An analytic theory of convolutional neural network inverse problems solvers

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 *physics-aware* and *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*).

Problem Deep learning models, specifically convolutional neural networks, are incredibly powerful for many imaging inverse problems, such as denoising, deblurring or tomography. However, they are often treated as black boxes because we do not fully understand the math behind why they work so well. Solution To solve this, we developed a mathematical formula that accurately mimics how these neural networks operate. By incorporating the natural rules these networks follow---such as focusing on small, local image areas---our formula reveals how these models reconstruct images. Instead of magically inventing new visual data, they act like master puzzle solvers, rebuilding an image by intelligently stitching together a patchwork of tiny visual examples they memorized during their training. Impact This discovery is significant because it turns an unpredictable black box into a transparent, predictable tool. By understanding exactly how these networks make decisions, developers can design safer and more reliable AI systems for critical future applications, such as accelerated medical MRI scans.