[Paper] Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networks

Source: arXiv - 2512.23643v1

Overview

The paper introduces a new theoretical framework for training deep neural networks (DNNs) that can approximate both a probability distribution’s score function and any of its higher‑order derivatives at the same time. By relaxing the usual “bounded‑support” assumption, the authors show that accurate approximation is possible even for distributions that stretch out to infinity, while still avoiding the dreaded curse of dimensionality.

Key Contributions

• Unified approximation theory for the score function and all its derivatives (not just the first‑order gradient).
• Error bounds that match the best known rates in the literature without requiring bounded support of the data distribution.
• Dimension‑free guarantees: the bounds do not blow up as the ambient dimension grows, making the results applicable to high‑dimensional data with low‑dimensional intrinsic structure.
• Extension to arbitrary derivative order, opening the door to higher‑order score‑based methods (e.g., Stein operators, higher‑order Langevin dynamics).
• Constructive proof technique that yields explicit network architectures (depth, width, activation choices) needed to achieve the stated accuracy.

Methodology

1. Problem Setup – The score function of a density (p(x)) is (\nabla \log p(x)). The authors consider a family of target densities that may have unbounded support but possess a low‑dimensional manifold‑like structure (e.g., data lying near a subspace).
2. Network Design – They use standard feed‑forward ReLU (or smooth) networks and carefully control the growth of the weights so that the network output remains well‑behaved on the tails of the distribution.
3. Approximation Strategy
   • Approximate the log‑density (\log p(x)) by a neural network (f_\theta(x)).
   • Show that the same network simultaneously approximates its gradient (\nabla f_\theta(x)) and higher‑order derivatives (\nabla^{(k)} f_\theta(x)).
   • The analysis leverages recent results on approximation of Sobolev functions by DNNs, combined with a novel decomposition that isolates the low‑dimensional component of the data.
4. Error Analysis – By measuring error in Sobolev norms (which capture both function value and derivative errors), they derive bounds that depend only on the intrinsic dimension and the smoothness of the target log‑density, not on the ambient dimension.

Results & Findings

• Approximation error for the score and its (k)-th derivative scales as (\mathcal{O}(N^{-s/d_{\text{intr}}})), where (N) is the number of network parameters, (s) the smoothness order, and (d_{\text{intr}}) the intrinsic dimension.
• No curse of dimensionality: the rate does not contain the ambient dimension (d).
• The derived bounds are tight: they match existing lower‑bounds for first‑order score approximation under bounded‑support assumptions.
• The theory works for any prescribed derivative order (k), showing that deeper networks can faithfully capture higher‑order score information without a penalty in sample complexity.

Practical Implications

• Score‑based generative modeling (e.g., diffusion models, score‑matching GANs) can now be justified for data that lives on manifolds or has heavy tails, expanding their applicability to domains like physics simulations, finance, or high‑resolution image synthesis.
• Higher‑order Stein methods: practitioners can design estimators that use second‑ or third‑order score information for variance reduction, hypothesis testing, or Bayesian inference, knowing that a single DNN can provide all required derivatives.
• Efficient training: because the same network yields multiple derivative orders, developers can avoid training separate models for each order, saving compute and memory.
• Robustness to out‑of‑distribution tails: the unbounded‑support guarantee means models are less likely to catastrophically fail when encountering rare but extreme inputs—a common concern in safety‑critical systems.
• Low‑dimensional data handling: the dimension‑free rates suggest that even very high‑dimensional datasets (e.g., 3D point clouds, genomics) can be tackled as long as the underlying structure is low‑dimensional, encouraging the use of score‑based techniques in those fields.

Limitations & Future Work

• The results are theoretical; the paper does not provide empirical validation on real datasets, so practical performance remains to be demonstrated.
• The construction assumes knowledge of the intrinsic dimension and smoothness parameters, which may be hard to estimate in practice.
• The analysis focuses on ReLU‑type activations; extending to other architectures (e.g., transformers, convolutional nets) is left open.
• Future research could explore adaptive network designs that automatically discover low‑dimensional structure, and training algorithms that directly minimize the derived Sobolev‑norm errors rather than standard likelihood or score‑matching losses.

Authors

• Konstantin Yakovlev
• Nikita Puchkin

Paper Information

• arXiv ID: 2512.23643v1
• Categories: math.NA, cs.LG, math.ST, stat.ML
• Published: December 29, 2025
• PDF: Download PDF