Multi-coil MRI benchmark

Given: The finite complex image space, per-slice ESPIRiT-estimated coil sensitivity maps, the fixed Fourier/subsampling operators, raw coil measurement vectors, the magnitude-MVUE ground-truth convention for the reported metrics, and the train/search/test split stated in the Example.

Benchmark risk, compute budgets, and uncertainty records: a benchmark fixes its observation model, noise description, loss/metric, evaluation law, and tuning protocol.

Scope of InverseBench findings: the empirical MRI results are for the specified multi-coil knee data and test split, not a theorem for other image populations.

For an explicit conditional finite-data benchmark, supply the actual nonempty training, validation, and test lists of the stated sizes, including realized observations, target images, operator metadata, and valid discrete forward values. Let $X=U=C$ be the finite set of labelled cases, with its full power-set sigma-algebra; labels distinguish repeated images. Let $Q={\mathbb{R}}^{{320}^2}$ with Euclidean distance and Borel sets, and let $Y$ be the observation-and-operator-metadata space, a finite disjoint union of finite-dimensional Borel spaces if coil counts or data shapes vary; include each case's coil sensitivities and mask in its observation metadata. For case $i$, set $q(i)=a_i$ (its magnitude-MVUE reference image), $F(i)=f_i$ (its discrete noiseless forward data), and $K(\cdot \mid i)={\delta }_{y_i}$ (its recorded data, with the same metadata). For the coil data, the conditional residual law is the point mass at the recorded residual $y_i^{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}-f_i^{\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}$; it is not asserted Gaussian, independent of the state, or a law for unobserved acquisitions. Thus $K$ describes replay conditional on these realized lists, not a physical law for repeated MRI acquisitions. Set $P_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}=N^{-1}{\sum }_{i\in T}{\delta }_{(y_i,a_i)}$, where $T$ is the test index set and $N=94$; use the analogous uniform empirical validation law. Take $P_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}$ to be the point mass at the entire supplied fitting record, including training/tuning lists and seeds. This conditional specialization supplies laws, not just sample counts, and is not a reconstruction of the paper's full evaluation protocol.

Supply each candidate's complete fitting/tuning procedure and deployed measurable rule $g:Y\to Q$, with all seeds fixed. Fix the loss $\mathrm{\ell }(a,b)=\mathrm{\parallel }a-b{\mathrm{\parallel }}_2^2$ and a radius $r>0$ before testing. The uncertainty diagnostic is empirical coverage $N^{-1}{\sum }_{i\in T}\mathbf{1}\{\mathrm{\parallel }g(y_i)-a_i{\mathrm{\parallel }}_2\le r\}$, targeting coverage of the balls of radius $r$ centered at $g(y)$ under $P_{\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t}}$. Evaluate every test case once and average its loss. Before fitting or evaluation, supply finite upper bounds for every applicable forward, adjoint, gradient, ODE-evaluation, iteration/sampling, and runtime count, marking inapplicable entries as not applicable, together with hardware, precision, software/model versions, seeds, tuning data, and stopping tolerances. A procedure must return within these bounds or record failure; its deployed rule returns the zero image on failure. Retain that case in both aggregates and report every status and realized count. These supplied procedures and budget records are parameters of this mathematical specialization; the coverage statistic is not a reported InverseBench calibration result.