Sensing plays a vital role in modern technology, particularly in communications, automotive radar, and remote sensing. One of the main restrictions in sensing systems is identifying the parameters of interest with limited resources like power and sensor numbers by leveraging prior knowledge about signals to capture only a portion of high-dimensional data. This dissertation focuses on two foundational principles: sparse sampling geometries for data collection and algorithms for solving inverse problems to extract desired information from these measurements.
The first part of the dissertation examines the superiority of sparse sensing strategies in source localization tasks under strict time snapshot and sensor constraints. Key contributions include a rigorous non-asymptotic analysis of the Coarray ESPRIT algorithm, demonstrating that sparse arrays maintain high resolution and accuracy even with limited temporal snapshots compared to traditional uniform sensing geometries. The dissertation also analyzes an interpolation technique for array synthesis under single snapshot and positive source conditions, showing that sparse arrays can be effectively used by solving a simple convex feasibility search problem. Additionally, the research provides minimax error rates, comparing the performance of ULAs and nested arrays, and establishing theoretical bounds that demonstrate the superiority of sparse arrays and validate the empirical advantages of sparse arrays. The study also examines random sparse arrays, proposing new sampling schemes to ensure large contiguous coarrays, which are crucial for effective sensing.
Beyond array signal processing, the dissertation extends the analysis of these spatio-temporal tradeoffs to other non-linear inverse problems. It introduces Khatri-Rao dictionary learning for tensor data, demonstrating global identifiability and providing sample complexity bounds. Furthermore, it explores the training of polynomial neural networks via low-rank tensor recovery, proposing a novel iterative algorithm that improves efficiency and generalizes previous approaches. Lastly, it addresses the joint support recovery of sparse vectors from 1-bit measurements, offering an adaptive method that operates effectively in extreme spatial compression regimes.
