Tensor Train Variety Overview

• Tensor train variety is a projective algebraic variety that parametrizes tensors with fixed TT-rank via contracted core tensors and unfolding matrices.
• Its structure is defined by vanishing minors, explicit dimension formulas, and stratification into smooth and singular loci for precise rank estimation.
• TT-varieties enable efficient energy minimization, tensor completion, and quantum marginal feasibility through advanced algebraic and numerical algorithms.

A tensor train variety is a projective algebraic variety parametrizing all tensors admitting a fixed TT-rank structure: a decomposition into contracted core tensors along a chain, with rank constraints encoded via matrix unfoldings. Originating as the geometric underpinning of the TT-format for high-dimensional tensor analysis and quantum many-body applications (matrix product states), the TT-variety gives a precise description not only of the manifold of TT-decomposable tensors but also of the singularities, tangent cones, ideal generators, and dimension. Fundamental algebraic-geometric and geometric-invariant-theory techniques yield both structure theorems and algorithms for energy minimization, completion, and feasibility within this variety (Borovik et al., 7 Dec 2025, Vermeylen et al., 19 Feb 2024, Vermeylen et al., 2023, Krämer, 2017, Bernardi et al., 2021, Vermeylen et al., 2023, Semnani et al., 2021).

1. Algebraic-Geometric Definition and Characterization

Given mode sizes $k_1,\dots,k_d$ and a TT-rank vector $\mathbf{r} = (r_1,\dots,r_{d-1})$, a tensor $T \in \mathbb{C}^{k_1 \times \cdots \times k_d}$ admits a TT-decomposition via core tensors $G^{(1)},\dots,G^{(d)}$, where

$$G^{(1)} \in \mathbb{C}^{k_1 \times r_1}, \;\; G^{(2)} \in \mathbb{C}^{r_1 \times k_2 \times r_2}, \ldots, G^{(d)} \in \mathbb{C}^{r_{d-1} \times k_d}.$$

The contraction yields $T$ by summing over the internal "bond" indices. The TT-variety $V$ is the Zariski closure of the image of the rational map

$(G^{(1)},\dots,G^{(d)}) \mapsto T,$

yielding $$V = V_{(k_1,\ldots,k_d), (r_1,\ldots,r_{d-1})} \subset \mathbb{P}(\mathbb{C}^{k_1} \otimes \cdots \otimes \mathbb{C}^{k_d})$$ (Borovik et al., 7 Dec 2025). Equivalently, $T \in V$ iff all matricizations $$T^{(i)} \in \mathbb{C}^{(k_1 \cdots k_i) \times (k_{i+1} \cdots k_d)}$$ satisfy $\operatorname{rank}(T^{(i)}) \leq r_i$ for $i=1,\ldots,d-1$.

Set-theoretically, $V$ is defined by vanishing of all $(r_i+1) \times (r_i+1)$ minors of $T^{(i)}$. Conjecturally, these minors generate the full ideal and form a Gröbner basis for an appropriate order (Conjecture 2.9 in (Borovik et al., 7 Dec 2025)). For special cases (binary tensors, Segre), reduced Gröbner bases are established.

2. Dimension Theory and Stratification

The dimension of the TT-variety is furnished by parameter counting in the cores, subtracting gauge freedoms:

$$\dim V = \sum_{i=1}^{d} r_{i-1} k_i r_i - \sum_{i=1}^{d-1} r_i^2$$

with $r_0 = r_d = 1$ (Bernardi et al., 2021, Vermeylen et al., 19 Feb 2024). This coincides with the general upper bound for tensor network varieties (path graphs), proven to be sharp in the supercritical regime, where $r_{i-1} r_i \leq k_i$ for all $i$ (Bernardi et al., 2021).

The TT-variety admits a stratification: the smooth stratum with fixed TT-rank is a manifold of this dimension, while the full variety is the Zariski closure over lower ranks, with singularities manifesting when multiple ranks drop simultaneously (Vermeylen et al., 19 Feb 2024).

3. Segre Product Cases and Birational Parametrization

For uniform bond dimensions $r_i = 1$, the TT-variety reduces to the Segre embedding $$\mathbb{P}^{k_1-1} \times \cdots \times \mathbb{P}^{k_d-1} \to \mathbb{P}^{k_1 \cdots k_d - 1}$$ (Borovik et al., 7 Dec 2025, Semnani et al., 2021). More generally, if nontrivial ranks are "separated blocks" (no two adjacent $r_i > 1$), and certain saturation bounds hold, Segre-type factorization arises (Theorem 2.7 in (Borovik et al., 7 Dec 2025)).

