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The Star Geometry of Critic-Based Regularizer Learning
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Studies optimal critic-based regularizers over star-body gauges, showing the adversarial (Wasserstein) objective equals a dual mixed volume with a data-dependent star body, and proposes a Hellinger-based loss with a small MNIST denoising experiment.
Problem
Understanding what structure regularizers learned via critic-based (adversarial/variational) losses have and how it relates to the clean and noisy data distributions, which is largely unknown theoretically.
Approach
Restricts regularizers to star body gauges and uses star geometry and dual Brunn-Minkowski theory to rewrite the adversarial Wasserstein objective as a dual mixed volume, giving exact optimal regularizers in some cases; also derives Hellinger (alpha=1/2) divergence-based losses with similar dual mixed volume interpretations.
Key results
- On MNIST denoising (sigma^2=0.05), the Hellinger-based regularizer achieves 23.06 PSNR / 0.0052 MSE, vs 22.32 PSNR for the adversarial regularizer and 20.52 PSNR for TV.
- A further-tuned adversarial regularizer reaches 24.14 PSNR / 0.0041 MSE, slightly outperforming the Hellinger-based loss.
- Under distributional containment assumptions, the adversarial objective's unique minimizer over unit-volume star bodies is characterized as a dilate of a data-dependent star body L_{r,n}.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the Hellinger-based loss instead of the adversarial loss with recommended regularization strength | Deep convolutional neural network regularizer · MNIST denoising | PSNR: 22.32 -> 23.06 | Helped |
| Use the Hellinger-based loss instead of the adversarial loss with recommended regularization strength | Deep convolutional neural network regularizer · MNIST denoising | MSE: 0.0064 -> 0.0052 | Helped |
| Tune the regularization parameter for the adversarial regularizer instead of using the recommended strength | Adversarial regularizer · MNIST denoising | PSNR: 22.32 -> 24.14 | Helped |
| Tune the regularization parameter for the adversarial regularizer instead of using the recommended strength | Adversarial regularizer · MNIST denoising | MSE: 0.0064 -> 0.0041 | Helped |
| Use the Hellinger-based loss instead of the tuned adversarial loss | Deep convolutional neural network regularizer · MNIST denoising | PSNR: 24.14 -> 23.06 | Hurt |
| Use the Hellinger-based loss instead of the tuned adversarial loss | Deep convolutional neural network regularizer · MNIST denoising | MSE: 0.0041 -> 0.0052 | Hurt |
| Use the Hellinger-based learned regularizer instead of total variation regularization | MNIST denoising · MNIST denoising | PSNR: 20.52 -> 23.06 | Helped |
| Use the Hellinger-based learned regularizer instead of total variation regularization | MNIST denoising · MNIST denoising | MSE: 0.0091 -> 0.0052 | Helped |
Surprising findings
- The Leaky ReLU (positively homogeneous) regularizer outperforms networks with non-homogeneous activations such as Tanh or GELU in denoising.
Limitations
- Main existence and uniqueness results rely on regularity and containment assumptions on densities (radial map positive and continuous) that fail for empirical distributions and singular inverse-problem noise distributions.
- Global minimizers of the Hellinger-inspired loss are not precisely characterized.
- Empirical evaluation is limited to a simple MNIST denoising setting and does not test inverse problems.
Experimental setup
Datasets: MNIST
Benchmarks: MNIST denoising
Metrics: PSNR MSE
Baselines: adversarial regularization total variation regularization
Builds on: adversarial regularization Wasserstein distance dual formulation dual mixed volumes (Lutwak) variational representation of alpha-divergences total variation regularization
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