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On defective spans of singular vector tuples beyond the boundary format

This paper investigates tensor spaces beyond the boundary format by relating the codimension of the span of singular vector tuples to cohomological kernels, identifying an infinite family of defective order-three tensors, and proposing a classification linked to Koszul cohomology.

Original authors: Ettore Teixeira Turatti, Emanuele Ventura

Published 2026-05-22

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CC BY 4.0

Original authors: Ettore Teixeira Turatti, Emanuele Ventura

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Tensors as Multi-Dimensional Puzzles

Imagine you have a matrix (a 2D grid of numbers). You know that any matrix can be broken down into its "skeleton" using something called Singular Value Decomposition (SVD). Think of this like taking a complex 3D object and realizing it's just a stack of flat, simple sheets. This is a very reliable rule for 2D grids.

Now, imagine a tensor. This is a generalization of a matrix into 3D, 4D, or even higher dimensions. It's like a Rubik's cube made of numbers, or a stack of matrices. Mathematicians have been trying to find a similar "skeleton" for these multi-dimensional objects. They look for special sets of numbers called singular vector tuples.

Think of these tuples as the "key points" or "landmarks" on the tensor. If you find all the landmarks, you can try to build the whole tensor out of them.

The "Boundary" and the "Beyond"

For a long time, mathematicians knew a specific rule: if your tensor is "balanced" (a condition the authors call the boundary format), the landmarks you find are enough to perfectly reconstruct the whole tensor. It's like having a map where the landmarks you found cover the entire territory.

However, this paper asks: What happens when the tensor is unbalanced? What if it's "beyond the boundary"? In these cases, the tensor is skewed or stretched in a way that breaks the old rules. The authors wanted to know: If we find all the landmarks in these weird, unbalanced shapes, do they still cover the whole shape, or is there a gap?

The Main Discovery: The "Gap" in the Map

The authors discovered that for most unbalanced tensors, the landmarks do cover the whole shape. The map is complete.

However, they found a very specific, strange family of 3D tensors (3D cubes of numbers) where the map is broken.

The Analogy: Imagine you are trying to paint a wall using a set of specific brushstrokes (the landmarks). Usually, if you use all the brushstrokes, you cover the whole wall.
The Defect: The authors found a specific type of wall (a tensor of size $2 \times n \times (n+2)$ ) where, no matter how many brushstrokes you use, you leave a large, unpainted hole in the middle. The "span" (the area covered by the landmarks) is much smaller than it should be based on the number of landmarks you have.

They call this a "defective" behavior. It's like having a keyring with 10 keys, but only 3 of them actually fit any lock, and the other 7 are useless for opening the door. In this specific family of tensors, the "useless" keys are the majority, leaving a huge gap.

How They Figured It Out: The "Cohomology" Machine

To prove this, the authors didn't just look at the numbers; they used a high-powered mathematical tool called cohomology.

The Analogy: Think of cohomology as a sophisticated "stress test" or a "quality control scanner" for the mathematical structure.
They built a machine (a mathematical map called $\alpha_T$ ) that takes the information from the landmarks and checks if it fills the space.
In most cases, this machine works perfectly (it has "full rank"), meaning the landmarks fill the space.
In their specific defective family, the machine breaks down completely. It outputs zero. This mathematical "zero" proves that the landmarks are failing to cover the space, confirming the existence of the gap.

The "Koszul" Connection

In the final section, the authors connect this broken machine to something called Koszul cohomology.

The Analogy: Imagine you have a complex knot. You want to know if the knot is truly tangled or if it's just a loose loop. Koszul cohomology is like a specific type of magnifying glass that looks at the "threads" of the knot to see how they interact.
The authors suggest that the reason the map is broken in their specific case is deeply tied to how these mathematical "threads" (syzygies) interact in a way that creates a knot that cannot be untangled into a full shape.

Summary of Claims

The Question: Do the "landmarks" (singular vector tuples) of a tensor always cover the whole tensor, even when the tensor is unbalanced?
The General Answer: Yes, usually they do.
The Exception: There is an infinite family of 3D tensors (specifically $2 \times n \times (n+2)$ ) where the landmarks fail to cover the tensor. The "gap" is huge (codimension $n-1$ ).
The Proof: They used advanced algebraic geometry (Bott-Borel-Weil theorem) to show that a specific mathematical map becomes zero for these tensors, proving the gap exists.
The Conjecture: They guess that only these specific cases (and a few tiny variations) have this "broken map" problem. Everywhere else, the landmarks work perfectly.

In short: The paper finds a specific, weird shape of multi-dimensional data where the usual rules for breaking it down fail, leaving a large empty space that the data points cannot fill. They proved exactly when and why this happens.

Technical Summary: On Defective Spans of Singular Vector Tuples Beyond the Boundary Format

Problem Statement
The paper investigates the spectral theory of tensors, specifically focusing on the relationship between the singular vector $k$ -tuples of a tensor $T$ and the linear subspace they span. In the context of tensor decomposition, a singular vector $k$ -tuple generalizes the concept of singular pairs for matrices. For a general tensor $T$ , let $Z_T$ denote the set of its singular vector $k$ -tuples (viewed as points in the projective space $\mathbb{P}(V)$ ), and let $\langle Z_T \rangle$ be their projective span.

