Solving Quadratic Equations via PhaseLift When There Are About as Many Equations as Unknowns

• Published in: Foundations of computational mathematics Vol. 14; no. 5; pp. 1017 - 1026
• Main Authors: Candès, Emmanuel J., Li, Xiaodong
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.10.2014; Springer; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Computer Science; Economics; Equations, Quadratic; Foundations; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematical problems; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Optimization; Quadratic equations; Recovery; Texts; Vectors (mathematics); United States; 49N30; Deviation inequalities for random matrices; 60F10; 90C25; Phase retrieval; Semidefinite relaxations of nonconvex quadratic programs; PhaseLift; 62H12
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-013-9162-z

This note shows that we can recover any complex vector exactly from on the order of n quadratic equations of the form |〈 a i , x 0 〉| 2 = b i , i =1,…, m , by using a semidefinite program known as PhaseLift. This improves upon earlier bounds in Candès et al. (Commun. Pure Appl. Math. 66:1241–1274, 2013 ), which required the number of equations to be at least on the order of n log n . Further, we show that exact recovery holds for all input vectors simultaneously, and also demonstrate optimal recovery results from noisy quadratic measurements; these results are much sharper than previously known results.