Super-resolution is generally referred to as the task of recovering fine details from coarse information. Motivated by applications such as single-molecule imaging, radar imaging, etc., we consider parameter estimation of complex exponentials from their modulations with unknown waveforms, allowing for non-stationary blind super-resolution. This problem, however, is ill-posed since both the parameters associated with the complex exponentials and the modulating waveforms are unknown. To alleviate this, we assume that the unknown waveforms live in a common low-dimensional subspace. Using a lifting trick, we recast the blind super-resolution problem as a structured low-rank matrix recovery problem. Atomic norm minimization is then used to enforce the structured low-rankness, and is reformulated as a semidefinite program that is solvable in polynomial time. We show that, up to scaling ambiguities, exact recovery of both of the complex exponential parameters and the unknown waveforms is possible when the waveform subspace is random and the number of measurements is proportional to the number of degrees of freedom in the problem. Numerical simulations support our theoretical findings, showing that non-stationary blind super-resolution using atomic norm minimization is possible.
Our work is most closely related to the recent works @cite_22 @cite_17 . In @cite_17 , a biconvex problem for simultaneous sparse recovery and unknown gain calibration is studied. In their work, a subspace model is employed for the unknown gains to make the problem well-posed. It is worth mentioning that they use @math minimization as a convex program, which is different from ours. Then, a sample complexity bound that is suboptimal is derived for sparse recovery and self-calibration. Inspired by @cite_17 , @cite_22 considers a super-resolution problem that has a similar setup to @cite_27 , except that the point spread function is assumed unknown. By employing a subspace model for the point spread function, an atomic norm minimization program is formulated for simultaneous super-resolution of point sources and recovery of the unknown point spread function. The atomic norm minimization problem therein is recast as an SDP. The sample complexity bound derived there, however, is suboptimal. As we explain in , our work further generalizes the model in @cite_22 to the non-stationary case, where the point spread functions can vary with the point sources.
@cite_27 @cite_22 @cite_17
2166207339 2003571173 2041507040 2096586829
We investigate the problem of reconstructing a high-dimensional nonnegative sparse vector from lower-dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes can be more efficient both with respect to signal sensing as well as reconstruction complexity. Known constructions use the adjacency matrices of expander graphs, which often lead to recovery algorithms which are much more efficient than minimization. However, prior constructions of sparse measurement matrices rely on expander graphs with very high expansion coefficients which make the construction of such graphs difficult and the size of the recoverable sets very small. In this paper, we introduce sparse measurement matrices for the recovery of nonnegative vectors, using perturbations of the adjacency matrices of expander graphs requiring much smaller expansion coefficients, hereby referred to as minimal expanders. We show that when minimization is used as the reconstruction method, these constructions allow the recovery of signals that are almost three orders of magnitude larger compared to the existing theoretical results for sparse measurement matrices. We provide for the first time tight upper bounds for the so called weak and strong recovery thresholds when minimization is used. We further show that the success of optimization is equivalent to the existence of a “unique” vector in the set of solutions to the linear equations, which enables alternative algorithms for minimization. We further show that the defined minimal expansion property is necessary for all measurement matrices for compressive sensing, (even when the non-negativity assumption is removed) therefore implying that our construction is tight. We finally present a novel recovery algorithm that exploits expansion and is much more computationally efficient compared to minimization. This paper considers the problem of recovering an unknown sparse p×p matrix X from an m×m matrix Y=AXB T , where A and B are known m×p matrices with m≪p. The main result shows that there exist constructions of the sketching matrices A and B so that even if X has O(p) nonzeros, it can be recovered exactly and efficiently using a convex program as long as these nonzeros are not concentrated in any single row column of X. Furthermore, it suffices for the size of Y (the sketch dimension) to scale as m = O(√(# nonzeros in X) × log p). The results also show that the recovery is robust and stable in the sense that if X is equal to a sparse matrix plus a perturbation, then the convex program we propose produces an approximation with accuracy proportional to the size of the perturbation. Unlike traditional results on sparse recovery, where the sensing matrix produces independent measurements, our sensing operator is highly constrained (it assumes a tensor product structure). Therefore, proving recovery guarantees require nonstandard techniques. Indeed, our approach relies on a novel result concerning tensor products of bipartite graphs, which may be of independent interest. This problem is motivated by the following application, among others. Consider a p×n data matrix D, consisting of n observations of p variables. Assume that the correlation matrix X:=DD T is (approximately) sparse in the sense that each of the p variables is significantly correlated with only a few others. Our results show that these significant correlations can be detected even if we have access to only a sketch of the data S=AD with A ∈ R m×p . We investigate a compressive sensing framework in which the sensors introduce a distortion to the measurements in the form of unknown gains. We focus on blind calibration, using measures performed on multiple unknown (but sparse) signals and formulate the joint recovery of the gains and the sparse signals as a convex optimization problem. We divide this problem in 3 sub-problems with different conditions on the gains, specifially i) gains with different amplitude and the same phase, ii) gains with the same amplitude and different phase and iii) gains with different amplitude and phase. In order to solve the first case, we propose an extension to the basis pursuit optimization which can estimate the unknown gains along with the unknown sparse signals. For the second case, we formulate a quadratic approach that eliminates the unknown phase shifts and retrieves the unknown sparse signals. An alternative form of this approach is also formulated to reduce complexity and memory requirements and provide scalability with respect to the number of input signals. Finally for the third case, we propose a formulation that combines the earlier two approaches to solve the problem. The performance of the proposed algorithms is investigated extensively through numerical simulations, which demonstrates that simultaneous signal recovery and calibration is possible with convex methods when sufficiently many (unknown, but sparse) calibrating signals are provided. This paper studies the problem of recovering a sparse signal x ∈ ℝn from highly corrupted linear measurements y = Ax + e ∈ ℝm, where e is an unknown error vector whose nonzero entries may be unbounded. Motivated by an observation from face recognition in computer vision, this paper proves that for highly correlated (and possibly overcomplete) dictionaries A, any sufficiently sparse signal x can be recovered by solving an l1 -minimization problem min ||x||1 + ||e||1 subject to y = Ax + e. More precisely, if the fraction of the support of the error e is bounded away from one and the support of a: is a very small fraction of the dimension m, then as m becomes large the above l1 -minimization succeeds for all signals x and almost all sign-and-support patterns of e. This result suggests that accurate recovery of sparse signals is possible and computationally feasible even with nearly 100 of the observations corrupted. The proof relies on a careful characterization of the faces of a convex polytope spanned together by the standard crosspolytope and a set of independent identically distributed (i.i.d.) Gaussian vectors with nonzero mean and small variance, dubbed the "cross-and-bouquet" (CAB) model. Simulations and experiments corroborate the findings, and suggest extensions to the result.