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Normalizing Flows are Capable Generative Models
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Proposes TarFlow, a Transformer-based block autoregressive normalizing flow trained by maximum likelihood with Gaussian noise augmentation, score-based denoising, and guidance, achieving SOTA likelihood on ImageNet 64x64 and competitive FID as a standalone NF.
Problem
Normalizing flows have fallen behind other generative model families in image likelihood and sample quality; it is unclear whether NFs are fundamentally limited or just lack a suitable trainable architecture and sampling techniques.
Approach
Stack alternating-direction causal Vision Transformer blocks on image patches as block autoregressive flows (a Transformer generalization of MAF), trained end-to-end with MLE, plus Gaussian noise augmentation, training-free score-based denoising via Tweedie's formula, and a CFG-style guidance applicable to both conditional and unconditional NFs.
Key results
- Achieves 2.99 BPD on unconditional ImageNet 64x64 test set, beating the prior best NFDM (3.20) and the previous flow-based SOTA by a large margin.
- Conditional ImageNet 64x64 FID of 2.66 with the 2-1024-8-8 configuration, better than BigGAN (4.06) and the reported NF baselines.
- Conditional ImageNet 128x128 FID of 5.03 with noise sigma 0.15, competitive with GAN baselines and approaching diffusion results (e.g., ADM-G 2.97).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use TarFlow instead of NFDM | ImageNet 64x64 test likelihood · ImageNet 64x64 test | BPD: 3.20 -> 2.99 | Helped |
| apply guidance with weight 2 instead of guidance 0 (ablation) | TarFlow · ImageNet 64x64 | FID: 25.3 -> 5.7 | Helped |
| apply guidance with weight 2 instead of guidance 0 (ablation) | VP variant of TarFlow · ImageNet 64x64 | FID: 81.5 -> 51.0 | Helped |
| apply guidance with weight 2 instead of guidance 0 (ablation) | Channel-coupling variant of TarFlow · ImageNet 64x64 | FID: 50.3 -> 20.4 | Helped |
| replace non-volume-preserving transformations with volume-preserving transformations (ablation) | TarFlow · ImageNet 64x64 | FID: 5.7 -> 51.0 | Hurt |
| remove causal masks (ablation) | TarFlow · ImageNet 64x64 | FID: 5.7 -> 20.4 | Hurt |
| increase training noise σ from 0.05 to 0.15 (ablation) | TarFlow · Conditional ImageNet 128x128 | FID: 5.29 -> 5.03 | Helped |
| use σ=0.15 rather than σ=0.05 without score-based denoising (ablation) | TarFlow · ImageNet 64x64 | FID | Hurt |
| apply score-based denoising after sampling (ablation) | Noise-augmented TarFlow · ImageNet 64x64 | FID | Helped |
| increase the number of flow blocks or Transformer layers per flow (ablation) | TarFlow · Conditional ImageNet 64x64 | FID | Helped |
| use one flow block (T=1) (ablation) | TarFlow · Conditional ImageNet 64x64 | FID: FID 267 | Hurt |
| use linearly increasing guidance weights by position instead of uniform guidance weights (ablation) | TarFlow · ImageNet 64x64 | FID | Helped |
Surprising findings
- Narrow uniform dequantization noise causes sampling numerical failures, while moderate Gaussian noise (sigma about 0.05) gives much better samples.
- Score-based denoising using only the flow model itself (via Tweedie's formula) removes the injected noise without extra modules.
Limitations
- Sampling is inherently sequential along the autoregressive direction, making generation slower than parallel models.
- Unconditional FID (18.42) lags behind unconditional diffusion and flow-matching baselines on ImageNet 64x64.
- Training used only up to 8 flow blocks and models were trained for limited epochs; the authors note longer training may further improve results.
Experimental setup
Datasets: ImageNet 64x64 ImageNet 128x128 AFHQ 256x256
Benchmarks: ImageNet 64x64 test BPD ImageNet 64x64 conditional FID ImageNet 128x128 conditional FID ImageNet 64x64 unconditional FID AFHQ 256x256
Metrics: bits per dimension FID
Baselines: Glow Flow++ Sparse Transformer Routing Transformer VDM Flow Matching NFDM EDM iDDPM ADM BigGAN BigGAN-deep
Builds on: MAF IAF classifier-free guidance Tweedie's formula Vision Transformer
Closest papers by meaning
- Normalizing Flows with Iterative Denoising (ICML 2026 · similarity 0.94)
- STARFlow: Scaling Latent Normalizing Flows for High-resolution Image Synthesis (NeurIPS 2025 · similarity 0.91)
- Bidirectional Normalizing Flow: From Data to Noise and Back (CVPR 2026 · similarity 0.91)
- Normalizing Flows With Multi-Scale Autoregressive Priors (CVPR 2020 · similarity 0.90)
- Gradient Boosted Normalizing Flows (NeurIPS 2020 · similarity 0.90)
- Normalizing Trajectory Models (NeurIPS 2026 · similarity 0.88)
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