. Author manuscript; available in PMC: 2017 Aug 18.

Published in final edited form as: J Am Stat Assoc. 2016 Aug 18;111(514):621–633. doi: 10.1080/01621459.2015.1021005

Algorithm 2.

Algorithm of Structured Matrix Completion with unknown r

Input: A₁₁ ∈ ℝ^m^₁×^m^₂, $A_{12}^{m_{1} \times (p_{2} - m_{2})}, A_{21}^{(p_{1} - m_{1}) \times m_{2}}$ . Thresholding level: T_R, (or T_C).
Calculate the SVD A_•1 = U⁽¹⁾Σ⁽¹⁾V^(1)⊤, A_1• = U⁽²⁾Σ⁽²⁾V^(2)⊤.
Calculate Z₁₁ ∈ ℝ^m^₁×^m^₂, Z₁₂ ∈ ℝ^m^₁×(^p^₂−^m^₂), Z₂₁ ∈ ℝ⁽^p^₁−^m^₁)×^m^₂ $Z_{11} = U^{(2) ⊤} A_{11} V^{(1)}, Z_{12} = U^{(2) ⊤} A_{12}, Z_{21} = A_{21} V^{(1)} .$
for s = min(m₁, m₂): −1: 1 do (Use iteration to find r̂)
Calculate D_R,s ∈ ℝ⁽^p^₁−^m^₁)^×^s (or D_C,s ∈ ℝ^s^×(^p^₂−^m^₂)) by solving linear equation system, $D_{R, s} = Z_{21, [:, 1 : s]} Z_{11, [1 : s, 1 : s]}^{- 1} (or D_{C, s} = Z_{11, [1 : s, 1 : s]}^{- 1} Z_{12, [1 : s, :]})$
if Z_11,[1:_s_,1:_s_] is not singular and ||D_R,s|| ≤ T_R (or ||D_C,s|| ≤ T_C) then
r̂ = s; break from the loop;
end if
end for
if (r̂ is not valued) then r̂ = 0.
end if
Finally we calculate the estimate as
${\hat{A}}_{22} = Z_{21, [:, 1 : \hat{r}]} Z_{11, [1 : \hat{r}, 1 : \hat{r}]}^{- 1} Z_{12, [1 : \hat{r}, :]}$