An unbiased approach to compressed sensing
Artikel i vetenskaplig tidskrift, 2020

In compressed sensing a sparse vector is approximately retrieved from an under-determined equation system Ax = b. Exact retrieval would mean solving a large combinatorial problem which is well known to be NP-hard. For b of the form Ax(0) + epsilon, where x(0) is the ground truth and epsilon is noise, the 'oracle solution' is the one you get if you a priori know the support of x(0), and is the best solution one could hope for. We provide a non-convex functional whose global minimum is the oracle solution, with the property that any other local minimizer necessarily has high cardinality. We provide estimates of the type parallel to(x) over cap - x(0)parallel to(2) <= C parallel to epsilon parallel to(2) that are significantly lower than for competing methods or theorems, and our theory relies on soft assumptions on the matrix A, in comparison with standard results in the field. The framework also allows to incorporate a priori information on the cardinality of the sought vector. In this case we show that despite being non-convex, our cost functional has no spurious local minima and the global minima is again the oracle solution, thereby providing the first method which is guaranteed to find this point for reasonable levels of noise, without resorting to combinatorial methods.

non-smooth optimization

regularization

non-convex optimization

compressed sensing

Författare

Marcus Carlsson

Lunds universitet

Daniele Gerosa

Lunds universitet

Carl Olsson

Chalmers, Elektroteknik, Signalbehandling och medicinsk teknik, Digitala bildsystem och bildanalys

Lunds universitet

Inverse Problems

0266-5611 (ISSN)

Vol. 36 11 115014

Ämneskategorier

Beräkningsmatematik

Reglerteknik

Matematisk analys

DOI

10.1088/1361-6420/abbd7f

Mer information

Senast uppdaterat

2020-12-04