Robust 1-Bit Compressive Sensing via Binary Stable Embeddings of Sparse Vectors (1104.3160v4)

Abstract: The Compressive Sensing (CS) framework aims to ease the burden on analog-to-digital converters (ADCs) by reducing the sampling rate required to acquire and stably recover sparse signals. Practical ADCs not only sample but also quantize each measurement to a finite number of bits; moreover, there is an inverse relationship between the achievable sampling rate and the bit depth. In this paper, we investigate an alternative CS approach that shifts the emphasis from the sampling rate to the number of bits per measurement. In particular, we explore the extreme case of 1-bit CS measurements, which capture just their sign. Our results come in two flavors. First, we consider ideal reconstruction from noiseless 1-bit measurements and provide a lower bound on the best achievable reconstruction error. We also demonstrate that i.i.d. random Gaussian matrices describe measurement mappings achieving, with overwhelming probability, nearly optimal error decay. Next, we consider reconstruction robustness to measurement errors and noise and introduce the Binary $\epsilon$-Stable Embedding (B$\epsilon$SE) property, which characterizes the robustness measurement process to sign changes. We show the same class of matrices that provide almost optimal noiseless performance also enable such a robust mapping. On the practical side, we introduce the Binary Iterative Hard Thresholding (BIHT) algorithm for signal reconstruction from 1-bit measurements that offers state-of-the-art performance.

• The paper establishes a 1-bit sensing method that uses sign-only measurements to enable effective sparse signal recovery.
• It derives a theoretical lower bound on reconstruction error and introduces Binary ε-Stable Embedding to ensure noise resilience.
• A new Binary Iterative Hard Thresholding algorithm leveraging random Gaussian matrices achieves near-optimal reconstruction performance.

Robust 1-Bit Compressive Sensing via Binary Stable Embeddings of Sparse Vectors

This paper explores a novel approach within the Compressive Sensing (CS) framework, emphasizing the shift from traditional sampling rate improvements to manipulating the bit depth of each measurement, particularly focusing on 1-bit measurements. In the context of CS, the reduction of the required sampling rate is a critical factor that eases the burden on analog-to-digital converters (ADCs). However, existing sampling methodologies also impose a high bit-depth requirement due to quantization, thereby creating a bottleneck due to limited speed and increasing power consumption.

Core Contributions

The primary contribution of this paper is the establishment of a 1-bit Compressive Sensing framework. This approach entails extreme quantization, where only the sign of each measurement is captured. The paper demonstrates that sufficient information for signal reconstruction remains, despite the dramatic reduction in bit depth.

1. Lower Bound on Reconstruction Error: A theoretical lower bound on reconstruction performance is derived, which indicates the best possible accuracy achievable by any measurement mapping with only 1-bit quantization.
2. Robustness to Noise and Quantization Errors: The paper introduces the Binary $\epsilon$-Stable Embedding (B$\epsilon$SE) property that characterizes the signal reconstruction robustness despite sign changes. This property ensures closely preserved geometric relationships during binary measurement mapping.
3. Random Gaussian Matrices as Effective Measurement Mappings: Random Gaussian matrices are shown to achieve near-optimal error decay in reconstructions from noiseless one-bit measurements. This finding implies that particular random matrices can preserve signal structure efficiently, providing the desired isometric embedding.
4. Binary Iterative Hard Thresholding Algorithm: A new reconstruction algorithm, Binary Iterative Hard Thresholding (BIHT), is developed. This algorithm offers state-of-the-art performance for reconstructing signals from 1-bit measurements, ensuring both model adherence and measurement consistency.

Implications and Future Developments

The paper’s findings highlight several practical and theoretical implications:

• Efficiency in Signal Acquisition: By focusing on measurement sign rather than amplitude, the paper lays the groundwork for new ultra-low-power, high-speed ADC designs that perform effectively at low sampling rates and reduce hardware costs.
• Robust Signal Recovery Mechanisms: The introduction of BIHT and the Binary $\epsilon$-Stable Embedding signifies a robust step forward for AI applications that require resilient and reliable signal recovery, even in the presence of noise or quantization imperfections.
• Extension to Broader Signal Classes: While this paper focuses on sparse signals, future research could extend these results to more general signal classes, fostering broader applications across AI and signal processing domains.
• Theoretical Exploration of Noise Resilience: Examining further the resilience of these 1-bit techniques to various noise models may provide deeper insights into their practical applicability in real-world systems where such imperfections are unavoidable.

Ultimately, this research advances the capabilities of compressive sensing by highlighting the trade-offs between measurement rate and depth, introducing efficient embeddings, and providing robust algorithms, thus opening pathways for significant hardware and theoretical advancements in the field.