Tensor train cores, bond ranks, and canonical gauges

Let $A\in {\mathbb{F}}^{n_1\times \cdots \times n_d}$ with $d\ge 2$. A tensor-train (TT) representation consists of integers $r_0=r_d=1$, internal bond sizes $r_1,\dots ,r_{d-1}\in {\mathbb{N}}_0$, and cores $G_k\in {\mathbb{F}}^{r_{k-1}\times n_k\times r_k}(1\le k\le d)$ such that $A_{i_1,\dots ,i_d}={\sum }_{{\alpha }_1=1}^{r_1}\cdots {\sum }_{{\alpha }_{d-1}=1}^{r_{d-1}}{G}_1(1,i_1,{\alpha }_1)G_2({\alpha }_1,i_2,{\alpha }_2)\cdots G_d({\alpha }_{d-1},i_d,1)\mathrm{.}$ An empty bond index set gives an empty sum, hence the zero tensor; in particular, all internal bond sizes zero represent zero. A bond size $r_k$ is the storage dimension contracted between cores $k$ and $k+1$. The cut-$k$ unfolding groups modes $1,\dots ,k$ against $k+1,\dots ,d$ as fixed in Finite tensor arrays and the Frobenius inner product.

For an internal bond $k$, an invertible matrix $S\in {\mathbb{F}}^{r_k\times r_k}$ gives a gauge transformation by replacing the two adjacent cores with ${\tilde{G}}_k(\alpha ,i,\beta )={\sum }_{\gamma =1}^{r_k}{G}_k(\alpha ,i,\gamma )S_{\gamma \beta },{\tilde{G}}_{k+1}(\beta ,j,\eta )={\sum }_{\gamma =1}^{r_k}(S^{-1}{)}_{\beta \gamma }{G}_{k+1}(\gamma ,j,\eta )\mathrm{.}$ Their contraction is unchanged because ${\sum }_{\beta }{S}_{\gamma \beta }(S^{-1}{)}_{\beta {\gamma }^{\mathrm{\prime }}}={\delta }_{\gamma ,{\gamma }^{\mathrm{\prime }}}$.

The left matricization of core $G_k$ has row index $({\alpha }_{k-1},i_k)$ and column index ${\alpha }_k$. The core is left-isometric if this matrix $L_k$ satisfies $L_k^{\ast }{L}_k=I_{r_k}$. The right matricization has row index ${\alpha }_{k-1}$ and column index $(i_k,{\alpha }_k)$; the core is right-isometric if its matrix $R_k$ satisfies $R_k{R}_k^{\ast }=I_{r_{k-1}}$. A TT is in mixed canonical gauge about core $\mathrm{\ell }$ when cores $k<\mathrm{\ell }$ are left-isometric and cores $k>\mathrm{\ell }$ are right-isometric.

The diagram shows the order-three pattern. Each box is one core, each horizontal wire is a bond index summed over in the entry formula, and each vertical leg is a free physical index. Thus leaving the physical legs open and contracting the two bond wires denotes exactly the core product defining $A_{i_1,i_2,i_3}$; the diagram is only a notation for that finite sum.