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Local Curvature Smoothing with Stein's Identity for Efficient Score Matching
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Proposes LCSS, a score matching variant that uses Stein's identity over a Gaussian neighborhood to replace the Jacobian trace, avoiding both DSM's affine-SDE constraint and SSM's random-projection variance.
Problem
Score matching for score-based diffusion models requires an expensive Jacobian trace; existing alternatives either learn an approximate (denoising) score tied to affine SDEs or suffer high variance and instability from random projections.
Approach
Regularize score matching with local curvature smoothing, express it as an expectation over a Gaussian centered at the data point, and apply Stein's identity so the Jacobian trace becomes an inner product of the score with (x'-x)/sigma^2, giving an unbiased-in-form, random-projection-free objective that learns the true score and does not require affine SDE coefficients.
Key results
- On CIFAR-10, LCSS reaches FID 4.90 vs DSM 4.45 for NCSN++ and BPD 2.47 vs 2.64 for DDPM++ (LCSS better on DDPM++ variants, worse on NCSN++ variants).
- Training time on FFHQ (NCSNv2, 1000 iterations) is 1075 ms for LCSS vs 1838 (SSM), 1367 (FD-SSM), 1381 (DSM).
- SSM and FD-SSM collapse to all-black images on CIFAR-10 at 95k and 495k steps, while LCSS keeps producing images; LCSS generates 1024x1024 CelebA-HQ samples.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LCSS instead of SSM | MLP · Checkerboard | training time (ms): 497 -> 419 | Helped |
| Use LCSS instead of DSM | NCSNv2 · FFHQ 256×256 | training time (ms): 1381 -> 1075 | Helped |
| Replace DSM with LCSS | NCSN++ variants · CIFAR-10 | IS: 9.86 -> 9.88; 9.86 -> 9.95 | Helped |
| Replace DSM with LCSS | NCSN++ variants · CIFAR-10 | FID: 4.45 -> 4.90; 4.29 -> 4.72 | Hurt |
| Replace DSM with LCSS | NCSN++ variants · CIFAR-10 | BPD: 3.62 -> 4.17; 3.38 -> 3.61 | Hurt |
| Replace DSM with LCSS | DDPM++ variants · CIFAR-10 | BPD: 2.64 -> 2.47; 2.64 -> 2.58 | Helped |
| Replace DSM with LCSS | DDPM++ variants · CIFAR-10 | FID: 4.81 -> 5.06; 4.49 -> 4.61 | Hurt |
| Replace DSM with LCSS | DDPM++ variants · CIFAR-10 | IS: 9.62 -> 9.63; 9.58 -> 9.80 | Helped |
| Train with LCSS instead of SSM or FD-SSM | NCSNv2 · CIFAR-10, CelebA 64×64, and FFHQ 256×256 | sample generation stability | Helped |
| Train with LCSS instead of DSM | NCSNv2 · AFHQ 256×256 | sample quality | Helped |
| Set the first loss-term coefficient γ to 0.5 (ablation) | LCSS-trained NCSNv2 · FFHQ 256×256 | sample quality | Hurt |
| Set the first loss-term coefficient γ to 10 (ablation) | LCSS-trained NCSNv2 · FFHQ 256×256 | sample quality | Mixed |
Surprising findings
- LCSS surpasses DSM in Inception score on CIFAR-10 but underperforms in FID for NCSN++ models, so quantitative results are mixed.
- DSM training on FFHQ+AFHQ-scale settings showed a sharp quality decline around 210k steps, while LCSS did not.
Limitations
- Only existing affine SDEs (VE and subVP) were used, so the claimed flexibility for non-affine SDEs is not demonstrated.
- Quantitative results versus DSM are mixed: worse FID on NCSN++ and worse BPD on NCSN++ variants on CIFAR-10.
- Stein identity derivation relies on interchangeability assumptions and a first-order approximation (O(epsilon^2) ignored).
Experimental setup
Datasets: CIFAR-10 CelebA CelebA-HQ FFHQ AFHQ Checkerboard
Benchmarks: CIFAR-10 CelebA-HQ 1024x1024 FFHQ AFHQ CelebA Checkerboard
Metrics: FID Inception score bits per dimension elapsed training time visual sample quality
Baselines: SSM FD-SSM DSM
Builds on: denoising score matching sliced score matching finite-difference sliced score matching NCSN score SDE
Closest papers by meaning
- What’s the score? Automated Denoising Score Matching for Nonlinear Diffusions (ICML 2024 · similarity 0.88)
- Score Distillation via Reparametrized DDIM (NeurIPS 2024 · similarity 0.87)
- Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models (NeurIPS 2024 · similarity 0.87)
- Neural Network-Based Score Estimation in Diffusion Models: Optimization and Generalization (ICLR 2024 · similarity 0.87)
- Training Diffusion-based Generative Models with Limited Data (ICML 2025 · similarity 0.86)
- Beyond Scores: Proximal Diffusion Models (NeurIPS 2025 · similarity 0.86)
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