Tractable downfall of basis pursuit in structured sparse optimization

The problem of finding the sparsest solution to a linear underdetermined system of equations, often appearing, e.g., in data analysis, optimal control, system identification, or sensor selection problems, is considered. This non-convex problem is commonly solved by convexification via $\ell_1$-norm minimization, known as basis pursuit (BP). In this work, a class of structured matrices, representing the system of equations, is introduced for which (BP) tractably fails to recover the sparsest solution. In particular, this enables efficient identification of matrix columns corresponding to unrecoverable non-zero entries of the sparsest solution and determination of the uniqueness of such a solution. These deterministic guarantees complement popular probabilistic ones and provide insights into the a priori design of sparse optimization problems. As our matrix structures appear naturally in optimal control problems, we exemplify our findings based on a fuel-optimal control problem for a class of discrete-time linear time-invariant systems. Finally, we draw connections of our results to compressed sensing and common basis functions in geometric modeling.

💡 Research Summary

The paper investigates the circumstances under which the widely used ℓ₁‑norm relaxation, known as Basis Pursuit (BP), fails to recover the sparsest solution of an underdetermined linear system. While probabilistic guarantees such as the Restricted Isometry Property (RIP) have been the cornerstone of compressed‑sensing theory, they are often inconclusive or overly conservative for structured matrices that arise in practical applications, especially in discrete‑time optimal control.

The authors introduce a deterministic failure criterion based on the ℓ₁‑norm of transformed columns of the measurement matrix V. For a matrix V∈ℝ^{m×n} (m<n) they define
 p_k := ‖V(:,1:m)^{-1} V(:,k)‖₁, k=1,…,n.
If there exists an index k such that p_k < 1, then any non‑zero entry of the true ℓ₀‑optimal vector u* at position k cannot be recovered by BP. This condition is both necessary and sufficient for failure in the sense that BP will necessarily produce a solution with at least one additional non‑zero component.

Checking the condition directly is computationally hard for arbitrary V, but the authors show that for a broad class of structured matrices the verification becomes tractable. The key structural property is total positivity (TP) or, more generally, k‑sign consistency. A matrix X is called (strictly) k‑totally positive if all its r‑th multiplicative compound matrices X