Sparse non-negative super-resolution — simplified and stabilised

Abstract

The convolution of a discrete measure, x " ř k i"1 aiδt i , with a local window function, φps´tq, is a common model for a measurement device whose resolution is substantially lower than that of the objects being observed. Super-resolution concerns localising the point sources tai, tiu k i"1 with an accuracy beyond the essential support of φps´tq, typically from m samples ypsjq " ř k i"1 aiφpsj´tiq`δj , where δj indicates an inexactness in the sample value. We consider the setting of x being non-negative and seek to characterise all non-negative measures approximately consistent with the samples. We first show that x is the unique non-negative measure consistent with the samples provided the samples are exact, i.e. δj " 0, m ě 2k`1 samples are available, and φps´tq generates a Chebyshev system. This is independent of how close the sample locations are and does not rely on any regulariser beyond non-negativity; as such, it extends and clarifies the work by Schiebinger et al. in [1] and De Castro et al. in [2], who achieve the same results but require a total variation regulariser, which we show is unnecessary.Moreover, we characterise non-negative solutionsx consistent with the samples within the bound ř m j"1 δ 2 j ď δ 2 . Any such non-negative measure is within Opδ 1{7 q of the discrete measure x generating the samples in the generalised Wasserstein distance. Similarly, we show using somewhat different techniques that the integrals ofx and x over pti´ǫ, ti`ǫq are similarly close, converging to one another as ǫ and δ approach zero. We also show how to make these general results, for windows that form a Chebyshev system, precise for the case of φps´tq being a Gaussian window. The main innovation of these results is that non-negativity alone is sufficient to localise point sources beyond the essential sensor resolution and that, while regularisers such as total variation might be particularly effective, they are not required in the non-negative setting.

Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems

“…The authors establish a sampling theorem for Gaussian and Ricker-wavelet convolution kernels, which characterizes what sampling patterns yield exact recovery under a minimum-separation condition. Other works have analyzed the Gaussian deconvolution problem under nonnegativity constraints [33,75], and also for randomized measurements [67]. All of these works exploit the properties of specific measurement operators.…”

Sparse Inverse Problems over Measures: Equivalence of the Conditional Gradient and Exchange Methods

“…This is indeed a difficult task: Even given the red curve φ ẋ (from whichẏ is sampled), it is hard to see that there is an impulse located atṫ 3 . Solving Program (1.1) with x T V ≤ b for large enough b uniquely recovers x, as proved in [1]. In this paper, we describe Algorithms 3.1 and 4.1 to solve Program (1.1), and establish their equivalence.…”

“…is the total variation of measure x, see for example [1]. 1 We are particularly interested in the case where L : C m → R is a differentiable loss function and Φ : I → C m is a continuous function. Note that Program (1.1) is an infinite-dimensional problem and that the constraints ensure that the problem is bounded.…”

“…To that end, Program (1.1) searches for a nonnegative measure x supported on I that minimizes the loss L( I Φ(t)x(dt) − y), while encouraging its sparsity through the total variation constraint x T V ≤ 1. Under certain conditions on Φ and when L = 1 2 · 2 2 , a minimizer x of Program (1.1) is a robust estimate of the true measureẋ in the sense that d( x,ẋ) ≤ c · L(y −ẏ) for a known factor c and in a certain metric d [1,15,7,16].…”

“…3Chebyshev systems are widely used in classical approximation theory and generalize the notion of ordinary polynomials. Many functions form Chebyshev systems, for example sinusoids or translated copies of the Gaussian window, and we refer the interested reader to[46,47,1] for more on their properties and applications. Let C I ⊂ C m be the convex hull of {Φ(t)} t∈I ∪ {0}, namely(7.1) C I := I Φ(t)x(dt) : x ∈ B + (I), x T V ≤ 1 .…”