Learning High-Order Equilibrium Dynamics with Sinkhorn-Constrained Heterogeneous Priors

Deep equilibrium models (DEMs) provide an effective framework for solving inverse problems by modeling the reconstruction process as finding the fixed point of a learned operator. However, they are fundamentally limited by the dilemma between expressivity and stability, i.e., incorporating more expressive or heterogeneous priors violating the contractivity required for convergence. The implicit equilibrium solver could suffer from unstable solving dynamics or degenerated solutions by simply integrating multiple priors that represent complementary image statistics, even well captured by varying neural architectures. To address this challenge, we propose a manifold-constrained hyper-connected deep equilibrium model (MHC-DEM) that achieves multi-prior fusion with a structured transport process. Specifically, we construct a learnable interaction matrix via Sinkhorn normalization that lies on the Birkhoff polytope to project a query distribution into convex fusion weights, and consequently, enforce doubly-stochastic constraints on prior interactions to simultaneously ensure Lipschitz constraints with bounded Lipschitz constants and prevent collapse to a single dominant prior. Furthermore, we design a high-order equilibrium solver based on the stable multi-prior fusion mechanism. We introduce an inertial second-order update in an augmented state space to accelerate convergence, and embed a short inner optimization trajectory within each equilibrium step to perform multi-stage spatial refinement with non-shared parameters. We demonstrate in theory that the proposed multi-prior fusion preserves the contractivity property of neural network based operator represented by MHC-DEM. Extensive experiments show that MHC-DEM achieves consistent gains over existing methods across diverse inverse problems, including image restoration and interferometric imaging, with stable convergence guarantees.