Algorithm 2.

Low-rank MNE block coordinate descent algorithm (locally optimal domain)

1:	inputs: maximum rank r, maximum range for regularization parameters ϵmax, number of regularization parameter grid points ngrid, training set indices Ttrain⊆{1, … , N} and cross-validation set indices TCV⊂{1, … , N} where Ttrain∩TCV = ∅, paired data samples (st, yt) for all t ∈ Ttrain∪TCV, initial guess for weights a, h, U, and V, set πk for all k = 1, … , r, maximum iterations Mmax, convergence precision δp, maximum failures to find a better solution σmax
2:	initialization: J ← UVT, Lbest ← L(a, h, U, V)|TCV (evaluated over data indices t ∈ TCV), regularization grid resolution δϵ ← ϵmax/ngrid, early completion switch σ ← 1
3:
4:	for m ← 1, … , Mmax do
5:	for k ← 1, … , r do
6:	a′ ← a, h′ ← h, ${\text{u}}^{\prime }\leftarrow {\text{U}}_{•,k}$, ${\text{v}}^{\prime }\leftarrow {\text{V}}_{•,k}$
7:	J ← J − u′ v′T                                                                                                 ⊳ remove block k from J
8:	for n ← 0, … , ngrid do
9:	ϵk ← nδϵ
10:	a′, h′, u′, v′ ← Solve the block k subproblem (Equation 13) using an
11:	interior-point method algorithm with inputs a′, h′, u′, v′, J,
12:	ϵk, πk, (st, yt):∀t ∈ Ttrain
13:	$L\prime =L(a\prime ,\text{h}\prime ,\text{J}+\text{u}\prime \text{v}{\prime }^{\text{T}}){|}_{T_{\text{CV}}}$
14:	if $L^{\prime }<L_{\text{best}}-{\delta }_p$ then
15:	$L_{\text{best}}\leftarrow L^{\prime }$
16:	a ← a′, h ← h′, ${\text{U}}_{•,k}\leftarrow {\text{u}}^{\prime }$, ${\text{V}}_{•,k}\leftarrow {\text{v}}^{\prime }$
17:	σ ← 0
18:	else if $L^{\prime }\le L(a,\text{h},\text{U},\text{V}){|}_{T_{\text{CV}}}$ then
19:	a ← a′, h ← h′, ${\text{U}}_{•,k}\leftarrow {\text{u}}^{\prime }$, ${\text{V}}_{•,k}\leftarrow {\text{v}}^{\prime }$ (or skip for monotonic convergence)
20:	$\text{J}\leftarrow \text{J}+{\text{U}}_{•,k}{\text{V}}_{•,k}^{\text{T}}$                                                                                      ⊳ include block k's solution in J
21:	if σ = σmax then
22:	break                                                                                                               ⊳ optimization has finished
23:	σ ← σ + 1
24:
25:	outputs: a, h, J