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 + ϵ, where x 0 is the ground truth and ϵ 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 ‖ x ̂ − x 0 ‖ 2 ⩽ C ‖ ϵ ‖ 2 with constants C 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.

Low rank recovery problems have been a subject of intense study in recent years. While the rank function is useful for regularization it is difficult to optimize due to its non-convexity and discontinuity. The standard remedy for this is to exchange the rank function for the convex nuclear norm, which is known to favor low rank solutions under certain conditions. On the downside the nuclear norm exhibits a shrinking bias that can severely distort the solution in the presence of noise, which motivates the use of stronger non-convex alternatives. In this paper we study two such formulations. We characterize the critical points and give sufficient conditions for a low rank stationary point to be unique. Moreover, we derive conditions that ensure global optimality of the low ranks stationary point and show that these hold under moderate noise levels.

Given underdetermined measurements of a Positive Semi-Definite (PSD) matrix X of known low rank K, we present a new algorithm to estimate X based on recent advances in non-convex optimization schemes. We apply this in particular to the phase retrieval problem for Fourier data, which can be formulated as a rank 1 PSD matrix recovery problem. Moreover, we provide theory for how oversampling affects the stability of the lifted inverse problem.

Cardinality and rank functions are ideal ways of regularizing under-determined linear systems, but optimization of the resulting formulations is made difficult since both these penalties are non-convex and discontinuous. The most common remedy is to instead use the $$\ell ^1$$ ℓ 1 - and nuclear norms. While these are convex and can therefore be reliably optimized, they suffer from a shrinking bias that degrades the solution quality in the presence of noise. This well-known drawback has given rise to a fauna of non-convex alternatives, which usually features better global minima at the price of maybe getting stuck in undesired local minima. We focus in particular penalties based on the quadratic envelope, which have been shown to have global minima which even coincide with the “oracle solution,” i.e., there is no bias at all. So, which one do we choose, convex with a definite bias, or non-convex with no bias but less predictability? In this article, we develop a framework which allows us to interpolate between these alternatives; that is, we construct sparsity inducing penalties where the degree of non-convexity/bias can be chosen according to the specifics of the particular problem.

In R 3 we consider the vector fieldsbe a function such that X 1 f, X 2 f ∈ L p (R 3 + ) for some p > 1. In this paper, we prove that the restriction of f to the plane z = 0 belongs to a suitable Besov space that is defined using the Carnot-Carathéodory metric associated with X 1 and X 2 and the related perimeter measure.