Exact Support Recovery for Sparse Spikes Deconvolution (1306.6909v3)

Abstract: This paper studies sparse spikes deconvolution over the space of measures. We focus our attention to the recovery properties of the support of the measure, i.e. the location of the Dirac masses. For non-degenerate sums of Diracs, we show that, when the signal-to-noise ratio is large enough, total variation regularization (which is the natural extension of the L1 norm of vectors to the setting of measures) recovers the exact same number of Diracs. We also show that both the locations and the heights of these Diracs converge toward those of the input measure when the noise drops to zero. The exact speed of convergence is governed by a specific dual certificate, which can be computed by solving a linear system. We draw connections between the support of the recovered measure on a continuous domain and on a discretized grid. We show that when the signal-to-noise level is large enough, the solution of the discretized problem is supported on pairs of Diracs which are neighbors of the Diracs of the input measure. This gives a precise description of the convergence of the solution of the discretized problem toward the solution of the continuous grid-free problem, as the grid size tends to zero.

• The paper demonstrates that under non-degenerate dual certificate conditions, exact support recovery of Dirac spikes is achievable even with minimal noise.
• It extends discrete L1 techniques to continuous domains via total variation regularization, bridging rigorous theory with practical signal processing applications.
• The analysis confirms that discretized solutions converge to continuous spikes with linearly proportional amplitude recovery, enhancing super-resolution methods.

Sparse Spikes Deconvolution Using Total Variation Regularization and Support Recovery

The paper by Vincent Duval and Gabriel Peyré addresses the robust mathematical framework of sparse spikes deconvolution by leveraging total variation (TV) regularization over continuous domains, effectively extending concepts well-embedded in discrete signal processing. Traditional sparsity-inducing techniques such as the L_1 norm are adapted to the continuous case using TV, which serves as the natural analogue for Radon measures. The paper's core contribution is the detailed analysis of the exact support recovery of Dirac spikes convolved with a known kernel under sufficiently large signal-to-noise conditions.

Theoretical Framework and Results

The authors begin by establishing a foundational understanding of sparse spikes deconvolution, emphasizing the importance of support recovery—identifying the exact positions of spikes amidst noise and blur. This necessitates a refined consideration of the dual problem associated with TV regularization, specifically focusing on the construction of dual certificates. Their main result provides conditions under which a non-degenerate dual certificate enables exact support recovery, even as noise approaches zero. This certificate can be ascertained by computing the so-called vanishing derivative pre-certificate.

A rigorous analysis details the convergence behavior of solutions obtained on a discretized grid to those on a continuous domain as the grid size approaches zero. This convergence reveals, under auspicious circumstances, that the recovered measure's support is composed of locations proximal to the original Dirac spikes—a critical finding for the fields of super-resolution and high-fidelity signal recovery.

Numerical Implications and Robustness

This rigorous theoretical framework establishes that when the non-degenerate source condition is satisfied, and both the regularization parameter and noise level are small, the reconstructed Diracs match the original measure count and formation. More strikingly, the amplitudes of these Diracs converge linearly proportional to noise level, providing predictive power for estimation accuracy.

Additionally, addressing the common computational constraint of working with discrete grids, Duval and Peyré show how the support of the discretized solution hones in on pairs of Diracs adjacent to the continuous case. This provides a methodological bridge between continuous mathematical theory and practical application constraints, thus offering major implications for sparse signal processing in practice.

Broader Context and Future Directions

From a broader perspective, the paper offers substantial theoretical and practical advancements in the understanding of sparse deconvolution within the continuous setting. The main contribution lies not only in elucidating the conditions necessary for exact support recovery but also in offering practical algorithms and conditions that can be readily implemented.

Future pursuits may adapt these foundations to non-stationary filtering scenarios or multi-dimensional cases, exploiting the robustness of this framework in higher-order analyses. Given these advancements, the paper paved pathways for addressing more complex inverse problems in signal processing, tightly connecting theory with computational practice and offering a profound impact on future AI developments in fields requiring precise signal reconstruction.