Algorithm 2.

Alternating least squares algorithm.

Input: CP-format vectors F and G, with ranks $R_{\psi }<R_{\mathbf{G}}$;

Output: out$\mathbf{F}$, such that $∥{}^{out}\mathbf{F}-\mathbf{G}∥$ is minimized

(1) Initialization:

construct $\mathbf{B}$ and $\mathbf{P}$ matrices

(2) for $\alpha =1\dots N_{ALS}$:

for $c=1\dots D$: (Loop over coordinates)

(a) downdate: $B_{r^{\prime },r}^{\ne c}\leftarrow B_{r^{\prime },r}∕B_{r^{\prime },r}^c$ for all $r^{\prime }$, r

(b) downdate: $P_{r^{\prime },r}^{\ne c}\leftarrow P_{r^{\prime },r}∕P_{r^{\prime },r}^c$ for all $r^{\prime }$, r

(c) compute $b_r^{\left(i_c,c\right)}$ for all ic, r (Eq. (10))

(d) solve linear system for ${\mathbf{x}}^{\left(i_c,c\right)}$ (Eq. (11));

replace $f_{i_c}^{\left(r,c\right)}\leftarrow x_r^{\left(i_c,c\right)}$ for all ic, r

(e) normalize $s_r^{\mathbf{F}}\leftarrow ∥\mathbf{f}{}^{\left(r,c\right)}∥$; $\mathbf{f}{}^{\left(r,c\right)}\leftarrow \frac{{\mathbf{f}}^{\left(r,c\right)}}{∥{\mathbf{f}}^{\left(r,c\right)}∥}$ for all r;

(f) update: $B_{r^{\prime },r}\leftarrow B_{r^{\prime },r}^{\ne c}\cdot B_{r^{\prime },r}^c$ for all $r^{\prime }$, r

(g) update: $P_{r^{\prime },r}\leftarrow P_{r^{\prime },r}^{\ne c}\cdot P_{r^{\prime },r}^c$ for all $r^{\prime }$, r