Algorithm 1.

The framework of TENDER algorithm.

Input: K0 = r1 = 0, t1 = C = 0, τ

Output: Flow-scaled residue functions $K\in {\mathbb{R}}^{T\times N_1\times N_2\times N_3}$.

for n = 1, 2, …, N do

C = C + 1

(1) Steepest gradient descent Kg = rn + sn+1AT (C − Arn)

where $s^{n+1}=\frac{\mathit{vec}{(Q)}^T\mathit{vec}(Q)}{\mathit{vec}{(\mathit{AQ})}^T\mathit{vec}(\mathit{AQ})}$, Q ≡ AT (Arn − C), vec(·) vectorizes a matrix

(2) Proximal map:

if C = τ (Acceleration Step) then

Kn = proxγ(2α ‖K‖TENDER)(fold(Kg)), C=0

end if

(3) Update t, $r{t}^{n+1}=(1+\sqrt{1+4{(t^n)}^2})∕2$, rn + 1 = Kn + ((tn − 1)/tn+1)(Kn − Kn−1)

end for