Description
This work is a comparative study of the theoretical implementation and computational performance of two interior point methods namely the affine scaling algorithm (variation of Karmarkar’s Projective Scaling Algorithm) and the primal dual central path algorithm applied to the compressed sensing problem. Compressed sensing is a paradigm in signal and image processing which allows reconstruction of signals from fewer measurements than what traditional paradigms in digital signal processing would suggest. These measurements are random linear combinations of the entries of the signal vector. The reconstruction is possible due to the knowledge of sparsity in signal representation that a whole host of man made and natural signals possess when represented with respect to certain basis. Convex Optimization on the other hand has been revolutionized by the advent of interior point methods both in terms of computational complexity and practical applicability. One of the different ways in which compressed sensing can be done is the so called basis pursuit framework, wherein we can reconstruct the signal which has a sparse representation by minimizing p[superscript th] powers of l[subscript p](0 ≤ p ≤ 1) quasinorms subject to affine constraints. The geometry of these norms is such that minimizing them subject to affine constraints encourages sparse solutions. These types of minimization problems can be solved using the interior point family of methods, the two aforementioned methods belonging to this family being the focus of this work. We study the theoretical nuances and find the similarities and dissimilarities in the application of these two methods on the compressed sensing problem. This is followed by numerical experiments where we reconstruct sparse signals of given length from very few measurements using both the methods. We shall look at the numerical performance statistics of both the methods in terms of percentage of signals reconstructed with varying levels of sparsity and the number of measurements. We compare the performance of the affine scaling solver written by the author with the standard l₁-magic solver which makes use of the primal dual central path algorithm. We finally conclude this work by revisiting some of the crucial concepts again and putting them in perspective after having seen the numerical results
