Robust 1-bit compressed sensing and sparse logistic regression: A convex programming approach (1202.1212v3)

Abstract: This paper develops theoretical results regarding noisy 1-bit compressed sensing and sparse binomial regression. We show that a single convex program gives an accurate estimate of the signal, or coefficient vector, for both of these models. We demonstrate that an s-sparse signal in R^n can be accurately estimated from m = O(slog(n/s)) single-bit measurements using a simple convex program. This remains true even if each measurement bit is flipped with probability nearly 1/2. Worst-case (adversarial) noise can also be accounted for, and uniform results that hold for all sparse inputs are derived as well. In the terminology of sparse logistic regression, we show that O(slog(n/s)) Bernoulli trials are sufficient to estimate a coefficient vector in R^n which is approximately s-sparse. Moreover, the same convex program works for virtually all generalized linear models, in which the link function may be unknown. To our knowledge, these are the first results that tie together the theory of sparse logistic regression to 1-bit compressed sensing. Our results apply to general signal structures aside from sparsity; one only needs to know the size of the set K where signals reside. The size is given by the mean width of K, a computable quantity whose square serves as a robust extension of the dimension.

• The paper establishes a unified convex optimization framework that accurately recovers s-sparse signals from minimal 1-bit measurements.
• It demonstrates robust recovery under extreme noise, including adversarial conditions and nearly 50% random bit flips.
• The approach bridges compressed sensing and sparse logistic regression, offering theoretical guarantees close to optimal.

Robust 1-bit Compressed Sensing and Sparse Logistic Regression: A Convex Programming Approach

This paper addresses two essential problems in modern data analysis: 1-bit compressed sensing (CS) and sparse logistic regression. The authors provide theoretical guarantees through a unified convex programming approach. The paper demonstrates that an $s$-sparse signal can be accurately estimated from $m = O(s\log (n/s))$ single-bit measurements and that such estimation is feasible even under adversarial noise conditions or when each bit is randomly flipped with probability nearly 1/2.

Key Contributions

1. Unified Framework via Convex Programming: The paper establishes a single convex optimization model that effectively estimates the signal or coefficient vector in both 1-bit CS and sparse logistic regression. This approach works under a generalized linear model framework, where the link function may be unknown.
2. Handling of Noise: The proposed method accounts for various noise models, ranging from nearly full adversarial noise to random bit flips. This robustness against noise is demonstrated theoretically, enhancing the potential applicability of 1-bit CS in practical scenarios.
3. Sparse Logistic Regression: The authors extend their results to logistic regression, providing estimates for the number of Bernoulli trials needed to estimate a coefficient vector accurately. This establishes the first known theoretical connection between sparse logistic regression and 1-bit CS.
4. Generalized Signal Structures: Beyond sparsity, the method applies to other signal structures characterized by their mean width, a notion borrowed from high-dimensional geometry. This flexibility allows for broader applicability, including low-rank matrix recovery.
5. Theoretical Guarantees and Optimality: Through rigorous analysis, the authors argue that their results, particularly the dependence on the mean width and number of measurements, are close to optimal. This work leverages advanced techniques, including random hyperplane tessellations, to provide these guarantees.

Mathematical and Practical Implications

• The paper highlights how dimensionality reduction through signal structure—particularly sparsity—can lead to effective data recovery even in extremely quantized settings.
• The robust handling of noise establishes a foundation for further research in developing quantization techniques that could improve efficiency in data transmission and storage without significant degradation of performance.
• The convex programming approach offers a computationally feasible solution, broadening the potential use cases where these theoretical insights can be applied in practice.

Speculating on Future Developments

Future research might extend these results to finite-bit quantization levels, bridging the gap between 1-bit and more traditional multi-bit CS, potentially offering insights into more general quantization schemes. Additionally, exploring non-Gaussian measurement models could lead to more generalized applicability in practical engineering problems, such as signal processing and machine learning tasks where Gaussian assumptions are unrealizable.

In summary, this paper marks a significant step in understanding and leveraging extreme quantization in compressed sensing and logistic regression through advanced mathematical techniques and convex optimization. The robust noise handling and applicability across varying signal structures make the findings compelling for theoretical and practical advancements in the field.