  Real-valued functions on geometric data -- such as node attributes on a graph
-- can be optimized using descriptors from persistent homology, allowing the
user to incorporate topological terms in the loss function. When optimizing a
single real-valued function (the one-parameter setting), there is a canonical
choice of descriptor for persistent homology: the barcode. The operation
mapping a real-valued function to its barcode is differentiable almost
everywhere, and the convergence of gradient descent for losses using barcodes
is relatively well understood. When optimizing a vector-valued function (the
multiparameter setting), there is no unique choice of descriptor for
multiparameter persistent homology, and many distinct descriptors have been
proposed. This calls for the development of a general framework for
differentiability and optimization that applies to a wide range of
multiparameter homological descriptors. In this article, we develop such a
framework and show that it encompasses well-known descriptors of different
flavors, such as signed barcodes and the multiparameter persistence landscape.
We complement the theory with numerical experiments supporting the idea that
optimizing multiparameter homological descriptors can lead to improved
performances compared to optimizing one-parameter descriptors, even when using
the simplest and most efficiently computable multiparameter descriptors.
