mlsanityData to 14 Oct 2026

On the Interpolation Effect of Score Smoothing in Diffusion Models

arXiv 2502.19499 ↗First posted 26 Feb 2025cs.LG math.OC stat.MLcited by 15

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

Diffusion models have achieved remarkable progress in various domains with an intriguing ability to produce new data that do not exist in the training set. In this work, we study the hypothesis that such creativity arises from the neural network backbone learning a smoothed version of the empirical score function, which guides the denoising dynamics to generate data points that interpolate the training data. Focusing mainly on settings where the training set lies uniformly in a one-dimensional subspace, we elucidate the interplay between score smoothing and the denoising dynamics with analytical solutions and numerical experiments, demonstrating how smoothing the score function can cause the denoised data samples to interpolate the training set along the subspace. Moreover, we present theoretical and empirical evidence that learning score functions with neural networks - either with or without explicit regularization - can naturally achieve a similar effect, including when the data belong to simple nonlinear manifolds.

Key facts. Cited by 15 papers in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 65 of its 90 references that are in the map. First posted on arXiv 26 Feb 2025.

Builds on

65 of 90 references.

Closest ancestors · 44

Same direction, newest first

  1. 2025Understanding Representation Dynamics of Diffusion Models via Low-Dimensional Modeling9 Feb
  2. 2025Memorization and Regularization in Generative Diffusion Models27 Jan
  3. 2024The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications12 Dec
  4. 2024Towards a Mechanistic Explanation of Diffusion Model Generalization28 Nov
  5. 2024Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure31 Oct
  6. 2024Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality24 Oct
  7. 2024Shallow diffusion networks provably learn hidden low-dimensional structure15 Oct
  8. 2024Linear Convergence of Diffusion Models Under the Manifold Hypothesis11 Oct
  9. 2024Losing dimensions: Geometric memorization in generative diffusion11 Oct
  10. 2024Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion8 Oct
  11. 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions27 Sep
  12. 2024Evaluating the design space of diffusion-based generative models18 Jun
32 more
  1. 2024A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models5 Jun
  2. 2024Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models23 May
  3. 2024Learning general Gaussian mixtures with efficient score matching29 Apr
  4. 2024Learning Mixtures of Gaussians Using Diffusion Models29 Apr
  5. 2024Convergence Analysis of Probability Flow ODE for Score-based Generative Models15 Apr
  6. 2024Dynamical Regimes of Diffusion Models28 Feb
  7. 2024Minimax Optimality of Score-based Diffusion Models: Beyond the Density Lower Bound Assumptions23 Feb
  8. 2024Analyzing Neural Network-Based Generative Diffusion Models through Convex Optimization3 Feb
  9. 2024Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization28 Jan
  10. 2024A Good Score Does not Lead to A Good Generative Model10 Jan
  11. 2023On the Generalization Properties of Diffusion Models3 Nov
  12. 2023Closed-Form Diffusion Models19 Oct
  13. 2023On Memorization in Diffusion Models4 Oct
  14. 2023Generalization in diffusion models arises from geometry-adaptive harmonic representations4 Oct
  15. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization7 Aug
  16. 2023Learning Mixtures of Gaussians Using the DDPM Objective3 Jul
  17. 2023On the Generalization of Diffusion Model24 May
  18. 2023Diffusion Models Generate Images Like Painters: an Analytical Theory of Outline First, Details Later4 Mar
  19. 2023Diffusion Models are Minimax Optimal Distribution Estimators3 Mar
  20. 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data14 Feb
  21. 2022Scalable Diffusion Models with Transformers19 Dec
  22. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions3 Nov
  23. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions22 Sep
  24. 2022Convergence of denoising diffusion models under the manifold hypothesis10 Aug
  25. 2022Convergence for score-based generative modeling with polynomial complexity13 Jun
  26. 2022Score-Based Generative Models Detect Manifolds2 Jun
  27. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  28. 2021Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling1 Jun
  29. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  30. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  31. 2020Denoising Diffusion Implicit Models6 Oct
  32. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 21

