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Exploring Invariance in Images through One-way Wave Equations
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Shows that images can be encoded into a single vector and reconstructed at high fidelity via a first-order norm+linear autoregressive process (FINOLA), interpreted as a set of learned one-way wave equations with latent speeds.
Problem
Understanding how images can be compactly represented and reconstructed without explicit positional information, and whether images share an underlying invariant structure in latent space.
Approach
Encode each image into one or more vectors placed at the center of a feature map and propagate them with a shared first-order norm+linear autoregression along x and y; diagonalizing the learned matrices reveals decorrelated one-way wave equations with (complex) learnable speeds, with multi-path aggregation relaxing the constraint.
Key results
- Single-path FINOLA with C=3072 reaches 25.8 PSNR on ImageNet validation at 256x256; multi-path with C=2048 and M=16 reaches 29.1 (30.0 with scattered initial positions).
- Multi-path FINOLA (2048x16, 32768 latent) outperforms DCT (27.5 at 30720 latent) and DWT (28.7 at 55953 latent) only partially; it beats DWT and DCT at smaller latents.
- Masked FINOLA with Mobile-Former MF-W2880 reaches 83.9 top-1 on ImageNet fine-tuning with 57M params and 9.9G MAdds, comparable to MAE and SimMIM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase the number of FINOLA paths from one to four (4×1024) (ablation) | FINOLA · ImageNet-1K validation, 8×8–256×256 feature maps | PSNR: 25.4 -> 25.9; 25.8 -> 26.2; 25.8 -> 26.2; 25.7 -> 26.1; 25.3 -> 25.4; 24.6 -> 24.8 | Helped |
| Replace repetition with norm+linear autoregression (ablation) | FINOLA · ImageNet-1K validation, 16×16 and 64×64 feature maps | PSNR: 16.1 -> 25.8; 13.3 -> 25.7 | Helped |
| Add normalization to the linear autoregressive process (ablation) | FINOLA · ImageNet-1K validation, 16×16 feature map | PSNR: 25.4 -> 25.8 | Helped |
| Add normalization to the linear autoregressive process (ablation) | FINOLA · ImageNet-1K validation, 64×64 feature map | PSNR: not converge -> 25.7 | Helped |
| Replace layer normalization with batch normalization (ablation) | FINOLA · ImageNet-1K reconstruction validation | PSNR: 25.8 -> 16.3 | Hurt |
| Replace the linear component with two MLP layers and GELU (ablation) | FINOLA · ImageNet-1K reconstruction, C=512, 1024, and 3072 | PSNR: 22.2 -> 22.4; 23.7 -> 23.8; 25.8 -> 25.8 | No clear effect |
| Increase latent channels from 64 to 4096 (ablation) | Single-path FINOLA · ImageNet-1K validation reconstruction | PSNR: 18.2 -> 25.9 | Helped |
| Increase encoder size from 5.0M to 67.6M parameters (ablation) | Single-path FINOLA · ImageNet-1K validation reconstruction | PSNR: 24.4 -> 26.1 | Helped |
| Use multiple FINOLA paths rather than a single path at similar latent sizes | FINOLA · ImageNet-1K validation reconstruction | PSNR: DCT 20.6 vs FINOLA 24.8; DCT 23.5 vs FINOLA 27.1; DCT 25.6 vs FINOLA 28.0; DCT 27.5 vs FINOLA 29.1 | Helped |
| Use FINOLA multi-path instead of Conv-AE with the same latent size | Image reconstruction · ImageNet-1K validation, latent sizes 2048 and 8192 | PSNR: 24.6 -> 24.8; 26.0 -> 27.1 | Helped |
| Use complex rather than real wave speeds (ablation) | FINOLA multi-path (C=1024, M=4) · ImageNet-1K validation reconstruction | PSNR: 25.1 -> 26.1 | Helped |
| Scatter multiple initial conditions at different positions instead of placing them at the center (ablation) | FINOLA multi-path · ImageNet-1K validation reconstruction | PSNR: 28.0 -> 28.9; 29.1 -> 30.0 | Helped |
Surprising findings
- Learned matrices A and B are invertible and their product AB^-1 is diagonalizable with complex eigenvalues across configurations.
- Reconstruction from the mean embedding yields a gray image, and most of the embedding space is occupied by noisy images.
Limitations
- The invariance is revealed empirically without theoretical proof.
- Only multi-path FINOLA is studied, which covers a subspace of the one-way wave equation solutions.
- Gains over strong compression baselines like DTCWT are not consistent; performance lags Stable Diffusion at largest latent size.
Experimental setup
Datasets: ImageNet-1K Kodak
Benchmarks: ImageNet-1K validation (reconstruction PSNR) Kodak ImageNet-1K fine-tuning
Metrics: PSNR bits per pixel top-1 accuracy MAdds parameters
Baselines: DCT DWT DTCWT Convolutional auto-encoder JPEG MAE SimMIM MoCo-v3 Stable Diffusion autoencoder VQGAN DALL-E
Builds on: Mobile-Former MAE SimMIM PixelCNN
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