  This article develops a new theoretical basis for decomposing signals that
are formed by the linear superposition of a finite number of modes. Each mode
depends nonlinearly upon several parameters; we seek both these parameters and
the weights within the superposition. The particular focus of this article is
upon solving the decomposition problem when the number of modes is not known in
advance. A sheaf-theoretic formalism is introduced that describes all modal
decomposition problems, and it is shown that minimizing the local consistency
radius within this sheaf is guaranteed to solve them. Since the modes may or
may not be well-separated, a greedy algorithm that identifies the most distinct
modes first may not work reliably. This article introduces a novel mathematical
formalism, aggregation sheaves, and shows how they characterize the behavior of
greedy algorithms that attempt to solve modal decomposition problems.
