The blooming progress made in deep learning-based image restoration has been largely attributed to the availability of high-quality, large-scale datasets and advanced network structures. However, optimization functions such as L1 and L2 are still de facto. In this study, we propose to investigate new optimization functions to improve image restoration performance. Our key insight is that ``random weight network can be acted as a constraint for training better image restoration networks''. However, not all random weight networks are suitable as constraints. We draw inspiration from Functional theory and show that alternative random weight networks should be represented in the form of a strict mathematical manifold. We explore the potential of our random weight network prototypes that satisfy this requirement: Taylor's unfolding network, invertible neural network, central difference convolution, and zero-order filtering. We investigate these prototypes from four aspects: 1) random weight strategies, 2) network architectures, 3) network depths, and 4) combinations of random weight networks. Furthermore, we devise the random weight in two variants: the weights are randomly initialized only once during the entire training procedure, and the weights are randomly initialized in each training epoch. Our approach can be directly integrated into existing networks without incurring additional training and testing computational costs. We perform extensive experiments across multiple image restoration tasks, including image denoising, low-light image enhancement, and guided image super-resolution to demonstrate the consistent performance gains achieved by our method. Upon acceptance of this paper, we will release the code.