Plotting the Relationship Curve Between Signal Sparsity K and Reconstruction Success Probability

In compressive sensing theory, the relationship between signal sparsity K and reconstruction success probability serves as a crucial performance metric. Experimental simulations are typically conducted to plot their relationship curves, providing visual demonstrations of how different reconstruction algorithms perform under varying sparsity levels. The experimental workflow generally involves the following implementation steps: First, sparse coefficient vectors need to be generated where the number of non-zero elements equals K (typically implemented using random position selection with Gaussian or Bernoulli distributed amplitudes). Then, linear projection of the signal is performed through a measurement matrix (commonly implemented using random Gaussian or Bernoulli matrices). Next, reconstruction algorithms such as OMP (Orthogonal Matching Pursuit) or CoSaMP (Compressive Sampling Matching Pursuit) are employed for signal recovery, where the algorithm iteratively identifies support sets and solves least-squares problems. Finally, the reconstruction success probability is statistically calculated by comparing the recovered signal with the original. When plotting the curve, the horizontal axis typically represents sparsity K (often normalized as K/N where N is the signal length), while the vertical axis shows the reconstruction success probability. To obtain smooth curves, multiple independent experiments must be conducted for each K value, taking the probability mean across trials. As K increases, the success probability generally demonstrates a declining trend, with different algorithms showing distinct descent rates - this reflects each algorithm's sensitivity to signal sparsity variations. These curves hold significant importance for comparing algorithm performance and determining system operational boundaries. In practical applications, engineers often seek to identify the maximum tolerable sparsity K that maintains a guaranteed reconstruction probability threshold, which can be determined by analyzing the inflection points of these performance curves.