An analytic theory of convolutional neural network inverse problems solvers
This provides theoretical insights for researchers and practitioners using CNNs in imaging, though it is incremental as it builds on existing MMSE frameworks.
The paper tackled the lack of theoretical understanding of convolutional neural networks (CNNs) in solving imaging inverse problems by deriving an analytic formula for a constrained Minimum Mean Square Error estimator that matches CNN outputs with PSNR values greater than or equal to 25 dB across various datasets and architectures.
Supervised convolutional neural networks (CNNs) are widely used to solve imaging inverse problems, achieving state-of-the-art performance in numerous applications. However, despite their empirical success, these methods are poorly understood from a theoretical perspective and often treated as black boxes. To bridge this gap, we analyze trained neural networks through the lens of the Minimum Mean Square Error (MMSE) estimator, incorporating functional constraints that capture two fundamental inductive biases of CNNs: translation equivariance and locality via finite receptive fields. Under the empirical training distribution, we derive an analytic, interpretable, and tractable formula for this constrained variant, termed Local-Equivariant MMSE (LE-MMSE). Through extensive numerical experiments across various inverse problems (denoising, inpainting, deconvolution), datasets (FFHQ, CIFAR-10, FashionMNIST), and architectures (U-Net, ResNet, PatchMLP), we demonstrate that our theory matches the neural networks outputs (PSNR $\gtrsim25$dB). Furthermore, we provide insights into the differences between \emph{physics-aware} and \emph{physics-agnostic} estimators, the impact of high-density regions in the training (patch) distribution, and the influence of other factors (dataset size, patch size, etc).