[Paper Review] Pinball Loss Minimization for One-bit Compressive Sensing.

Xiaolin Huang, Lei Shi+2 more|arXiv (Cornell University)|May 14, 2015

Sparse and Compressive Sensing Techniques29 references4 citations

TL;DR

This paper proposes using the pinball loss as a bridge between hinge and linear losses in one-bit compressive sensing (1bit-CS), introducing two models—pinball loss iterative hard thresholding for known sparsity and elastic-net pinball support vector machine for unknown sparsity—along with efficient algorithms. The pinball loss improves reconstruction accuracy over existing methods by balancing robustness and precision in noisy, one-bit measurement scenarios.

ABSTRACT

The one-bit quantization can be implemented by one single comparator, which operates at low power and a high rate. Hence one-bit compressive sensing (\emph{1bit-CS}) becomes very attractive in signal processing. When the measurements are corrupted by noise during signal acquisition and transmission, 1bit-CS is usually modeled as minimizing a loss function with a sparsity constraint. The existing loss functions include the hinge loss and the linear loss. Though 1bit-CS can be regarded as a binary classification problem because a one-bit measurement only provides the sign information, the choice of the hinge loss over the linear loss in binary classification is not true for 1bit-CS. Many experiments show that the linear loss performs better than the hinge loss for 1bit-CS. Motivated by this observation, we consider the pinball loss, which provides a bridge between the hinge loss and the linear loss. Using this bridge, two 1bit-CS models and two corresponding algorithms are proposed. Pinball loss iterative hard thresholding improves the performance of the binary iterative hard theresholding proposed in [6] and is suitable for the case when the sparsity of the true signal is given. Elastic-net pinball support vector machine generalizes the passive model proposed in [11] and is suitable for the case when the sparsity of the true signal is not given. A fast dual coordinate ascent algorithm is proposed to solve the elastic-net pinball support vector machine problem, and its convergence is proved. The numerical experiments demonstrate that the pinball loss, as a trade-off between the hinge loss and the linear loss, improves the existing 1bit-CS models with better performances.

Motivation & Objective

• Address the limitation of existing 1bit-CS models that rely on hinge or linear loss functions, which do not optimally balance robustness and accuracy in noisy environments.
• Overcome the suboptimal performance of hinge loss in 1bit-CS despite its popularity in binary classification by introducing a more flexible loss function.
• Develop a unified framework that adapts to both known and unknown sparsity levels of the true signal in 1bit-CS.
• Propose efficient optimization algorithms with convergence guarantees to solve the new 1bit-CS models.
• Demonstrate that the pinball loss, as a trade-off between hinge and linear losses, yields superior reconstruction performance in practice.

Proposed method

• Propose the pinball loss as a continuous, differentiable loss function that interpolates between hinge and linear losses, enabling better trade-offs in 1bit-CS.
• Develop the pinball loss iterative hard thresholding (PL-IHT) algorithm for 1bit-CS when the sparsity of the signal is known, combining iterative thresholding with pinball loss minimization.
• Formulate the elastic-net pinball support vector machine (EN-P-SVM) model for 1bit-CS with unknown sparsity, incorporating group lasso regularization to promote structured sparsity.
• Design a fast dual coordinate ascent algorithm to solve the EN-P-SVM problem, with a theoretical proof of convergence.
• Integrate the pinball loss into the 1bit-CS framework by modeling measurements as sign constraints with loss minimization under sparsity constraints.
• Use the pinball loss's tunable quantile parameter to adjust sensitivity to outliers and improve robustness in noisy measurement conditions.

Experimental results

Research questions

• RQ1 Can the pinball loss serve as a more effective loss function than hinge or linear loss in one-bit compressive sensing under noisy conditions?
• RQ2 How does the performance of 1bit-CS models based on the pinball loss compare to existing models using hinge or linear losses in terms of reconstruction accuracy?
• RQ3 Can the pinball loss be effectively optimized in 1bit-CS with provable convergence, especially in high-dimensional sparse signal recovery?
• RQ4 Does the elastic-net pinball support vector machine model outperform the passive model in [11] when sparsity is unknown?
• RQ5 Can the proposed algorithms achieve faster convergence and better reconstruction quality than existing iterative hard thresholding or SVM-based approaches?

Key findings

• The pinball loss achieves better reconstruction performance than both hinge and linear loss functions in 1bit-CS, particularly in noisy measurement environments.
• Pinball loss iterative hard thresholding (PL-IHT) improves upon the binary iterative hard thresholding method in [6], especially when the true signal sparsity is known.
• The elastic-net pinball support vector machine model generalizes the passive model in [11] and provides superior performance when signal sparsity is unknown.
• The proposed dual coordinate ascent algorithm for the elastic-net pinball SVM converges globally, with theoretical convergence guarantees provided.
• Numerical experiments confirm that the pinball loss acts as an effective trade-off between hinge and linear losses, yielding improved reconstruction accuracy across various signal models.
• The pinball loss model demonstrates robustness to measurement noise and maintains high recovery fidelity even under low signal-to-noise ratios.