[Paper Review] Adapting to unknown noise level in sparse deconvolution

Claire Boyer, Yohann de Castro+1 more|arXiv (Cornell University)|Jun 15, 2016

Sparse and Compressive Sensing Techniques23 references3 citations

TL;DR

This paper introduces the Concomitant Beurling Lasso (CBLasso), a novel convex optimization method for sparse deconvolution that jointly estimates the target Radon measure and unknown noise level in super-resolution problems. By incorporating a concomitant estimation framework akin to the square-root Lasso, CBLasso achieves theoretical guarantees—such as consistent noise level estimation and minimax optimal prediction/localization error—without requiring prior knowledge of the noise variance, using only the sample size to tune the regularization parameter.

ABSTRACT

In this paper, we study sparse spike deconvolution over the space of complex-valued measures when the input measure is a finite sum of Dirac masses. We introduce a modified version of the Beurling Lasso (BLasso), a semi-definite program that we refer to as the Concomitant Beurling Lasso (CBLasso). This new procedure estimates the target measure and the unknown noise level simultaneously. Contrary to previous estimators in the literature, theory holds for a tuning parameter that depends only on the sample size, so that it can be used for unknown noise level problems. Consistent noise level estimation is standardly proved. As for Radon measure estimation, theoretical guarantees match the previous state-of-the-art results in Super-Resolution regarding minimax prediction and localization. The proofs are based on a bound on the noise level given by a new tail estimate of the supremum of a stationary non-Gaussian process through the Rice method.

Motivation & Objective

• To address sparse deconvolution in super-resolution when the noise level is unknown, a common challenge in imaging and signal processing.
• To develop a convex optimization framework that simultaneously estimates the sparse signal (Radon measure) and the unknown noise variance.
• To provide theoretical guarantees—such as consistent noise level estimation and minimax optimal recovery—without requiring prior knowledge of the noise level.
• To extend the Beurling Lasso (BLasso) to a concomitant formulation that is robust and adaptive to unknown noise, improving on existing methods.

Proposed method

• Proposes the Concomitant Beurling Lasso (CBLasso) as a jointly convex optimization problem minimizing a data-fitting term scaled by the noise estimate, a penalty on the total variation of the measure, and a regularization term on the noise level.
• Uses a rescaling of the data-fitting term by the noise estimate σ and adds a σ/2 regularization to avoid degenerate solutions and ensure homogeneity.
• Employs a dual formulation based on the Lagrangian and the perspective function to derive the dual problem, enabling theoretical analysis via duality and saddle-point conditions.
• Applies the Rice method to derive a new tail estimate for the supremum of a stationary non-Gaussian process, which is critical for bounding the noise level in high-dimensional settings.
• Establishes strong duality and shows that the dual problem reduces to maximizing ⟨y, λc⟩ over a constrained set Dn, enabling recovery guarantees.
• Demonstrates that the dual polynomial in the CBLasso setup cannot have constant modulus under certain tuning conditions, which is essential for exact recovery.

Experimental results

Research questions

• RQ1 Can a convex optimization method jointly estimate the sparse signal and the unknown noise level in super-resolution deconvolution?
• RQ2 Does the proposed CBLasso estimator achieve consistent estimation of the noise level without prior knowledge of its value?
• RQ3 Can theoretical guarantees—such as minimax optimal prediction and localization error—be established for the CBLasso under unknown noise?
• RQ4 What is the role of the concomitant formulation in improving robustness and adaptivity in sparse deconvolution?

Key findings

• The CBLasso estimator achieves consistent estimation of the unknown noise level σ₀, with theoretical guarantees that hold uniformly over the noise level.
• Theoretical analysis shows that the prediction and localization errors of the CBLasso match the state-of-the-art minimax rates for super-resolution, even when the noise level is unknown.
• The tuning parameter λ depends only on the sample size n, enabling practical use without requiring knowledge of σ₀.
• A new tail estimate for the supremum of a stationary non-Gaussian process is derived using the Rice method, which is instrumental in bounding the noise level and proving consistency.
• The dual polynomial in the CBLasso cannot have constant modulus under the condition λ ∈ [λmin(y), λmax(y)], which prevents overfitting and supports exact recovery.
• The CBLasso solution satisfies ˆσ = ∥y − Fn(ˆµ)∥₂ / √n, ensuring that the estimated noise level matches the residual norm, a key property for concomitant estimation.