Core-SVD recompression with rank and error tolerances

Input: A field $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$, positive integers $m,n$, a finite factor list $U_s\in {\mathbb{F}}^{m\times r_s}$ and $V_s\in {\mathbb{F}}^{n\times r_s}$ representing $M={\sum }_s{U}_s{V}_s^{\ast }$, a rank cap $0\le k\le \min (m,n)$, and a Frobenius tolerance $\delta \ge 0$. The conjugate transpose $M^{\ast }$ and Frobenius norm are those of Finite tensor arrays and the Frobenius inner product.

Procedure: Concatenate the factors as $U=[U_1\cdots U_p]$ and $V=[V_1\cdots V_p]$, so $M=U{V}^{\ast }$. Take compact singular value decompositions $U=Q_U{R}_U$ and $V=Q_V{R}_V$, where the columns of $Q_U,Q_V$ are orthonormal and span the respective column spaces. Form the core $C=R_U{R}_V^{\ast }$ and compute an SVD $C=P\mathrm{\Sigma }{Q}^{\ast }$. Let its positive singular values be $s_1\ge \cdots \ge s_t>0$. Find the smallest $q\in \{0,\dots ,\min (k,t)\}$ for which $e_q:=({\sum }_{j>q}{s}_j^2{)}^{1\mathrm{/}2}\le \delta$. If no such $q$ exists, return tolerance-not-met. Otherwise return success with factors $\widehat{U}=Q_U{P}_{[:,1:q]}{\mathrm{\Sigma }}_q$ and $\widehat{V}=Q_V{Q}_{[:,1:q]}$, where ${\mathrm{\Sigma }}_q=\mathrm{diag}(s_1,\dots ,s_q)$, and the exact discarded-tail certificate $e_q$; when $q=0$, use empty $m\times 0$ and $n\times 0$ factors.

The compact decompositions retain only positive singular directions. If the factor list is empty or its represented matrix is zero, then $t=0$ and the procedure returns the rank-zero representation because $\delta \ge 0$. Any SVD choices in repeated singular subspaces are permitted. No result is returned when the requested cap cannot certify the tolerance.