Signal Recovery on Incoherent Manifolds
This work addresses signal recovery in high-dimensional inverse problems for applications like imaging or sensing, representing a significant extension beyond current linear models.
The paper tackles the problem of recovering unknown signals from noisy linear measurements when the signal is a sum of two components from nonlinear sub-manifolds, introducing SPIN, a first-order projected gradient method that provably recovers the components under incoherence and restricted isometry conditions, matching or exceeding state-of-the-art performance.
Suppose that we observe noisy linear measurements of an unknown signal that can be modeled as the sum of two component signals, each of which arises from a nonlinear sub-manifold of a high dimensional ambient space. We introduce SPIN, a first order projected gradient method to recover the signal components. Despite the nonconvex nature of the recovery problem and the possibility of underdetermined measurements, SPIN provably recovers the signal components, provided that the signal manifolds are incoherent and that the measurement operator satisfies a certain restricted isometry property. SPIN significantly extends the scope of current recovery models and algorithms for low dimensional linear inverse problems and matches (or exceeds) the current state of the art in terms of performance.