Towards a Unified Theoretical Framework for Splitting-based Self-Supervised MRI Reconstruction

The demand for high-resolution, non-invasive imaging continues to drive innovation in magnetic resonance imaging (MRI), but long acquisition times remain a major practical limitation. Although deep learning-based reconstruction methods have enabled accelerated imaging, their predominant supervised paradigm relies on fully-sampled reference data that are difficult to acquire in practice. Self-supervised learning (SSL) has therefore emerged as a promising alternative, among which splitting methods are a widely used strategy. However, most existing splitting-based methods are empirically designed, and a unified theoretical understanding remains limited. In this work, we introduce UNITS (Unified Theory for Splitting-based self-supervision), a general theoretical framework for splitting-based self-supervised MRI reconstruction. Theoretically, we show that the self-supervised risk can be expressed as a weighted supervised risk. Consequently, self-supervision admits the same pointwise Bayes-optimal predictor as supervised learning. We further relate the training residual to the prediction bias, revealing how different sampling mechanisms affect training behavior. UNITS makes a broad class of existing methods interpretable as special cases within a common framework, and provides a general design space through sampling stochasticity and flexible data utilization. Together, these contributions establish UNITS as a theoretical foundation, a practical paradigm, and a benchmark for interpretable, generalizable, and applicable self-supervised MRI reconstruction.