Representing Vineyard Modules

Time-series of persistence diagrams, known as vineyards, have shown to be useful in diverse applications. A natural algebraic version of vineyards is a time series of persistence modules equipped with interleaving maps between the persistence modules at different time values. We call this a vineyard module. In this paper we will set up the framework for representing vineyards modules via families of matrices and outline an algorithmic way to change the bases of the persistence modules at each time step within the vineyard module to make the matrices in this representation as simple as possible. With some reasonable assumptions on the vineyard modules, this simplified representation of the vineyard module can be completely described (up to isomorphism) by the underlying vineyard and a vector of finite length. We first must set up a lot of preliminary results about changes of bases for persistence modules where we are given $\epsilon$-interleaving maps for sufficiently close $\epsilon$. While this vector representation is not in general guaranteed to be unique we can prove that it will be always zero when the vineyard module is isomorphic to the direct sum of vine modules. This new perspective on vineyards provides an interesting and yet tractable case study within multi-parameter persistence.