Stability Bounds for the Unfolded Forward-Backward Algorithm

TL;DR

The paper analyzes the stability of an unfolded forward-backward (proximal-gradient) network for linear degradation inverse problems, focusing on perturbations in observed data and, critically, in the data bias. It formulates a virtual leakage-enabled network, derives computable Lipschitz bounds and α-averagedness conditions using eigen-decomposition of the degradation and regularization operators, and connects these results to the actual unfolded network. Numerically, it demonstrates that leakage can improve stability and that multi-layer unrolling yields tighter bounds than separable analyses. The findings offer a rigorous framework for designing robust unrolled networks with proximal activations, informing parameter choices and potential extensions to more general proximal schemes. Overall, the work provides practical, theoretically grounded guarantees on the robustness of unfolded inverse-problem solvers against data perturbations.

Abstract

We consider a neural network architecture designed to solve inverse problems where the degradation operator is linear and known. This architecture is constructed by unrolling a forward-backward algorithm derived from the minimization of an objective function that combines a data-fidelity term, a Tikhonov-type regularization term, and a potentially nonsmooth convex penalty. The robustness of this inversion method to input perturbations is analyzed theoretically. Ensuring robustness complies with the principles of inverse problem theory, as it ensures both the continuity of the inversion method and the resilience to small noise - a critical property given the known vulnerability of deep neural networks to adversarial perturbations. A key novelty of our work lies in examining the robustness of the proposed network to perturbations in its bias, which represents the observed data in the inverse problem. Additionally, we provide numerical illustrations of the analytical Lipschitz bounds derived in our analysis.

Figures (6)

• Figure 1: Global architecture of the $m$-layers neural network \ref{['def:oldmodelNN']}.
• Figure 2: Stationary case: Lipschitz constant of the VNN as a function of $\lambda$ and $\eta$. First row: $\theta_1$, second one: $\theta_{15}/2^{14}$. First column: numerical value, second one: white if $\theta_m/2^{m-1} \le 1$.
• Figure 3: Stationary case: Difference $(\theta_1)^{15}-\theta_{15}/2^{14}$ for the VNN in the stationary case as a function of $\lambda$ and $\eta$.
• Figure 4: Stationary case: Lipschitz constant of the network with inputs $(x_0,b_0)$ and output $x_{15}$ as a function of $\lambda$ and $\eta$. Top left: $\overline{\theta}_{15}/2^{14}$. Top right: white when $\overline{\theta}_{15}/2^{14}<1$. Bottom: Difference $\theta_{15}-\overline{\theta}_{15}$.
• Figure 5: Stationary case: Lipschitz constant $\widehat{\theta}_{15}/2^{14}$as a function of $\lambda$ and $\eta$. First row: $F = 0$, second one: $F = {\rm 1 l}$. First column: numerical value, second one: white if $\widehat{\theta}_{15}/2^{14} \le 1$.

Theorems & Definitions (27)

• Definition 1
• Proposition 2.1
• Remark 3.1
• Proposition 4.1
• Lemma 4.1
• proof
• Remark 4.1
• Proposition 4.2
• proof
• Remark 4.2