On the dense open locus where each unfolding $T^{(i)}$ has maximal rank $r_i$, the TT-variety is birational to a product of Grassmannians, via recursive skeleton decompositions (Theorem 3.3 in (Borovik et al., 7 Dec 2025)):

$$\prod_{i=1}^{d-1} \mathrm{Gr}(r_i, r_{i-1} k_i) \times \mathbb{C}^{r_{d-1} \times k_d} \dashrightarrow V.$$

This birationality also underpins dimension counts and establishes irreducibility.

4. Tangent Cones, Singular Locus, and Rank Estimation

At boundary points (where some TT-ranks are strictly less than their upper limits), the tangent cone of $V$ admits an explicit block-TT parametrization (Vermeylen et al., 2023, Vermeylen et al., 19 Feb 2024, Vermeylen et al., 2023):

• The tangent space to the fixed-rank manifold corresponds to variations within current ranks ("top-left block").
• Additional blocks describe normal directions that enable rank increase, key to rank-adaptive optimization.

Approximate projection algorithms onto the tangent cone, satisfying rigorous angle conditions, facilitate gradient-based methods on $V$, with improved constants compared to earlier variants (Vermeylen et al., 2023). Rank-adaptive solvers exploit this structure to increase or decrease TT-rank efficiently during tensor completion and approximation problems, guided by singular-value gaps in the block unfoldings of the gradient (Vermeylen et al., 2023). TT-rounding is proven to act as an $\omega$-approximate projection (Theorem 4.2 in (Vermeylen et al., 19 Feb 2024)), crucial for stability in adaptive algorithms.

5. Feasibility of TT-Singular Tuples and Quantum Marginal Connection

The TT-format gives rise to the tensor feasibility problem (TFP): determining which sets of singular-value tuples are admissible for a TT decomposition (Krämer, 2017). This problem is equivalent to the quantum marginal problem (QMP), where the spectra of reduced density matrices for pure states in multipartite quantum systems are prescribed.

Feasibility of TT-singular values is characterized via polyhedral cones specified by Horn-type inequalities, trace constraints, and an intertwining with Knutson–Tao honeycomb objects. Linear programming algorithms and hive-based heuristics are available for feasibility testing and explicit core construction (Krämer, 2017).

6. Orthogonal Tensor Trains: Algebraic Structure and Comparison

A subclass, orthogonal tensor train varieties, arise by restricting core tensors to orthogonally decomposable forms (Semnani et al., 2021). The length-2 orthogonal TT variety is cut out by mode-pair symmetry, two families of quadratic contraction symmetry equations, and a single global degree-$n$ determinantal constraint. For $n=2$, these equations form a prime ideal, matching the predicted variety dimension.

Contrasted with general TT-varieties (whose defining relations can be highly nontrivial and include higher-degree determinantal constraints across unfoldings), the orthogonal case admits more tractable, compact algebraic descriptions.

7. Applications and Numerical Algebraic Geometry

TT-varieties underpin state-of-the-art high-dimensional numerical methods, including:

• Energy minimization through Rayleigh quotient optimization over TT-varieties, with all critical points enumerated by Rayleigh–Ritz degree using homotopy continuation (Borovik et al., 7 Dec 2025).
• Benchmarking against Alternating Linear Scheme and Density Matrix Renormalization Group protocols for quantum chemistry computations (Borovik et al., 7 Dec 2025).
• Rank-adaptive tensor completion and approximation, leveraging stratification, tangent cone parametrization, and rank estimation to avoid overfitting and ensure computational efficiency (Vermeylen et al., 19 Feb 2024, Vermeylen et al., 2023, Vermeylen et al., 2023).

Ongoing work explores explicit degree formulas, Gröbner basis generation, and further geometric criteria for singular locus and variety stratification.

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Table: Core Structural Properties of TT-Varieties

Property	TT-Variety	Segre Product Special Case
Defining equations	$(r_i+1)\times(r_i+1)$ minors of each unfolding	Simple monomial equations
Dimension (generic)	$$\sum_{i=1}^d r_{i-1}k_i r_i - \sum_{i=1}^{d-1} r_i^2$$	$\sum_{i=1}^d k_i - d$
Birational parametrization	Product of Grassmannians $\prod \mathrm{Gr}(r_i, r_{i-1}k_i)$	Identity map on product of projective spaces
Singular locus	Drop in rank in any flattening	Absent if $r_i=1$

Further technical details, including explicit algorithmic pseudocode, advanced horn inequalities, and quantum marginal reductions, can be found in (Borovik et al., 7 Dec 2025, Bernardi et al., 2021, Krämer, 2017, Vermeylen et al., 19 Feb 2024), and (Vermeylen et al., 2023).