Why l1 Is a Good Approximation to l0: A Geometric Explanation

In practice, we usually have partial information; as a result, we have several dierent possibilities consistent with the given measurements and the given knowledge. For example, in geosciences, several possible density distributions are consistent with the measurement results. It is reasonable to select the simplest among such distributions. A general solution can be described, e.g., as a linear combination of basic functions. A natural way to dene the simplest solution is to select one for which the number of the non-zero coecients ci is the smallest. The corresponding \‘0-optimization" problem is non-convex and therefore, dicult to solve. As a good approximation to this problem, Cand es and Tao proposed to use a solution to the convex ‘1 optimization problem P jcij ! min. In this paper, we provide a geometric