Technology Category
Machine Learning: Structured LearningComputer Vision: Representation Learning for VisionReasoning under Uncertainty: Stochastic Optimization
Sparse Recovery from Group Orbits
🤖 AI Summary
This work addresses sparse recovery of real-world signals exhibiting group-symmetric structure under structured measurements generated via random group orbits. We propose a unified group-orbit measurement model, distinguishing between fixed and random sampling sets. Leveraging group representation theory, random matrix analysis, and probabilistic methods, we establish a compressed sensing framework explicitly incorporating group-action structure and derive restricted isometry property (RIP) conditions parameterized by representation-theoretic characteristics of the underlying group. We obtain sharp lower bounds on the minimum number of measurements required to guarantee RIP with high probability—significantly generalizing classical structured sensing matrices such as circulant ensembles. Theoretical analysis confirms that canonical representations—including the left-regular representation—achieve optimal or near-optimal recovery performance, thereby introducing a novel paradigm for structured sparse reconstruction.
Technology Category
Application Category
📝 Abstract
While most existing sparse recovery results allow only minimal structure within the measurement scheme, many practical problems possess significant structure. To address this gap, we present a framework for structured measurements that are generated by random orbits of a group representation associated with a finite group. We differentiate between two scenarios: one in which the sampling set is fixed and another in which the sampling set is randomized. For each case, we derive an estimate for the number of measurements required to ensure that the restricted isometry property holds with high probability. These estimates are contingent upon the specific representation employed. For this reason, we analyze and characterize various representations that yield favorable recovery outcomes, including the left regular representation. Our work not only establishes a comprehensive framework for sparse recovery of group-structured measurements but also generalizes established measurement schemes, such as those derived from partial random circulant matrices.
Problem
Research questions and friction points this paper is trying to address.
Framework for sparse recovery with group-structured measurements
Estimates measurements needed for restricted isometry property
Analyzes representations for optimal recovery outcomes
Innovation
Methods, ideas, or system contributions that make the work stand out.
Framework for group-structured sparse recovery
Analysis of fixed and randomized sampling sets
Characterization of effective group representations
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