Single-cell multi-omics data provide a comprehensive molecular view of cells. However, single-cell multi-omics datasets consist of unpaired cells measured with distinct unmatched features across modalities, making data integration challenging. In this study, we present a novel algorithm, termed UnionCom, for the unsupervised topological alignment of single-cell multiomics integration. UnionCom does not require any correspondence information, either among cells or among features. It first embeds the intrinsic low-dimensional structure of each single-cell dataset into a distance matrix of cells within the same dataset and then aligns the cells across single-cell multi-omics datasets by matching the distance matrices via a matrix optimization method. Finally, it projects the distinct unmatched features across single-cell datasets into a common embedding space for feature comparability of the aligned cells. To match the complex nonlinear geometrical distorted low-dimensional structures across datasets, UnionCom proposes and adjusts a global scaling parameter on distance matrices for aligning similar topological structures. It does not require one-to-one correspondence among cells across datasets, and it can accommodate samples with dataset-specific cell types. UnionCom outperforms state-of-theart methods on both simulated and real single-cell multi-omics datasets. UnionCom is robust to parameter choices, as well as subsampling of features. UnionCom software is available at

Motivation Single-cell multi-omics sequencing data can provide a comprehensive molecular view of cells. However, effective approaches for the integrative analysis of such data are challenging. Existing manifold alignment methods demonstrated the state-of-the-art performance on single-cell multi-omics data integration, but they are often limited by requiring that single-cell datasets be derived from the same underlying cellular structure. Results In this study, we present Pamona, a partial Gromov-Wasserstein distance based manifold alignment framework that integrates heterogeneous single-cell multi-omics datasets with the aim of delineating and representing the shared and dataset-specific cellular structures across modalities. We formulate this task as a partial manifold alignment problem and develop a partial Gromov-Wasserstein optimal transport framework to solve it. Pamona identifies both shared and dataset-specific cells based on the computed probabilistic couplings of cells across datasets, and it aligns cellular modalities in a common low-dimensional space, while simultaneously preserving both shared and dataset-specific structures. Our framework can easily incorporate prior information, such as cell type annotations or cell-cell correspondence, to further improve alignment quality. We evaluated Pamona on a comprehensive set of publicly available benchmark datasets. We demonstrated that Pamona can accurately identify shared and dataset-specific cells, as well as faithfully recover and align cellular structures of heterogeneous single-cell modalities in a common space, outperforming the comparable existing methods. Availability Pamona software is available at https://github.com/caokai1073/Pamona.