Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structures

• URL: http://arxiv.org/abs/2606.19894v1
• Date: Thu, 18 Jun 2026 07:51:33 GMT
• Title: Score Approximation for Diffusion Models on Arbitrary Low-Dimensional Structures
• Abstract summary: This work establishes a universal score approximation theorem that works for any distribution supported on any compact set of upper Minkowski dimension $d$.<n>Using a novel discrete-mixture formulation, we prove that the score function can be approximated with a ReLU network whose complexity grows exponentially only with $d$.
• Score: 23.118804869806898
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The remarkable success of score-based diffusion models has spurred significant efforts to establish their theoretical foundations. However, existing complexity bounds for score approximation rely heavily on restrictive assumptions like Lipschitz continuous densities or smooth manifold supports, which are routinely violated by the singularities, sharp boundaries, and disjoint clusters inherent to real-world perceptual data. This work establishes a universal score approximation theorem that works for any distribution supported on any compact set of upper Minkowski dimension $d$. Using a novel discrete-mixture formulation, we prove that the score function can be approximated with a ReLU network whose complexity grows exponentially only with $d$, thus breaking the exponential curse of ambient dimensionality. Combined with existing theories on accurately solving the backward diffusion SDE for arbitrary compact distributions, our work shows that diffusion models readily adapt to irregular, non-smooth data structures, explaining their competence in real-world generative tasks.

Related papers

• Diffusion Models Are Statistically Optimal for Learning Low-Dimensional Multi-Modal Distributions [11.621528352410502]
  We study the sample complexity of diffusion models for learning distributions supported on a union of low-dimensional subspaces.<n>We show that diffusion models require at most $widetildeO(varepsilon-k vee 2)$ samples to achieve $varepsilon$ error in 1-Wasserstein distance, where $k$ is the intrinsic dimension.
  arXiv  Detail & Related papers  (2026-05-28T16:13:44Z)
• From Scores to Gibbs Correctors: Accelerating Uniform-Rate Discrete Diffusion Models [58.02657021435213]
  We propose a novel Gibbs-based corrector for discrete diffusion models, termed Gibbs-Accelerated Discrete Diffusion (GADD)<n>We show GADD achieves an overall sampling complexity of $mathcalO(mathrmpolylog (varepsilon-1)$, yielding the first such rate for diffusion-based samplers for uniform-rate discrete diffusion models.
  arXiv  Detail & Related papers  (2026-05-26T17:52:28Z)
• GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains [56.121725064621295]
  Score-based diffusion models in function spaces offer a principled framework for modelling function-valued data.<n>However, practical implementations have struggled to fully realise these benefits.<n>We propose a novel architecture that represents generalised convolutional kernels as finite graph kernels.
  arXiv  Detail & Related papers  (2026-05-05T08:33:52Z)
• Efficient Sampling with Discrete Diffusion Models: Sharp and Adaptive Guarantees [9.180350432640912]
  We study the sampling efficiency of score-based discrete diffusion models under a continuous-time Markov chain (CTMC) formulation.<n>For uniform discrete diffusion, we show that the $$-leaping algorithm achieves an complexity of order $tilde O(d/varepsilon)$.<n>For masking discrete diffusion, we introduce a modified $$-leaping sampler whose convergence rate is governed by an intrinsic information-theoretic quantity.
  arXiv  Detail & Related papers  (2026-02-16T18:48:17Z)
• Non-asymptotic error bounds for probability flow ODEs under weak log-concavity [6.661419982187023]
  This work establishes non-asymptotic convergence bounds in the 2-Wasserstein distance for a general class of probability flow ODEs.<n>Our results extend convergence theory to more realistic data distributions and practical ODE solvers.
  arXiv  Detail & Related papers  (2025-10-20T14:54:38Z)
• Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers [49.97755400231656]
  We present the first performance guarantee with explicit dimensional dependencies for general score-mismatched diffusion samplers.<n>We show that score mismatches result in an distributional bias between the target and sampling distributions, proportional to the accumulated mismatch between the target and training distributions.<n>This result can be directly applied to zero-shot conditional samplers for any conditional model, irrespective of measurement noise.
  arXiv  Detail & Related papers  (2024-10-17T16:42:12Z)
• Convergence of Score-Based Discrete Diffusion Models: A Discrete-Time Analysis [56.442307356162864]
  We study the theoretical aspects of score-based discrete diffusion models under the Continuous Time Markov Chain (CTMC) framework.<n>We introduce a discrete-time sampling algorithm in the general state space $[S]d$ that utilizes score estimators at predefined time points.<n>Our convergence analysis employs a Girsanov-based method and establishes key properties of the discrete score function.
  arXiv  Detail & Related papers  (2024-10-03T09:07:13Z)
• Convergence Analysis of Discrete Diffusion Model: Exact Implementation through Uniformization [17.535229185525353]
  We introduce an algorithm leveraging the uniformization of continuous Markov chains, implementing transitions on random time points. Our results align with state-of-the-art achievements for diffusion models in $mathbbRd$ and further underscore the advantages of discrete diffusion models in comparison to the $mathbbRd$ setting.
  arXiv  Detail & Related papers  (2024-02-12T22:26:52Z)
• Diffusion Models are Minimax Optimal Distribution Estimators [49.47503258639454]
  We provide the first rigorous analysis on approximation and generalization abilities of diffusion modeling. We show that when the true density function belongs to the Besov space and the empirical score matching loss is properly minimized, the generated data distribution achieves the nearly minimax optimal estimation rates.
  arXiv  Detail & Related papers  (2023-03-03T11:31:55Z)
• Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data [68.62134204367668]
  This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. The generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution.
  arXiv  Detail & Related papers  (2023-02-14T17:02:35Z)
• Stochastic spectral embedding [0.0]
  We propose a novel sequential adaptive surrogate modeling method based on "stochastic spectral embedding" (SSE) We show how the method compares favorably against state-of-the-art sparse chaos expansions on a set of models with different complexity and input dimension.
  arXiv  Detail & Related papers  (2020-04-09T11:00:07Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.