Higher-order SVD truncation

Work in exact arithmetic over $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$. Given $A\in {\mathbb{F}}^{n_1\times \cdots \times n_d}$ and integer rank caps $0\le r_k\le n_k$, compute an exact SVD of each mode unfolding $A_{(k)}$. Let $U_k$ consist of the first $q_k=\min (r_k,\mathrm{rank}{A}_{(k)})$ left singular vectors, with any orthonormal choice at a tied cutoff. Define $P_k=U_k{U}_k^{\ast }$, the orthogonal projection onto their span; for $q_k=0$, set $P_k=0$.

For auxiliary cores and intermediate arrays in this procedure, allow mode sizes $m_k\ge 0$: an array of shape $(m_1,\dots ,m_d)$ is a function from ${\prod }_{k=1}^d\{1,\dots ,m_k\}$ to $\mathbb{F}$. If a mode size is zero, this is the unique empty array. Extend the mode-product entry formula of Tucker format, multilinear rank, and mode products to these arrays, taking every empty sum to be zero.

Return the factors $U_k\in {\mathbb{F}}^{n_k\times q_k}$, the core $G=A{\times }_1{U}_1^{\ast }{\times }_2\cdots {\times }_d{U}_d^{\ast }$ of shape $(q_1,\dots ,q_d)$, and reconstruction $\widehat{A}=G{\times }_1{U}_1{\times }_2\cdots {\times }_d{U}_d=A{\times }_1{P}_1{\times }_2\cdots {\times }_d{P}_d\mathrm{.}$ If any $q_k=0$, return the corresponding $n_k\times 0$ empty factor, the empty core of the stated shape, and the zero reconstruction of the original physical shape, using the same formulas and empty-sum convention. The algorithm makes no finite-precision claim.