Gradient Descent Method On The Sphere With Sparsity Constraints

The 1-Bit compressive sensing problem considers recovering the original sparse signal from the sign of the signal measurements.Since the modulus lengths are missing,it is impossible to accurately recover the original signal from the measurement matrix and mea-surement information by solving the nonlinear equation directly.In the literature of 1-Bit compressed sensing,the non-convex optimization models and agorit.hms are widely used to recover the original sparse signal.The Binary Iterative Hard Thresholding algorithm based on the one-sided l2 norm is one of the classical algorithms in the field of 1-Bit compressive sensing.The algorithm combines gradient descent,spherical projection,sparseness and oth-er methods.Although the algorithm has very good experimental results,the convergence guarantee is still unclear.Inspired by the recent researches on 1-bit compressed sensing and sparse PCA,we study the minimization of continuous differentiable functions on a u-nit sphere with sparse constraints We propose an optimal condition based on the concept of stability,and use this condition to derive a numerical algorithm:a necessary condition for the convergence of the Iterative Hard Threshold algorithm(IHT)on the sphere.The algorithm is essentially a spherical gradient descent projection method,in which two steps of hard threshold truncation and spherical projection can be transformed into a nonlinear sparse approximation problem.Motived by some analysis methods of iterative hard thresh-old algorithm in compressove sensing theory,we provide the convergence analysis of Iterative Hard Threshold algorithm(IHT)on the sphere.Finally,we show that the Binary Iterative Hard Thresholding algorithm based on the one sided l2 norm is a special case of the algo-rithm proposed in the thesis.To our best knowledge,this is the first convergence analysis of the Binary Iterative Hard Threshold algorithm based on the one sided l2 norm and math-ematically demonstrate the effectiveness of the algorithm.Our theory provides theoretical support for the further applications of this algorithm.