HyDRA: Hybrid Denoising Regularization for Measurement-Only DEQ Training

Solving image reconstruction problems of the form $\mathbf{A} \mathbf{x} = \mathbf{y}$ remains challenging due to ill-posedness and the lack of large-scale supervised datasets. Deep Equilibrium (DEQ) models have been used successfully but typically require supervised pairs $(\mathbf{x},\mathbf{y})$. In many practical settings, only measurements $\mathbf{y}$ are available. We introduce HyDRA (Hybrid Denoising Regularization Adaptation), a measurement-only framework for DEQ training that combines measurement consistency with an adaptive denoising regularization term, together with a data-driven early stopping criterion. Experiments on sparse-view CT demonstrate competitive reconstruction quality and fast inference.