Polar Alignment and Atomic Decomposition

Abstract

Structured optimization uses a prescribed set of atoms to assemble a solution that fits a model to data. Polarity, which extends the familiar notion of orthogonality from linear sets to general convex sets, plays a special role in a simple and geometric form of convex duality. This duality correspondence yields a general notion of alignment that leads to an intuitive and complete description of how atoms participate in the final decomposition of the solution. The resulting geometric perspective leads to variations of existing algorithms effective for large-scale problems. We illustrate these ideas with many examples, including applications in matrix completion and morphological component analysis for the separation of mixtures of signals.

A derivative-free method for structured optimization problems

“…Then, to obtain the desired contradiction with (16) we can reason similarly as in the proof of Proposition 4, with minor changes that are now described. Define K 1 , K 2 and K 3 as in (4).…”

“…This black-box structured optimization problem is somehow related to sparse atomic decomposition (see, e.g., [8,16] and references therein). In such a context the atomic structure can be exploited when developing tailored solvers for the problem.…”