  To analyze the topological properties of the given discrete data, one needs
to consider a continuous transform called filtration. Persistent homology
serves as a tool to track changes of homology in the filtration. The outcome of
the topological analysis of data varies depending on the choice of filtration,
making the selection of filtration crucial. Filtration learning is an attempt
to find an optimal filtration that minimizes the loss function. Exact
Multi-parameter Persistent Homology (EMPH) has been recently proposed,
particularly for topological time-series analysis, that utilizes the exact
formula of rank invariant instead of calculating it. In this paper, we propose
a framework for filtration learning of EMPH. We formulate an optimization
problem and propose an algorithm for solving the problem. We then apply the
proposed algorithm to several classification problems. Particularly, we derive
the exact formula of the gradient of the loss function with respect to the
filtration parameter, which makes it possible to directly update the filtration
without using automatic differentiation, significantly enhancing the learning
process.
