Restricted Isometry Property

In the previous chapter, we introduce the problem of compressive sensing: how to find the sparse truth β ∗ $$\beta ^*$$ from the linear equation Y = 𝕏 β ∗ $$Y=\mathbb {X}\beta ^*$$ . Recall that we list three major questions for the compressive sensing: 1. What is the algorithm to recover β ∗ $$\beta ^*$$ ? 2. What kind of matrix 𝕏 $$\mathbb {X}$$ can guarantee the recovery? 3. How efficiently can we compress β ∗ $$\beta ^*$$ , i.e., how small can n be with respect to d? The first question is solved by the basis pursuit estimator β ^ = arg min β ∥ β ∥ 1 $$\widehat \beta = \operatorname *{\text{arg min}}_{\beta} \|\beta \|_1$$ such that Y = 𝕏 β $$Y = \mathbb {X}\beta $$ . The second question is partially answered in Theorem 6.6 of Chap. 6 , as we show that the cone condition ℂ ( S ) ⋂ Null ( 𝕏 ) = 0 $$\mathbb {C}(S)\bigcap \mathrm {Null}(\mathbb {X})=0$$ is a sufficient and necessary condition for the perfect recovery of basis pursuit in Theorem 6.6. However, the cone condition is not easy to use in practice. It is not straightforward to construct 𝕏 $$\mathbb {X}$$ starting from the cone condition. In this chapter, we will discuss another sufficient condition for perfect recovery, called restricted isometry property, which is stronger but easier to implement. We will talk about how to construct 𝕏 $$\mathbb {X}$$ based on this property and answer the third question.