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Variational regularization methods are commonly used to approximate solutions of inverse problems. In recent years, model-based variational regularization methods have often been replaced with data-driven ones such as the fields-of-expert model [32]. Training the parameters of such data-driven methods can be formulated as a bilevel optimization problem. In this paper, we compare the framework of bilevel learning for the training of data-driven variational regularization models with the novel framework of deep equilibrium models [3] that has recently been introduced in the context of inverse problems [13]. We show that computing the lower-level optimization problem within the bilevel formulation with a fixed point iteration is a special case of the deep equilibrium framework. We compare both approaches computationally, with a variety of numerical examples for the inverse problems of denoising, inpainting and deconvolution.
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Figure 1. Comparison between bilevel optimization and deep equilibrium models for each of the three considered inverse problems, namely denoising, inpainting, and deblurring, over all the range of possible parameters. These boxplots consider the loss of the trained models evaluated on the test dataset. We removed all the configurations with a final loss larger than $ 0.5 $, a value we arbitrarily chose by looking for an empirical relation between the loss and the image quality
Figure 2. Denoising the MNIST dataset. Visual comparison between bilevel method (left) and deep equilibrium model (right), with parameters $ \tau = 0.5 $, $ \gamma = 0.1 $, and $ \sigma = \text{(ReLU)} $. Images are taken from the test dataset. The first row shows the original images; the second row is the model input. The last row is the output of the trained models
Figure 3. Inpainting MNIST. Comparison between bilevel method (left) and deep equilibrium model (right), with parameters $ \tau = 0.5 $, $ \gamma = 1.0 $, and $ \sigma = \text{(Softshrink)} $. Images are taken from the test dataset. The first row shows the original image, the second row is the masked image, i.e., the input of the algorithm. The fourth row is the output of the trained models. Finally, the third row shows what happens when we apply the inpainting operator on the output. The fourth row is the output of the trained deep equilibrium optimization problem. Ideally, the difference between the second and third row should be small
Figure 4. Deblurring MNIST. Comparison between bilevel method (left) and deep equilibrium model (right), with parameters $ \tau = 0.5 $, $ \gamma = 0.5 $, and $ \sigma = \text{(Softshrink)} $. Images are taken from the test dataset. The first row shows the original images; the second row is the model input. The last row is the output of the trained models. The third row shows the model output after we apply the convolution kernel to it. Ideally, the difference between the second and the third rows should be small
Figure 5. Comparison of the loss error for the test dataset evaluated after each training epoch, for increasing values of noise levels in training (noise levels from top to bottom row: $ 0, 0.05, 0.1, 0.5, 1 $). Simulations are grouped by the tasks, namely denoising, inpainting, and deblurring (left, center, right columns). Each plot shows the simulation with the configurations that achieve the lowest final test loss
Figure 7. Reconstruction for the inpainting task for a bilevel optimization model whose parameters have been trained by minimizing the error of the reconstruction $ u^* $ w.r.t. the true image $ u^\dagger $ (naïve approach). We show how the reconstruction $ \{u^k\} $ changes for different values of the iteration $ k $. As we can see, the model trained with the naïve approach is not able to inpaint the masked area
Figure 8. Denoising CelebA; sample from the test dataset. The first row contains the original image $ u $ and the noisy image $ f^\delta $. From left to right in the second and third rows: reconstructed image with random initializations of the kernels (left), with parameters learned using bilevel learning (center), and parameters learned using the DEQ model (right)
Figure 10. Comparison of $ 11\times 11 $ kernels of $ A $ (first two columns on the left) and of $ C^\top $ (two columns on the right), shown before and after training on the denoising task using the DEQ model. Note that, in the first row, the pairs of kernels in the first-third columns and the second-fourth are the same; this is because we initialize the kernels so that $ C^\top = A^\top $ before training. After the training, they are different (second row)
Figure 12. Comparison of $ 3\times 3 $ kernels of $ A $ (first five columns on the left) and of $ C^\top $ (five columns on the right), shown before and after training on the denoising task using the DEQ model. Note that, in the first row, kernels are pairwise equal; this is because we initialize the kernels so that $ C^\top = A^\top $ before training. After the training, they are different (second row)
Figure 13. Deblurring CelebA, a sample from the test dataset. The first row contains the original image $ u $ and the noisy blurred image $ f^\delta $. From left to right in the second and third rows: reconstructed image with the optimal kernels found for the denoising task in the bilevel scenario (left), with parameters learned using bilevel learning (center-left), with parameters learned using DEQ model (center-right), and optimal kernels found for the denoising task in the DEQ scenario
Figure 15. Comparison of $ 11\times 11 $ kernels of $ A $ (first two columns on the left) and of $ C^\top $ (two columns on the right), shown before and after training on the deblurring task using the DEQ model. Note that, in the first row, the pairs of kernels in the first-third columns and second-fourth are the same; this is because we initialize the kernels so that $ C^\top = A^\top $ before training. After the training, they are different (second row)
Figure 17. Comparison of $ 3\times 3 $ kernels of $ A $ (first five columns on the left) and of $ C^\top $ (five columns on the right), shown before and after training on the deblurring task using the DEQ model. Note that, in the first row, kernels are pairwise equal; this is because we initialize the kernels so that $ C^\top = A^\top $ before training
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Comparison between bilevel optimization and deep equilibrium models for each of the three considered inverse problems, namely denoising, inpainting, and deblurring, over all the range of possible parameters. These boxplots consider the loss of the trained models evaluated on the test dataset. We removed all the configurations with a final loss larger than
Denoising the MNIST dataset. Visual comparison between bilevel method (left) and deep equilibrium model (right), with parameters
Inpainting MNIST. Comparison between bilevel method (left) and deep equilibrium model (right), with parameters
Deblurring MNIST. Comparison between bilevel method (left) and deep equilibrium model (right), with parameters
Comparison of the loss error for the test dataset evaluated after each training epoch, for increasing values of noise levels in training (noise levels from top to bottom row:
These histograms show how many simulations were finished within an hour as a function of the number of epochs. Each simulation is a different configuration of hyperparameters. We consider only those runs where the loss on the test dataset is smaller than
Reconstruction for the inpainting task for a bilevel optimization model whose parameters have been trained by minimizing the error of the reconstruction
Denoising CelebA; sample from the test dataset. The first row contains the original image
Comparison of
Comparison of
Comparison of five (out of thirty)
Comparison of
Deblurring CelebA, a sample from the test dataset. The first row contains the original image
Comparison of
Comparison of
Comparison of five (out of thirty)
Comparison of