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In this article, we introduce a new parameterized family of topological
invariants, taking the form of candidate decompositions, for multi-parameter
persistence modules. We prove that our candidate decompositions are
controllable approximations: when restricting to modules that can be decomposed
into interval summands, we establish theoretical results about the
approximation error between our candidate decompositions and the true
underlying module in terms of the standard interleaving and bottleneck
distances. Moreover, even when the underlying module does not admit such a
decomposition, our candidate decompositions are nonetheless stable invariants;
small perturbations in the underlying module lead to small perturbations in the
candidate decomposition. Then, we introduce MMA (Multipersistence Module
Approximation): an algorithm for computing stable instances of such invariants,
which is based on fibered barcodes and exact matchings, two constructions that
stem from the theory of single-parameter persistence. By design, MMA can handle
an arbitrary number of filtrations, and has bounded complexity and running
time. Finally, we present empirical evidence validating the generalization
capabilities and running time speed-ups of MMA on several data sets.