Pinball Loss Minimization for One-bit Compressive Sensing
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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 (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.
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