  Patchwork learning arises as a new and challenging data collection paradigm
where both samples and features are observed in fragmented subsets. Due to
technological limits, measurement expense, or multimodal data integration, such
patchwork data structures are frequently seen in neuroscience, healthcare, and
genomics, among others. Instead of analyzing each data patch separately, it is
highly desirable to extract comprehensive knowledge from the whole data set. In
this work, we focus on the clustering problem in patchwork learning, aiming at
discovering clusters amongst all samples even when some are never jointly
observed for any feature. We propose a novel spectral clustering method called
Cluster Quilting, consisting of (i) patch ordering that exploits the
overlapping structure amongst all patches, (ii) patchwise SVD, (iii) sequential
linear mapping of top singular vectors for patch overlaps, followed by (iv)
k-means on the combined and weighted singular vectors. Under a sub-Gaussian
mixture model, we establish theoretical guarantees via a non-asymptotic
misclustering rate bound that reflects both properties of the patch-wise
observation regime as well as the clustering signal and noise dependencies. We
also validate our Cluster Quilting algorithm through extensive empirical
studies on both simulated and real data sets in neuroscience and genomics,
where it discovers more accurate and scientifically more plausible clusters
than other approaches.
