Abstract:Spark plays a great role in studying uniqueness of sparse solutions of the underdetermined linear equations. In this article, w e derive a new lower bound of spark. As an application, we obtain a new criterion fo r the uniqueness of sparse solutions of linear equations.
Keywords: : :: Underdetermined, Linear equations, Sparse solution, Spark.
1.IntroductionRecent theoretical developments have generated a great deal of interest in sparse signal representation. The setup assumes a given dictionary of "elementary" signals, and models an input signal as a linear combination of dictionary elements, with the provision that the representation is sparse, i.e., involves only a few of the dictionary elements. Finding sparse representations ultimately requires solving for the sparsest solution of an underdetermined system of linear equations. Such models arise often in signal processing, image processing, and digital communications.Given an ܣ א ܴ ൈ ሺ݊ ൏ ݉ሻ full-rank matrix with no zero columns n R b∈ , the linear system b Ax = has infinitely many solutions when the system is underdetermined. Depending on the nature of source problems, we are often interested in finding a particular solution, and thus optimization methods come into a play through certain merit functions that measure the desired special structure of the solution. One of the recent interests is to find the sparsest solution of an underdetermined linear system, which has found many applications in signal and image processing which is an NP-hard discrete optimization problem [4]. The recent study in the field of compressed sensing nevertheless shows that not all cardinality minimization problems are equally hard, and there does exist a class of matrices A such that the problem (1.1) is computationally tractable. These matrices can be characterized by such concepts as the spark