Algorithm 1 Distributed matrix completion algorithm for recognition, on the ith processing node.

Input: Initial portion of the data matrix for the ith node, Di = D0i, and parameter, λ.

Output: ith portion of the completed matrix, Di

ℒi0, = 0, μk > 0, ρ = 1.1, Ei0 = 0

while not converged do

1. Fix all other variables and update ${\mathbf{\text{D}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}+\mathbf{\text{1}}}}=\underset{{\text{D}}_i}{argmin}\frac{1}{{\mu }_k}{\Vert {\mathbf{\text{D}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}}}\Vert }_{*}+\frac{1}{2}{\Vert {\mathbf{\text{D}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}}}-({\mathbf{\text{D}}}_{{\mathbf{0}}_{\mathbf{\text{i}}}}+{\mathbf{\text{E}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}}}-\frac{{ℒ}_{{\text{i}}_{\text{k}}}}{{\mu }_k})\Vert }_F^2$

by:

${\mathbf{\text{J}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}}}={\mathbf{\text{D}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{0}}}}-{\mathbf{\text{E}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}}}+{\mu }_k^{-1}{ℒ}_{{\text{i}}_{\text{k}}}$

ci(k) = JikT Jik

Send ci(k) to all the neighbors, ,

Receive all cj(k)s from the neighbors, ,

ci(k) = ci(k) + (k) Σj∈ (cj(k) − ci(k))

(V, $\frac{1}{N_p}{\mathbf{\sum }}^2$, VT) = svd(ci(k))

Ui = Jik VΣ−1

Dik+1= Ui τ[Σ] VT

2. Fix all other variables and update

${\mathbf{\text{E}}}_{{\mathbf{\text{X}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}+\mathbf{\text{1}}}}}=\underset{{\mathbf{\text{E}}}_{{\mathbf{\text{X}}}_{\mathbf{\text{i}}}}}{argmin}\frac{1}{{\mu }_k}f({\mathbf{\text{E}}}_{{\mathbf{\text{X}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}}}})+\frac{1}{2}{\Vert {\mathbf{\text{E}}}_{{\mathbf{\text{X}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}}}}-({\mathbf{\text{D}}}_{{\mathbf{\text{0X}}}_{\mathbf{\text{i}}}}+{\mathbf{\text{D}}}_{{\mathbf{\text{X}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}+\mathbf{\text{1}}}}}-\frac{{ℒ}_{{\text{i}}_{\text{k}}}}{{\mu }_k})\Vert }_F^2$

3. Fix all other variables and update

${\mathbf{\text{E}}}_{{\mathbf{\text{Y}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}+\mathbf{\text{1}}}}}=\underset{{\mathbf{\text{E}}}_{{\mathbf{\text{Y}}}_{\mathbf{\text{i}}}}}{argmin}\frac{1}{{\mu }_k}g({\mathbf{\text{E}}}_{{\mathbf{\text{Y}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}}}})+\frac{1}{2}{\Vert {\mathbf{\text{E}}}_{{\mathbf{\text{Y}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}}}}-({\mathbf{\text{D}}}_{{\mathbf{\text{0Y}}}_{\mathbf{\text{i}}}}+{\mathbf{\text{D}}}_{{\mathbf{\text{Y}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}+\mathbf{\text{1}}}}}-\frac{{ℒ}_{{\text{i}}_{\text{k}}}}{{\mu }_k})\Vert }_F^2$

4. Set the ${\mathbf{\text{E}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}}}={\left[\begin{array}{lll}{{\mathbf{\text{E}}}_{{\mathbf{\text{Y}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}}}}}^{\text{T}}& {{\mathbf{\text{E}}}_{{\mathbf{\text{X}}}_{{\mathbf{\text{i}}}_{\mathbf{\text{k}}}}}}^{\text{T}}& {\mathbf{0}}^{\text{T}}\end{array}\right]}^{\text{T}}$,

5. Update the multiplier, ℒi: ℒik+1 = ℒik + μk(Dik+1 − Di0 − Eik+1)

7. Update parameter, μk+1, as: μk+1= min(ρμk, 1010) and k = k + 1.

8. Check the convergence condition: (Dik − D0i −Eik → 0)

end while