  Unsupervised data representation and visualization using tools from topology
is an active and growing field of Topological Data Analysis (TDA) and data
science. Its most prominent line of work is based on the so-called Mapper
graph, which is a combinatorial graph whose topological structures (connected
components, branches, loops) are in correspondence with those of the data
itself. While highly generic and applicable, its use has been hampered so far
by the manual tuning of its many parameters-among these, a crucial one is the
so-called filter: it is a continuous function whose variations on the data set
are the main ingredient for both building the Mapper representation and
assessing the presence and sizes of its topological structures. However, while
a few parameter tuning methods have already been investigated for the other
Mapper parameters (i.e., resolution, gain, clustering), there is currently no
method for tuning the filter itself. In this work, we build on a recently
proposed optimization framework incorporating topology to provide the first
filter optimization scheme for Mapper graphs. In order to achieve this, we
propose a relaxed and more general version of the Mapper graph, whose
convergence properties are investigated. Finally, we demonstrate the usefulness
of our approach by optimizing Mapper graph representations on several datasets,
and showcasing the superiority of the optimized representation over arbitrary
ones.
