  We develop a conformal inference method to construct joint confidence regions
for structured groups of missing entries within a sparsely observed matrix.
This method is useful to provide reliable uncertainty estimation for
group-level collaborative filtering; for example, it can be applied to help
suggest a movie for a group of friends to watch together. Unlike standard
conformal techniques, which make inferences for one individual at a time, our
method achieves stronger group-level guarantees by carefully assembling a
structured calibration data set mimicking the patterns expected among the test
group of interest. We propose a generalized weighted conformalization framework
to deal with the lack of exchangeability arising from such structured
calibration, and in this process we introduce several innovations to overcome
computational challenges. The practicality and effectiveness of our method are
demonstrated through extensive numerical experiments and an analysis of the
MovieLens 100K data set.
