  We propose a novel framework for combining datasets via alignment of their
intrinsic geometry. This alignment can be used to fuse data originating from
disparate modalities, or to correct batch effects while preserving intrinsic
data structure. Importantly, we do not assume any pointwise correspondence
between datasets, but instead rely on correspondence between a (possibly
unknown) subset of data features. We leverage this assumption to construct an
isometric alignment between the data. This alignment is obtained by relating
the expansion of data features in harmonics derived from diffusion operators
defined over each dataset. These expansions encode each feature as a function
of the data geometry. We use this to relate the diffusion coordinates of each
dataset through our assumption of partial feature correspondence. Then, a
unified diffusion geometry is constructed over the aligned data, which can also
be used to correct the original data measurements. We demonstrate our method on
several datasets, showing in particular its effectiveness in biological
applications including fusion of single-cell RNA sequencing (scRNA-seq) and
single-cell ATAC sequencing (scATAC-seq) data measured on the same population
of cells, and removal of batch effect between biological samples.
