Before revision:  That is, given a  set T of merge trees, we would like to find a  basis set S such that each tree in T can be approximately reconstructed from a linear combination of merge trees in S. A set of high-dimensional vectors can be sketched via matrix sketching techniques such as principal component analysis and column subset selection.
After revision:  That is, given a large set T of merge trees, we would like to find a basis set S such that each tree in T can be approximately reconstructed from a linear combination of merge trees in S. A set of high-dimensional vectors can be sketched via matrix sketching techniques such as principal component analysis and column subset selection.