Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networks
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Abstract (arXiv)
We present a theory for simultaneous approximation of the score function and its derivatives, enabling the handling of data distributions with low-dimensional structure and unbounded support. Our approximation error bounds match those in the literature while relying on assumptions that relax the usual bounded support requirement. Crucially, our bounds are free from the curse of dimensionality. Moreover, we establish approximation guarantees for derivatives of any prescribed order, extending beyond the commonly considered first-order setting.
Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 33 of its 49 references that are in the map. First posted on arXiv 29 Dec 2025.
Builds on
33 of 49 references.
References · 33
- 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
- 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,185
- 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,067
- 2016Gaussian Error Linear Units (GELUs)cited by 3,431
- 2022Video Diffusion Modelscited by 2,280
- 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,339
- 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptionscited by 440
- 2021The Intrinsic Dimension of Images and Its Impact on Learningcited by 307
- 2022Convergence of denoising diffusion models under the manifold hypothesiscited by 237
- 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 230
- 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
- 2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Datacited by 171
21 more
- 2023Diffusion Models are Minimax Optimal Distribution Estimatorscited by 159
- 2017Sobolev Training for Neural Networkscited by 136
- 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
- 2021On the approximation of functions by tanh neural networkscited by 81
- 2024Accelerating Convergence of Score-Based Diffusion Models, Provablycited by 74
- 2019Deep ReLU network approximation of functions on a manifoldcited by 73
- 2024A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Modelscited by 64
- 2022Verifying the Union of Manifolds Hypothesis for Image Datacited by 62
- 2024Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensionscited by 60
- 2024Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Modelscited by 56
- 2018Non-asymptotic bounds for sampling algorithms without log-concavitycited by 35
- 2022Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activationscited by 29
- 2020Statistical guarantees for generative models without dominationcited by 27
- 2025Generalization error bound for denoising score matching under relaxed manifold assumptioncited by 18
- 2025Faster Diffusion Models via Higher-Order Approximationcited by 16
- 2020Approximations with deep neural networks in Sobolev time-spacecited by 13
- 2025Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Modelscited by 9
- 2023Differentiable Neural Networks with RePU Activation: with Applications to Score Estimation and Isotonic Regressioncited by 8
- 2016Quantifying the accuracy of approximate diffusions and Markov chainscited by 7
- 2017Mode-Seeking Clustering and Density Ridge Estimation via Direct Estimation of Density-Derivative-Ratioscited by 3
- 2025Approximation Capabilities of Feedforward Neural Networks with GELU Activationscited by 2
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Citing papers per year: 2026: 3.
3 papers.