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

Authors

Carlos Ramirez, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow
Miguel Argaez, The University of Texas at El PasoFollow

Publication Date

3-2013

Comments

Technical Report: UTEP-CS-13-18

To appear in Journal of Uncertain Systems, 2013, Vol. 7.

Abstract

In practice, we usually have partial information; as a result, we have several different 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 define the simplest solution is to select a one for which the number of the non-zero coefficients c_i is the smallest. The corresponding "l₀-optimization" problem is non-convex and therefore, difficult to solve. As a good approximation to this problem, Candes and Tao proposed to use a solution to the convex l₁ optimization problem |c₁| + ... + |c_n| --> min. In this paper, we provide a geometric explanation of why l₁ is indeed the best convex approximation to l₀.