A Reliable Iteration Algorithm for One-Bit Compressive Sensing on the Unit Sphere

The one-bit compressed sensing problem is of fundamental importance in many areas, such as wireless communication, statistics, and so on. However, the optimization of one-bit problem constrained on the unit sphere lacks an algorithm with rigorous mathematical proof of convergence and validity. In this paper, an iteration algorithm is established based on difference-of-convex algorithm for the one-bit compressed sensing problem constrained on the unit sphere, with iterating formula \({x^{k + 1}} = \mathop {\arg \min }\limits_{x \in \,C} \{ ||x|{|_1} + {\eta _1}||{x^k}|{|_1}\max (||x||_2^2,1) - 2{\eta _2}||{x^k}|{|_1}\langle x,{x^k}\rangle \} ,\) $$x^{k+1}=\underset{x\in C}{\arg \min }\{\mid \mid x\mid {\mid }_1+{\eta }_1\mid \mid x^k\mid {\mid }_1\max (\mid \mid x\mid {\mid }_2^2,1)-2{\eta }_2\mid \mid x^k\mid {\mid }_1〈x,x^k〉\},$$ where C is the convex cone generated by the one-bit measurements and \({\eta _1} > {\eta _2} > {1 \over 2}\) $${\eta }_1>{\eta }_2>\frac{1}{2}$$ . The new algorithm is proved to converge as long as the initial point is on the unit sphere and accords with the measurements, and the convergence to the global minimum point of the ℓ₁ norm is discussed.