  Undersampled inverse problems occur everywhere in the sciences including
medical imaging, radar, astronomy etc., yielding underdetermined linear or
non-linear reconstruction problems. There are now a myriad of techniques to
design decoders that can tackle such problems, ranging from optimization based
approaches, such as compressed sensing, to deep learning (DL), and variants in
between the two techniques. The variety of methods begs for a unifying approach
to determine the existence of optimal decoders and fundamental accuracy bounds,
in order to facilitate a theoretical and empirical understanding of the
performance of existing and future methods. Such a theory must allow for both
single-valued and multi-valued decoders, as underdetermined inverse problems
typically have multiple solutions. Indeed, multi-valued decoders arise due to
non-uniqueness of minimizers in optimisation problems, such as in compressed
sensing, and for DL based decoders in generative adversarial models, such as
diffusion models and ensemble models. In this work we provide a framework for
assessing the lowest possible reconstruction accuracy in terms of worst- and
average-case errors. The universal bounds bounds only depend on the measurement
model $F$, the model class $\mathcal{M}_1 \subseteq \mathcal{X}$ and the noise
model $\mathcal{E}$. For linear $F$ these bounds depend on its kernel, and in
the non-linear case the concept of kernel is generalized for undersampled
settings. Additionally, we provide multi-valued variational solutions that
obtain the lowest possible reconstruction error.
