Low Rank Matrix Recovery: Nuclear Norm Penalization

In this chapter, we discuss a problem of estimation of a large target matrix based 4 on a finite number of noisy measurements of linear functionals (often, random) 5 of this matrix. The underlying assumption is that the target matrix is of small 6 rank and the goal is to determine how the estimation error depends on the rank 7 as well as on other important parameters of the problem such as the number of 8 measurements and the variance of the noise. This problem can be viewed as a 9 natural noncommutative extension of sparse recovery problems discussed in the 10 previous chapters. As a matter of fact, low rank recovery is equivalent to sparse 11 recovery when all the matrices in question are diagonal. There are several important 12 instances of such problems, in particular, matrix completion [41, 45, 70, 123], 13 matrix regression [40, 90, 126] and the problem of density matrix estimation in 14 quantum state tomography [70, 71, 88]. We will study some of these problems 15 using general empirical processes techniques developed in the first several chapters. 16 Noncommutative Bernstein type inequalities established in Sect. 2.4 will play a 17 very special role in our analysis. The main results will be obtained for Hermitian 18 matrices. So called “Paulsen dilation” (see Sect. 2.4) can be then used to tackle 19 the case of rectangular matrices. Throughout the chapter, we use the notations 20 introduced in Sect. A.4.