Restricted Isometry Property

Junwei Lu
Additional contact information
Junwei Lu: Harvard University

Chapter Chapter 9 in Big Data Analysis, 2025, pp 53-57 from Springer

Abstract: Abstract 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.

Date: 2025
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-03161-7_9

Ordering information: This item can be ordered from
http://www.springer.com/9783032031617

DOI: 10.1007/978-3-032-03161-7_9

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().