[Paper Review] Simple Bounds for Noisy Linear Inverse Problems with Exact Side Information

Samet Oymak, Christos Thrampoulidis+1 more|arXiv (Cornell University)|Dec 2, 2013

Sparse and Compressive Sensing Techniques38 references16 citations

TL;DR

This paper provides sharp, non-asymptotic error bounds for noisy linear inverse problems when exact side information—either the value of a convex structure-inducing function $ f(\mathbf{x}_0) $ or the $ \ell_2 $-norm of the noise $ \|\mathbf{z}\| $—is available. It shows that under Gaussian measurements, the reconstruction error scales with the signal's sparsity (via Gaussian width of the tangent cone) rather than the ambient noise dimension, significantly improving estimation accuracy for structured signals like sparse vectors using Lasso or SOCP with side information.

ABSTRACT

This paper considers the linear inverse problem where we wish to estimate a structured signal $x$ from its corrupted observations. When the problem is ill-posed, it is natural to make use of a convex function $f(\cdot)$ that exploits the structure of the signal. For example, $\ell_1$ norm can be used for sparse signals. To carry out the estimation, we consider two well-known convex programs: 1) Second order cone program (SOCP), and, 2) Lasso. Assuming Gaussian measurements, we show that, if precise information about the value $f(x)$ or the $\ell_2$-norm of the noise is available, one can do a particularly good job at estimation. In particular, the reconstruction error becomes proportional to the "sparsity" of the signal rather than the ambient dimension of the noise vector. We connect our results to existing works and provide a discussion on the relation of our results to the standard least-squares problem. Our error bounds are non-asymptotic and sharp, they apply to arbitrary convex functions and do not assume any distribution on the noise.

Motivation & Objective

• To derive non-asymptotic, sharp error bounds for structured signal estimation in noisy linear inverse problems under exact side information.
• To generalize existing results on $ \ell_1 $-regularized recovery (Lasso) to arbitrary convex structure-inducing functions $ f(\cdot) $.
• To show that estimation error depends on the Gaussian width of the tangent cone to the function at $ \mathbf{x}_0 $, not the ambient dimension of noise.
• To establish that reconstruction error scales with signal sparsity rather than noise dimension when side information is available.

Proposed method

• Formulates two convex optimization problems: Lasso with known $ f(\mathbf{x}_0) $ and SOCP with known $ \|\mathbf{z}\| $, both leveraging exact side information.
• Uses Gaussian width $ \omega(\mathcal{C}) $ of the intersection of the tangent cone and the unit $ \ell_2 $-ball to characterize estimation error.
• Applies concentration of measure and Gaussian process techniques to bound the error in terms of $ \omega(\mathcal{C} \cap \mathcal{B}^{n-1}) $, where $ \mathcal{C} $ is the tangent cone at $ \mathbf{x}_0 $.
• Derives probabilistic bounds on the error via tail inequalities, assuming $ \mathbf{A} $ has i.i.d. $ \mathcal{N}(0, 1/m) $ entries.
• Introduces a geometric decomposition of the error using projections onto the noise and signal subspaces, and bounds the resulting terms via concentration and symmetry arguments.
• Employs a key lemma showing that a certain error difference term is non-negative with high probability, ensuring tightness of the final bound.

Experimental results

Research questions

• RQ1 Can the performance of Lasso and SOCP be significantly improved when exact side information about $ f(\mathbf{x}_0) $ or $ \|\mathbf{z}\| $ is available, even for non-sparse or non-$ \ell_1 $-structured signals?
• RQ2 Can sharp, non-asymptotic error bounds be derived that depend on the Gaussian width of the tangent cone rather than the ambient dimension?
• RQ3 Do these bounds remain tight and accurate even for small sample sizes and low sparsity levels?
• RQ4 How does the inclusion of exact side information reduce the estimation error compared to standard Lasso or SOCP without such knowledge?
• RQ5 Can the analysis be generalized beyond $ \ell_1 $-norms to arbitrary convex functions $ f(\cdot) $?

Key findings

• The reconstruction error for both Lasso with known $ f(\mathbf{x}_0) $ and SOCP with known $ \|\mathbf{z}\| $ is bounded by $ \mathcal{O}(\omega(\mathcal{C} \cap \mathcal{B}^{n-1})) $, where $ \mathcal{C} $ is the tangent cone at $ \mathbf{x}_0 $, independent of the ambient noise dimension.
• The error bound scales with the sparsity of the signal, as captured by the Gaussian width of the tangent cone, rather than with the number of noise components.
• For sparse signals with $ f(\cdot) = \|\cdot\|_1 $, the error is proportional to $ \|\mathbf{z}\| $, matching known results, but now proven with sharp constants and non-asymptotic guarantees.
• The bounds are non-asymptotic and hold with high probability $ 1 - \frac{5}{2}\exp(-t^2/26) $, even for small $ m $ and $ n $, under i.i.d. Gaussian measurements.
• The analysis shows that the error reduction due to side information is geometrically quantifiable, with the key term $ \kappa_2 - \kappa_1 - \kappa_3 \geq 0 $ holding with high probability.
• The derived bounds are sharp and do not assume any distribution on the noise beyond sub-Gaussian tails, making them robust and widely applicable.