A central object of study is the critical space $H_T$ , defined as the linear subspace of $V$ cut out by the equations defining singular vector $k$ -tuples. It is known that $\langle Z_T \rangle \subseteq \mathbb{P}(H_T)$ . Previous work by Draisma, Ottaviani, and Tocino [DOT18] established that for tensor spaces of "boundary format" (where the dimension of one factor equals the sum of the dimensions of the others), equality holds: $\langle Z_T \rangle = \mathbb{P}(H_T)$ . This result generalizes the Eckart-Young theorem for matrices.

The problem addressed in this paper is the behavior of this equality when the tensor space exceeds the boundary format. Specifically, the authors analyze the case where the tensor space $V = \mathbb{C}^{n_1+1} \otimes \cdots \otimes \mathbb{C}^{n_k+1}$ satisfies $n_k = 1 + \sum_{i=1}^{k-1} n_i$ , which represents the minimal case beyond the boundary format. The core question is whether $\langle Z_T \rangle = \mathbb{P}(H_T)$ holds generally in this regime or if "defective" cases exist where the span is strictly smaller than the critical space.

Methodology
The authors employ a combination of algebraic geometry, representation theory, and cohomological techniques:

Vector Bundles and Cohomology: The study utilizes the Friedland-Ottaviani vector bundle $E$ on the Segre product $P = \mathbb{P}(\mathbb{C}^{n_1+1}) \times \cdots \times \mathbb{P}(\mathbb{C}^{n_k+1})$ . The singular vector $k$ -tuples correspond to the zero locus of a global section $s_T$ of $E$ .
Koszul Complex: The resolution of the ideal sheaf of the zero locus is analyzed via the Koszul complex associated with $s_T$ . This allows the authors to relate the geometry of $Z_T$ to the cohomology of twists of the bundle $E$ .
Bott-Borel-Weil Theorem: The authors perform a detailed analysis of the cohomology groups $H^q(E(r))$ using the Bott-Borel-Weil theorem and Bott's algorithm for $SL(n+1)$. This involves calculating weights and determining vanishing conditions for cohomology groups based on the tensor format.
Cohomological Maps: The codimension of $\langle Z_T \rangle$ within $\mathbb{P}(H_T)$ is identified with the dimension of the kernel of a specific linear map $\alpha_T$ between cohomology groups:
$\alpha_T: H^n(E(n+1)) \to H^n(E(n))$
where $n = \sum_{i=1}^k n_i$ .
Representation Theory: The map $\alpha_T$ is analyzed as an equivariant map under the action of the product of special linear groups. By decomposing the domain and codomain into irreducible representations and applying Schur's Lemma, the authors determine conditions under which $\alpha_T$ must vanish or have maximal rank.

Key Contributions and Results

Characterization of Codimension: The paper establishes that the codimension of $\langle Z_T \rangle$ in $\mathbb{P}(H_T)$ is exactly the dimension of the kernel of the map $\alpha_T$ . This formalizes and extends previous steps found in [ST24].
Distinct Behavior by Factor Count:
- For $k+1 \geq 4$ factors, the cohomological vanishings are uniform.
- For $k+1 = 3$ factors (order-three tensors), the behavior is peculiar, with fewer cohomological vanishings, leading to different structural properties.
Defective Family Identified: The authors identify an infinite family of order-three tensors where the span is defective. Specifically, for a general tensor $T \in \mathbb{C}^2 \otimes \mathbb{C}^n \otimes \mathbb{C}^{n+2}$ , the map $\alpha_T$ vanishes identically. Consequently, the span $\langle Z_T \rangle$ has a codimension of $n-1$ in $\mathbb{P}(H_T)$ , which is the maximum possible codimension allowed by parameter counting. This is stated in Theorem 5.2.
Dimensional Analysis: Through combinatorial inequalities involving binomial coefficients, the authors determine specific cases where the dimensions of the domain and codomain of $\alpha_T$ suggest the map cannot be injective. These include cases like $(2, 2, 2, 5)$ , $(2, 2, 3, 6)$ , and $(2, 2, 4, 7)$ for $k+1=4$ , and specific small cases for $k+1=3$ .
Conjectures: Based on the analysis, the authors conjecture that outside of a finite list of specific "exceptional" cases (including the infinite family mentioned above), the map $\alpha_T$ $α_{T}$ has maximal rank, implying $\langle Z_T \rangle = \mathbb{P}(H_T)$ $⟨ Z_{T} ⟩ = P (H_{T})$ .
- Conjecture 7.2 addresses the 3-factor case ( $a, b \geq 2$ ).
- Conjecture 7.3 addresses the $k+1 \geq 4$ factor case.
Connection to Koszul Cohomology: In the final section, the authors draw a connection between their conjectures and the vanishing of certain Koszul cohomology modules $K_{1,1}(X, \mathcal{O}(1,1), W)$ , linking the defectiveness of the span to the syzygies of the Segre variety.

Significance
The paper provides a rigorous framework for understanding the limitations of singular vector tuples in tensor decomposition beyond the boundary format. While the boundary format guarantees that a tensor is determined by its singular vector tuples (via the equality of the span and critical space), this work demonstrates that this property fails for specific infinite families of tensors. The identification of the map $\alpha_T$ as the determinant of this equality offers a concrete cohomological tool for analyzing tensor ranks and decompositions. The results refine the understanding of the "best rank-one approximation problem" and the geometric structure of the critical space, highlighting that the generalization of the Eckart-Young theorem to higher-order tensors is not universal and depends critically on the tensor format. The connection to Koszul cohomology suggests deeper algebraic structures underlying these geometric phenomena.