Tensor-train rounding

Work in exact arithmetic over $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$. The input is a valid TT core list for a tensor of order $d\ge 2$ and exactly one of rank caps $c_1,\dots ,c_{d-1}\in {\mathbb{N}}_0$ or absolute tail tolerances ${\tau }_1,\dots ,{\tau }_{d-1}\in [0,\mathrm{\infty })$.

1. Left orthogonalize without truncation. For $k=1,\dots ,d-1$, reshape core $G_k$ into a matrix with rows $({\alpha }_{k-1},i_k)$ and columns ${\alpha }_k$. Compute an exact compact SVD $M=U\mathrm{\Sigma }{V}^{\ast }$, replace $G_k$ by the reshape of $U$, and absorb $\mathrm{\Sigma }{V}^{\ast }$ into the next core along their common bond. Omitting zero singular directions is exact. The resulting cores $G_1,\dots ,G_{d-1}$ are left-isometric.

2. Truncate from right to left. For $k=d,d-1,\dots ,2$, reshape the current $G_k$ into a matrix $M_k$ with rows ${\alpha }_{k-1}$ and columns $(i_k,{\alpha }_k)$, and compute an exact compact SVD $M_k=U_k{\mathrm{\Sigma }}_k{V}_k^{\ast }$. If its positive singular values are ${\sigma }_1\ge \cdots \ge {\sigma }_s>0$, choose $q_{k-1}=\min (c_{k-1},s)$ for rank caps, or the least $q_{k-1}\in \{0,\dots ,s\}$ with ${\sum }_{j>q_{k-1}}{\sigma }_j^2\le {\tau }_{k-1}^2$ for tolerances. Record ${\delta }_{k-1}=({\sum }_{j>q_{k-1}}{\sigma }_j^2{)}^{1\mathrm{/}2}$. Replace $G_k$ by the reshape of the first $q_{k-1}$ rows of $V_k^{\ast }$ and absorb $U_{k,1:q}{\mathrm{\Sigma }}_{k,1:q}$ into the previous core.

If $q_{k-1}=0$, the represented tensor becomes zero; set all remaining internal ranks to zero and their unperformed tails to zero. If the input represents zero, return the all-zero-rank representation and zero tails immediately. The output is the rounded core list and all recorded cut tails. No finite-precision error guarantee is part of this exact-arithmetic procedure.