s-Goodness for Low-Rank Matrix Recovery

and Applied Analysis 3 If there does not exist such y for some X as above, we set γ s (A, β) = +∞ and to be compatible with the special case given by [24] we write γ s (A), γ s (A) instead of γ s (A, +∞), γ s (A, +∞), respectively. From the above definition, we easily see that the set of values that γ takes is closed. Thus, when γ s (A, β) < +∞, for every matrix X ∈ Rm×n with s nonzero singular values, all equal to 1, there exists a vector y ∈ Rp such that 󵄩 󵄩 󵄩 󵄩 y 󵄩 󵄩 󵄩 󵄩d ≤ β, σ i (A ∗ y){ = 1, if σ i (X) = 1, ∈ [0, γ s (A, β)] , if σ i (X) = 0, i ∈ {1, 2, . . . , r} . (10) Similarly, for every matrixX ∈ Rm×n with s nonzero singular values, all equal to 1, there exists a vector y ∈ Rp such thatAy and X share the same orthogonal row and column spaces: 󵄩 󵄩 󵄩 󵄩 y 󵄩 󵄩 󵄩 󵄩d ≤ β, 󵄩 󵄩 󵄩 󵄩 A ∗ y − X 󵄩 󵄩 󵄩 󵄩 ≤ γ s (A, β) . (11) Observing that the set {A∗y : ‖y‖ d ≤ β} is convex, we obtain that if γ s (A, β) < +∞ then for every matrix X with at most s nonzero singular values and ‖X‖ ≤ 1 there exist vectors y satisfying (10) and there exist vectors y satisfying (11). 2.2. Basic Properties of G-Numbers. In order to characterize the s-goodness of a linear transformation A, we study the basic properties of G-numbers. We begin with the result that G-numbers γ s (A, β) and γ s (A, β) are convex nonincreasing functions of β. Proposition 3. For every linear transformation A and every s ∈ {0, 1, . . . , r}, G-numbers γ s (A, β) and γ s (A, β) are convex nonincreasing functions of β ∈ [0, +∞]. Proof. We only need to demonstrate that the quantity γ s (A, β) is a convex nonincreasing function ofβ ∈ [0, +∞]. It is evident from the definition that γ s (A, β) is nonincreasing for given A, s. It remains to show that γ s (A, β) is a convex function of β. In other words, for every pair β 1 , β 2 ∈ [0, +∞], we need to verify that γ s (A, αβ 1 + (1 − α) β 2 ) ≤ αγ s (A, β 1 ) + (1 − α) γ s (A, β 2 ) , ∀α ∈ [0, 1] . (12) The above inequality follows immediately if one of β 1 , β 2 is +∞. Thus, we may assume β 1 , β 2 ∈ [0, +∞). In fact, from the argument around (10) and the definition of γ s (A, ⋅), we know that for every matrix X = UDiag(σ(X))V with s nonzero singular values, all equal to 1, there exist vectors y 1 , y 2 ∈ Rp such that for k ∈ {1, 2} 󵄩 󵄩 󵄩 󵄩 y k 󵄩 󵄩 󵄩 󵄩d ≤ β k , σ i (A ∗ y k ) { = 1, if σ i (X) = 1, ∈ [0, γ s (A, β k )] , if σ i (X) = 0, i ∈ {1, 2, . . . , r} . (13) It is immediate from (13) that ‖αy 1 + (1 − α)y 2 ‖ d ≤ αβ 1 +(1− α)β 2 . Moreover, from the above information on the singular values ofAy 1 ,Ay 2 , we may setAy k = X + Y k , k ∈ {1, 2} such that