Clustering-independent analysis of genomic data using spectral simplicial theory
Fig 3
Simplicial complexes provide topological representations of a space.
They are generalizations of graphs containing higher dimensional elements such as triangles, tetrahedra, etc. In some cases, higher dimensional elements in a simplicial complex can convey information of a point cloud that is not captured by the underlying graph. For example, in (a) a Čech complex is constructed from intersections of fixed-radius balls centered at the points of a point cloud, which in this example are ordered according to the horizontal coordinate. Simplicial complexes enable the application of co-homological techniques to point cloud features (that is, to functions defined over the elements of the point cloud). Features can be defined over individual points (b, top), pairs of points (b, bottom), triplets, etc. Co-homological techniques, like those discussed in this work, can rank and classify point cloud features according to their amount of localization along topological structures (disconnected components, loops, cavities, etc.) of the underlying simplicial complex. In (c), two examples of point cloud features localized along a topologically trivial region (left) or a non-contractible loop (right) are shown.