems with improved accuracy over standard approaches.
Diffusion models on manifold-supported data admit score decompositions whose statistical rates are controlled by intrinsic dimension and curvature.
Constraining score estimators to a Sobolev ball yields minimax rates for density scores on the torus and, under a conjecture, for score-based generative models.
Discretized Föllmer processes supply hyper-parameter settings for DDPM samplers that recover state-of-the-art sampling error bounds with slight improvements.
Reusing source latent spaces in diffusion models under distribution shift produces target score error set by principal-angle misalignment and diffusion-time-amplified ambient noise.
Diffusion models require new generalization frameworks because memorization and novel generation are incompatible, so research should focus on what models learn before memorization begins.
citing papers explorer
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Finite Sample Bounds for Learning with Score Matching
First non-asymptotic sample complexity bounds for structure learning of polynomial exponential families via score matching, with polynomial dependence on model dimension.
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Statistical Convergence of Spherical First Hitting Diffusion Models
FHDMs achieve minimax optimal TV convergence rates for spherically supported Sobolev data distributions up to log factors, the first optimality result for random-time denoising diffusion models.
-
Transformers Learn the Optimal DDPM Denoiser for Multi-Token GMMs
Transformers converge globally to the optimal DDPM denoiser for multi-token GMMs via self-attention mean denoising, with explicit token and iteration requirements.
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Intrinsic Wasserstein Rates for Score-Based Generative Models on Smooth Manifolds
Score-based generative models attain intrinsic Wasserstein-1 sample rates of order n to the power of -(beta+1)/(d+2beta) on d-dimensional smooth manifolds with beta-Holder densities.
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Diffusion and Flow-based Copulas: Forgetting and Remembering Dependencies
Diffusion and flow processes forget dependencies to define valid copulas then learn to remember them for density estimation and sampling, outperforming prior copula methods on complex datasets.
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Diffusion Models Adapt to Low-Dimensional Structure Under Flexible Coefficient Choices
For a broad class of coefficients, diffusion models achieve Õ(k/ε) iteration complexity for ε-accurate TV sampling under low-dimensional structure, independent of ambient dimension.
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Diffusion Image Generation with Explicit Modeling of Data Manifold Geometry
MIND integrates discrete patch tokenization into diffusion score functions via soft top-k and dual-branch layers, achieving FID 22.73 (no guidance) and 2.06 (with guidance) on ImageNet-256 after 80 epochs, outperforming DiT and larger LlamaGen models.
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Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine
SiLD is a score-matching framework that learns both manifold projection and intrinsic density from a single objective, with proven sample complexity depending only on intrinsic dimension.
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Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems
CNNs achieve dimension-dependent Sobolev approximation rates on manifolds, and a spectral boundary loss using Laplace-Beltrami eigenmodes enables stable PINN solutions for elliptic problems with improved accuracy over standard approaches.
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Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity
Diffusion models on manifold-supported data admit score decompositions whose statistical rates are controlled by intrinsic dimension and curvature.
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Optimal score function estimation via derivatives constraints
Constraining score estimators to a Sobolev ball yields minimax rates for density scores on the torus and, under a conjecture, for score-based generative models.
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A note on connections between the F\"ollmer process and the denoising diffusion probabilistic model
Discretized Föllmer processes supply hyper-parameter settings for DDPM samplers that recover state-of-the-art sampling error bounds with slight improvements.
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On the Limits of Latent Reuse in Diffusion Models
Reusing source latent spaces in diffusion models under distribution shift produces target score error set by principal-angle misalignment and diffusion-time-amplified ambient noise.
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Understanding diffusion models requires rethinking (again) generalization
Diffusion models require new generalization frameworks because memorization and novel generation are incompatible, so research should focus on what models learn before memorization begins.
- Diffusion-based Denoising Beats Vanilla Score Matching in Parameter Estimation: A Theoretical Explanation