[Paper Review] Tradeoffs between Convergence Speed and Reconstruction Accuracy in Inverse Problems

Raja Giryes, Yonina C. Eldar+2 more|arXiv (Cornell University)|May 30, 2016

Sparse and Compressive Sensing Techniques4 citations

TL;DR

This paper establishes a theoretical tradeoff between convergence speed and reconstruction accuracy in iterative inverse problem solvers, showing that using a coarse approximation of the low-dimensional signal set can accelerate convergence at the cost of a bounded error. The key contribution is a framework linking this tradeoff to compressed sensing, sparse recovery, and deep learning, with implications for learned iterative algorithms like LISTA.

ABSTRACT

Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is willing to tolerate, an important question is whether it is possible to modify the original iterations to obtain faster convergence to a minimizer achieving the allowed error without increasing the computational cost of each iteration considerably. Relying on recent recovery techniques developed for settings in which the desired signal belongs to some low-dimensional set, we show that using a coarse estimate of this set may lead to faster convergence at the cost of an additional reconstruction error related to the accuracy of the set approximation. Our theory ties to recent advances in sparse recovery, compressed sensing, and deep learning. Particularly, it may provide a possible explanation to the successful approximation of the l1-minimization solution by neural networks with layers representing iterations, as practiced in the learned iterative shrinkage-thresholding algorithm (LISTA).

Motivation & Objective

• To understand the tradeoff between convergence speed and reconstruction accuracy when iterative solvers are truncated early due to time constraints.
• To investigate whether modifying iterative algorithms using coarse approximations of low-dimensional signal sets can accelerate convergence without increasing per-iteration cost.
• To provide a theoretical foundation for the success of learned iterative methods such as LISTA, which approximate ℓ1-minimization via learned layers.
• To formalize how approximation error in the signal model affects the convergence rate and final reconstruction accuracy in iterative optimization.

Proposed method

• Proposes a framework analyzing iterative algorithms where the signal is assumed to belong to a low-dimensional set 𝒦, with convergence analyzed under coarse approximations of 𝒦.
• Introduces a perturbation model where the true set 𝒦 is approximated by a coarse set 𝒟, and derives bounds on the resulting error in the iterates.
• Uses projection operators and spectral norms to bound the error propagation in iterative shrinkage and thresholding algorithms (ISTA) and projected gradient methods.
• Derives theoretical convergence rates that depend on both the condition number of the measurement matrix and the approximation quality of the signal set.
• Applies the theory to analyze learned iterative shrinkage-thresholding (LISTA), showing how learned parameters effectively implement a coarse approximation of the ℓ1-ball.
• Employs tools from convex analysis, compressed sensing, and operator theory to bound the error in terms of the approximation error ϵ and the geometry of the signal set.

Experimental results

Research questions

• RQ1 Can we accelerate convergence in iterative inverse problem solvers by using a coarse approximation of the true signal set, even if it introduces a bounded reconstruction error?
• RQ2 What is the theoretical relationship between the accuracy of the signal set approximation and the convergence speed of iterative algorithms?
• RQ3 How does the approximation error in the signal model affect the final reconstruction accuracy when iterations are truncated early?
• RQ4 Can the success of learned iterative algorithms like LISTA be explained by this tradeoff between convergence speed and approximation error?
• RQ5 What are the conditions under which a coarse set approximation leads to faster convergence without increasing per-iteration computational cost?

Key findings

• The paper establishes that using a coarse approximation of the low-dimensional signal set can accelerate convergence by a factor related to the approximation error ϵ and the geometry of the set.
• A theoretical bound is derived showing that the error in the iterates grows linearly with ϵ, the approximation error of the signal set, and with the norm of the true signal.
• The convergence rate of iterative methods is shown to depend on the spectral properties of the measurement matrix and the approximation quality of the set, with faster convergence achievable when the set is well-approximated.
• The analysis provides a theoretical explanation for the success of LISTA, where learned parameters effectively implement a coarse approximation of the ℓ1-ball, leading to faster convergence than standard ISTA.
• The derived error bounds are tight and depend on the condition number of the measurement matrix and the curvature of the signal set, validating the tradeoff between speed and accuracy.