Technology Category
Reasoning under Uncertainty: Stochastic OptimizationSearch and Optimization: Non-convex OptimizationConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
Tractable downfall of basis pursuit in structured sparse optimization
🤖 AI Summary
Basis Pursuit (BP) systematically fails in structured sparse optimization for underdetermined linear systems, yet existing analyses lack deterministic characterizations of ℓ₁-minimization unrecoverability.
Method: This paper introduces a novel class of structured matrices and integrates linear algebra, convex optimization, and structured matrix analysis to deterministically characterize the conditions under which BP fails—without requiring prior knowledge of the signal or measurement matrix. The framework certifies BP failure, identifies unrecoverable nonzero entries, and verifies solution uniqueness.
Contribution/Results: Applied to fuel-optimal control of discrete-time LTI systems, the proposed method efficiently detects and rigorously validates BP failure instances. It overcomes the limitations of probabilistic recovery guarantees by providing verifiable, deterministic theoretical tools and a practical certification paradigm for reliability analysis in sparse signal recovery.
Technology Category
Application Category
📝 Abstract
The problem of finding the sparsest solution to a linear underdetermined system of equations, often appearing, e.g., in data analysis, optimal control, and system identification 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, determination of the uniqueness of such a solution, and certification of (BP) failing to compute a sparsest solution without prior knowledge on its non-zero entry locations. These deterministic guarantees contrast with popular probabilistic ones and provide valuable 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.
Problem
Research questions and friction points this paper is trying to address.
BP fails to recover sparsest solutions for structured matrices
Identify unrecoverable non-zero entries in sparse solutions
Determine uniqueness and certify BP failure without prior knowledge
Innovation
Methods, ideas, or system contributions that make the work stand out.
Structured matrices for BP failure analysis
Efficient identification of unrecoverable columns
Deterministic guarantees for sparse optimization
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