[Paper Review] Soft Recovery With General Atomic Norms

Axel Flinth|arXiv (Cornell University)|May 10, 2017

Sparse and Compressive Sensing Techniques15 references3 citations

TL;DR

This paper introduces a general dual certificate condition for atomic norm minimization that guarantees soft recovery of structured signals—specifically, approximate localization of atoms in a sparse decomposition—under mild assumptions on the measurement operator. The key contribution is a theoretical framework ensuring that peaks in the atomic decomposition of the solution are close to those in the true signal, even in infinite-dimensional settings, with applications to super-resolution and matrix recovery.

ABSTRACT

This paper describes a dual certificate condition on a linear measurement operator $A$ (defined on a Hilbert space $\mathcal{H}$ and having finite-dimensional range) which guarantees that an atomic norm minimization, in a certain sense, will be able to approximately recover a structured signal $v_0 \in \mathcal{H}$ from measurements $Av_0$. Put very streamlined, the condition implies that peaks in a sparse decomposition of $v_0$ are close the the support of the atomic decomposition of the solution $v^*$. The condition applies in a relatively general context - in particular, the space $\mathcal{H}$ can be infinite-dimensional. The abstract framework is applied to several concrete examples, one example being super-resolution. In this process, several novel results which are interesting on its own are obtained.

Motivation & Objective

• To establish a general theoretical framework for soft recovery in structured inverse problems where exact recovery is not feasible.
• To address the instability of atomic norm minimization in coherent dictionaries by proving approximate recovery of atomic components.
• To extend existing compressed sensing guarantees beyond exact recovery to include approximate localization of signal atoms.
• To provide a unified analysis applicable to infinite-dimensional Hilbert spaces and diverse atomic structures.
• To derive concrete conditions under which the solution's atomic decomposition closely approximates the true signal's decomposition.

Proposed method

• Proposes a dual certificate condition on the measurement operator $ A $ that ensures soft recovery in the sense of proximity between true and recovered atomic components.
• Applies the framework to infinite-dimensional Hilbert spaces, allowing analysis of continuous dictionaries such as complex exponentials or Gaussian kernels.
• Uses a generalized atomic norm defined as $ \|v\|_{\mathcal{A}} = \inf\{ t > 0 \mid v \in t \overline{\text{conv}} \Phi \} $, where $ \Phi $ is the dictionary of atoms.
• Introduces a measurement model based on convolution with a filter $ \phi $, leading to a finite sampling operator $ T_M $, and derives conditions on $ \phi $ and $ M $ for stable recovery.
• Employs frame theory and $ \ell_{1,2} $-norm estimates to bound the number of measurements required for soft recovery.
• Analyzes the trade-off between separation $ \Delta_{\text{sep}} $, localization error $ \Delta $, and peak amplitude $ |c_{x_0}^0| $ via explicit bounds involving the Gaussian filter.

Experimental results

Research questions

• RQ1 Under what conditions on the measurement operator $ A $ can atomic norm minimization approximately recover the atomic decomposition of a structured signal?
• RQ2 How can one guarantee that the support of the solution's atomic decomposition is close to that of the true signal, even when exact recovery fails?
• RQ3 What is the relationship between the minimal separation between atoms, the amplitude of the components, and the localization error in the solution?
• RQ4 How does the choice of filter $ \phi $, such as a Gaussian, affect the number of measurements and the accuracy of soft recovery?
• RQ5 Can the theoretical framework be applied to infinite-dimensional settings, such as super-resolution with continuous frequency components?

Key findings

• The dual certificate condition ensures that if a peak in the true signal has sufficient amplitude and separation from others, its corresponding atom in the solution is localized within a neighborhood of radius $ \Delta $.
• For a Gaussian filter $ \phi $, the required minimal separation $ \Delta_{\text{sep}} $ scales as $ \Lambda \sqrt{2 \log \left( \frac{1 - |c_{x_0}^0|}{|c_{x_0}^0|(1 - \theta)} \right)} $, where $ \theta $ controls the confidence level.
• When $ |c_{x_0}^0| = 0.1 $ and $ \gamma = 0.5 $, a separation of $ \Delta_{\text{sep}} = 3.95\Lambda $ ensures localization within $ \Delta = 2.1\Lambda $.
• The number of measurements $ |M| $ scales as $ \left\lceil \frac{| one| \Lambda^{-2 + \alpha^{-1}}}{\gamma^{2 + \alpha^{-1}}} \right\rceil $, indicating dependence on signal length and filter width.
• The framework proves that soft recovery is possible even in coherent dictionaries, where exact recovery fails due to ill-conditioning.
• The results are robust to noise and apply to a wide range of atomic structures, including super-resolution and low-rank matrix recovery.