Achieving weighted exchangeability for structured calibration in SCMC

Develop a modification of the structured calibration assembly (Algorithm 2: Assembling the structured calibration set) used in Structured Conformalized Matrix Completion so that the joint distribution of the calibration groups (X^{cal}_1, …, X^{cal}_n) and the test group X^* satisfies weighted exchangeability in the sense of Tibshirani et al. (2019). Establish such a modification explicitly and demonstrate how it enables the simplified computation of conformalization weights associated with weighted exchangeability.

Background

In the proposed SCMC framework, the calibration groups are constructed to mimic the structure of the test group by sampling K observed entries from a single column repeatedly. This structured calibration breaks standard exchangeability assumptions, complicating the use of conformal inference techniques that rely on symmetry properties.

Under the weighted exchangeability framework of Tibshirani et al. (2019), conformalization weights simplify dramatically, but the authors note that their structured groups (X^{cal}_1, …, X^{cal}_n, X^*) are not weighted exchangeable. They therefore derive new leave-one-out exchangeability results and corresponding weights, which are more complex to evaluate.

The explicit uncertainty concerns whether and how Algorithm 2 could be modified to achieve weighted exchangeability, which would allow the direct application of the simplified conformalization weights from the covariate shift framework and potentially reduce computational burden.