Black-hole VLBI benchmark

Given: A nonnegative image, telescope pairs and projected baselines, the complex visibility and gain/noise model, closure-data transformations on their stated nonzero-visibility domains, and the data-generation split stated in the Example.

Benchmark risk, compute budgets, and uncertainty records: a benchmark instance fixes its forward/observation model, noise description, metric, and evaluation population.

Scope of InverseBench findings: the black-hole results are empirical for the paper's selected image/data setup and do not imply a general ranking or calibration result.

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. All closure values in these lists must satisfy the stated nonzero-visibility restrictions. Let $X=U=C$ be the finite set of labelled cases, with its full power-set sigma-algebra; labels distinguish repeated images. Let $Q={\mathbb{R}}^{{64}^2}$ with Euclidean distance and Borel sets, and let $Y$ be the finite-dimensional observation-and-operator-metadata space with Borel sets. For case $i$, set $q(i)=a_i$ (its target 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). Thus $K$ describes replay conditional on these realized lists, not fresh telescope noise. 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 $N=100$; 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; no shift alignment is used in this chosen squared-error score. 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.