Budgeted CP alternating least squares

Work in exact arithmetic over $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$. Inputs are a tensor $A\in {\mathbb{F}}^{n_1\times \cdots \times n_d}$, a proposed CP component count $R\in {\mathbb{N}}_0$, initial factor matrices $F_k\in {\mathbb{F}}^{n_k\times R}$, and a finite sweep budget $N_{\max }\in {\mathbb{N}}_0$. The objective is $\mathrm{\Phi }(F_1,\dots ,F_d)={\parallel A-{\sum }_{j=1}^R{F}_1(:,j)\otimes \cdots \otimes F_d(:,j)\parallel }_F^2\mathrm{.}$ On every exit, return the tuple $(F_1,\dots ,F_d,\widehat{A},\text{status})$, where $\widehat{A}={\sum }_{j=1}^R{F}_1(:,j)\otimes \cdots \otimes F_d(:,j)$ is evaluated from the factor matrices as they stand at that exit. If $R=0$, return the empty factor matrices, the zero model, and status “rank-zero-model” immediately. If $N_{\max }=0$, return the current factors and model with status “budget-exhausted”. Otherwise, for each of at most $N_{\max }$ sweeps, visit modes $k=1,\dots ,d$ in that order. With the other factors fixed, form $Z_k$ whose rows are indexed by all tuples $(i_{\mathrm{\ell }}{)}_{\mathrm{\ell }\mathrm{\ne }k}$ and whose column $j$ is $(Z_k{)}_{(i_{\mathrm{\ell }}{)}_{\mathrm{\ell }\mathrm{\ne }k},j}={\prod }_{\mathrm{\ell }\mathrm{\ne }k}(F_{\mathrm{\ell }}{)}_{i_{\mathrm{\ell }},j}\mathrm{.}$ The mode-$k$ unfolding model is $A_{(k)}\approx X{Z}_k^{\mathsf{T}}$. If $Z_k$ is not full column rank, halt and return the current factors and model with status “rank-deficient-design”; no factor has changed at this block. If it has full column rank, solve the equivalent adjoint problem ${\min }_{Y\in {\mathbb{F}}^{R\times n_k}}\mathrm{\parallel }{A}_{(k)}^{\ast }-\overline{Z_k}Y{\mathrm{\parallel }}_F$ and replace $F_k$ by $Y^{\ast }$. The matrix $\overline{Z_k}$ has full column rank, so the minimizer is unique by Full-column-rank matrix least squares has a unique minimizer. After each completed sweep, increment the sweep counter. Return the current factors and model with status “budget-exhausted” after the $N_{\max }$th completed sweep. This finite-budget procedure guarantees neither convergence nor a global best CP approximation.