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  3. 2022Hierarchical Text-Conditional Image Generation with CLIP Latentscited by 6,585
  4. 2022Flow Matching for Generative Modelingcited by 5,702
  5. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  6. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,700
  7. 2017FiLM: Visual Reasoning with a General Conditioning Layercited by 2,333
  8. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 939
  9. 2024WorldSimBench: Towards Video Generation Models as World Simulatorscited by 881
  10. 2023Extracting Training Data from Diffusion Modelscited by 732
  11. 2017The Implicit Bias of Gradient Descent on Separable Datacited by 655
  12. 2019Gradient Descent Maximizes the Margin of Homogeneous Neural Networkscited by 320
9 more
  1. 2020Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Losscited by 268
  2. 2018Gradient descent aligns the layers of deep linear networkscited by 182
  3. 2020Generative Modeling with Denoising Auto-Encoders and Langevin Samplingcited by 119
  4. 2019A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Casecited by 114
  5. 2019How do infinite width bounded norm networks look in function space?cited by 106
  6. 2024An analytic theory of creativity in convolutional diffusion modelscited by 104
  7. 2024Understanding Hallucinations in Diffusion Models through Mode Interpolationcited by 91
  8. 2024Optimal score estimation via empirical Bayes smoothingcited by 61
  9. 2024How Do Flow Matching Models Memorize and Generalize in Sample Data Subspaces?cited by 19

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 5; 2026: 10.

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  1. 2025Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptivecited by 20
  2. 2025Selective Underfitting in Diffusion Modelscited by 18
  3. 2025Does Generation Require Memorization? Creative Diffusion Models using Ambient Diffusioncited by 17
  4. 2026A Random Matrix Theory Perspective on the Consistency of Diffusion Modelscited by 6
  5. 2026Diffusion Model's Generalization Can Be Characterized by Inductive Biases toward a Data-Dependent Ridge Manifoldcited by 5
  6. 2025Kernel-Smoothed Scores for Denoising Diffusion: A Bias-Variance Studycited by 3
  7. 2026Substitution-Based Analysis of Structural Novelty for Generative Models of Materialscited by 3
  8. 2026An exact information theory of generalization phase transitions in Bayesian diffusion modelscited by 3
  9. 2025On The Hidden Biases of Flow Matching Samplerscited by 2
  10. 2026Diffusion Models Memorize in Training -- and Generalize in Inferencecited by 2
  11. 2026Manifold Generalization Provably Proceeds Memorization in Diffusion Modelscited by 2
  12. 2026The two clocks and the innovation window: When and how generative models learn rulescited by 1
3 more
  1. 2026Training-Free Generative Sampling via Moment-Matched Score Smoothingcited by 0
  2. 2026Score-Control for Hallucination Reduction in Diffusion Modelscited by 0
  3. 2026Scaling and Distilling Text Embeddings for Better Diffusibilitycited by 0

Similar, not cited

Close in wording, no citation either way

  1. 2026Smoothing the Score Function to Enhance Generalization in Diffusion Models
  2. 2026Asymptotic Learning Curves for Diffusion Models with Random Features Score and Manifold Data
  3. 2026Denoising Score Matching in Diffusion Models: The Effect of Rank Constraints on Generalization
  4. 2026Provably Learning Diffusion Models under the Manifold Hypothesis: Collapse and Refine
  5. 2024Efficient Denoising using Score Embedding in Score-based Diffusion Models
  6. 2026Deterministic Dynamics of Sampling Processes in Score-Based Diffusion Models with Multiplicative Noise Conditioning
  7. 2025Are We Really Learning the Score Function? Reinterpreting Diffusion Models Through Wasserstein Gradient Flow Matching
  8. 2025Denoising Score Matching with Random Features: Insights on Diffusion Models from Precise Learning Curves