  Identifying molecular signatures from complex disease patients with
underlying symptomatic similarities is a significant challenge in the analysis
of high dimensional multi-omics data. Topological data analysis (TDA) provides
a way of extracting such information from the geometric structure of the data
and identifying multiway higher-order relationships. Here, we propose an
application of Harmonic persistent homology, which overcomes the limitations of
ambiguous assignment of the topological information to the original elements in
a representative topological cycle from the data. When applied to multi-omics
data, this leads to the discovery of hidden patterns highlighting the
relationships between different omic profiles, while allowing for common tasks
in multi-omics analyses, such as disease subtyping, and most importantly
biomarker identification for similar latent biological pathways that are
associated with complex diseases. Our experiments on multiple cancer data show
that harmonic persistent homology effectively dissects multi-omics data to
identify biomarkers by detecting representative cycles predictive of disease
subtypes.
