One-bit compressed sensing by linear programming (1109.4299v5)

Abstract: We give the first computationally tractable and almost optimal solution to the problem of one-bit compressed sensing, showing how to accurately recover an s-sparse vector x in R^n from the signs of O(s log^2(n/s)) random linear measurements of x. The recovery is achieved by a simple linear program. This result extends to approximately sparse vectors x. Our result is universal in the sense that with high probability, one measurement scheme will successfully recover all sparse vectors simultaneously. The argument is based on solving an equivalent geometric problem on random hyperplane tessellations.

• The paper presents a linear programming method that recovers s-sparse signals using O(s log²(n/s)) one-bit measurements.
• It employs geometric insights with random hyperplane tessellations to ensure high-probability accuracy in signal recovery.
• The approach is universal and practical, demonstrating robust performance in low-precision data acquisition scenarios.

Compressed Sensing and One-bit Recovery through Linear Programming

The paper by Yaniv Plan and Roman Vershynin addresses the challenge of recovering sparse signals from heavily quantized, specifically one-bit, measurements. The paper explores one-bit compressed sensing, a variant where only the sign information of linear measurements of a signal is retained. This setup emerges naturally in various areas like analog-to-digital conversion and threshold-based data processing.

The authors present a method to accurately recover an $s$-sparse vector $\vx$ in $\R^n$ using $O(s \log^2(n/s))$ one-bit measurements. This approach significantly reduces the information required for signal reconstruction compared to standard compressed sensing, where infinite-precision measurements are the norm. Furthermore, the recovery is realized by a straightforward linear programming scheme.

Key Contributions

1. Linear Programming Formulation: Plan and Vershynin develop a linear programming-based algorithm which facilitates the recovery of sparse or approximately sparse signals. The paper establishes that the problem can be effectively addressed by a convex optimization task, providing sound theoretical foundations for a computationally feasible approach.
2. Geometric Argumentation: A notable aspect of the paper is its use of geometric insights, particularly in solving a related issue concerning random hyperplane tessellations. The authors prove that with $m = O(s \log(n/s))$ random hyperplanes, the problem space is sufficiently partitioned to guarantee an accurate signal recovery with high probability.
3. Uniform Results Across Signals: The results are universal, indicating that with high probability, a single measurement matrix will suffice for the accurate recovery of all sparse vectors. This feature underscores the practical potential of deploying the proposed scheme in real-world applications where handling multiple signals concurrently is necessary.
4. Theoretical Bounds: The paper presents strong probabilistic bounds on the performance of the method. The core theorem highlights an error that diminishes with the ratio of $s$ (the sparsity level) to $m$ (the number of measurements), reflecting robustness in the proposed approach even when signals deviate from ideal sparsity.
5. Discussion on Practicality and Extensions: The implications of using one-bit measurements are significant, especially in scenarios with constraints on measurement precision. The universality of the method and its reliance on a simple linear programming framework suggest promising directions for extending to non-Gaussian contexts or further reducing the logarithmic factor in the number of required measurements.

Implications and Future Directions

The implications of this research stretch across both theoretical and practical domains. Theoretically, it expands the framework of compressed sensing into the regime of low-precision measurements, a domain with inherent challenges yet substantial applicability. Practically, the approach offers a viable path for resource-constrained data acquisition systems.

Future work might focus on reducing the logarithmic overhead in the measurement count or addressing additional practical considerations like measurement noise and quantization errors beyond binary one-bit measurements. Further advancement might involve extending the framework to matrix recovery problems and other higher-dimensional signal reconstruction tasks.

In conclusion, this paper delivers significant strides in compressed sensing theory, offering insights into tractable recovery methods under quantization constraints. The successful marriage of linear programming with geometric reasoning offers a robust toolkit for navigating challenges posed by low-precision data acquisition systems, marking a meaningful addition to the compressed sensing literature.