Sequential tensor-train SVD

Work in exact arithmetic over $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$. The input is a tensor $A\in {\mathbb{F}}^{n_1\times \cdots \times n_d}$ with $d\ge 2$, and exactly one of:

• rank caps $c_1,\dots ,c_{d-1}\in {\mathbb{N}}_0$; or
• absolute tail tolerances ${\tau }_1,\dots ,{\tau }_{d-1}\in [0,\mathrm{\infty })$.

At cut $k$, flatten the current remainder with rows $({\alpha }_{k-1},i_k)$ and columns $(i_{k+1},\dots ,i_d)$, where ${\alpha }_0=1$, and compute an exact compact SVD $M_k=U_k{\mathrm{\Sigma }}_k{V}_k^{\ast }$. Its positive singular values are ordered decreasingly as ${\sigma }_1,\dots ,{\sigma }_s$. Under rank caps choose $q_k=\min (c_k,s)$. Under tolerances choose the least $q_k\in \{0,\dots ,s\}$ for which ${\sum }_{j>q_k}{\sigma }_j^2\le {\tau }_k^2$. This set is nonempty because $q_k=s$ gives tail zero. Record the discarded tail ${\delta }_k={\left({\sum }_{j>q_k}{\sigma }_j^2\right)}^{1\mathrm{/}2}\mathrm{.}$

Reshape the first $q_k$ columns of $U_k$ into core $G_k$ and reshape ${\mathrm{\Sigma }}_{k,1:q_k}{V}_{k,1:q_k}^{\ast }$ as the next remainder. After cut $d-1$, reshape the remaining matrix as $G_d$. If $q_k=0$, the remainder has zero row dimension; all later ranks and tails are set to zero and the remaining cores are the corresponding empty zero arrays. Thus the zero tensor has all internal ranks zero, a cap zero returns the zero tensor, and a tolerance large enough to select $q_k=0$ does the same. The output is the TT core list and the cut tails $({\delta }_k)$. Any SVD tie at a retained cutoff may be resolved by any orthonormal choice within the tied singular subspace.