Interval B-Tensors in High-Order Analysis

• Interval B-tensors are interval families of real tensors where every member meets B-tensor positivity criteria, generalizing matrix dominance to higher orders.
• Double B-tensors impose stricter inequalities that ensure robust eigenvalue localization and positive definiteness, which are vital for optimization under uncertainty.
• This framework is applied in polynomial optimization and complementarity problems, providing certification of stability and solution uniqueness amid data uncertainties.

An interval B-tensor is an interval family of real tensors in which every member is a B-tensor, understood as a generalization of B-matrix positivity/dominance across a set of possible tensors. The class of interval double B-tensors further strengthens this, requiring every member tensor in the interval to satisfy the more stringent double B-tensor inequalities. These classes give rise to robust frameworks for analyzing structural properties of tensors under data uncertainty, such as positive definiteness, eigenvalue localization, and the interval P-tensor property. Such theories generalize interval matrix analysis and are central to high-order tensor analysis, especially for applications in polynomial optimization and complementarity problems where uncertainty is present (Ye et al., 18 Jan 2026).

1. Definitions and Foundational Notions

Let $T_{m,n}$ denote the set of real $m$th-order, $n$-dimensional tensors $\mathcal{A} = (a_{i_1…i_m})$, where $i_j \in [n] = \{1,\ldots,n\}$. An interval tensor is denoted

$$\mathcal{A}^{I} = [\underline{\mathcal{A}},\overline{\mathcal{A}}],$$

where $$\underline{\mathcal{A}}, \overline{\mathcal{A}} \in T_{m,n}$$ with $$\underline{\mathcal{A}} \leq \overline{\mathcal{A}}$$ entrywise, and

$$\mathcal{A}^I = \{ \mathcal{A}\in T_{m,n} : \underline{\mathcal{A}} \leq \mathcal{A} \leq \overline{\mathcal{A}} \}.$$

The extreme points of $\mathcal{A}^I$ are the $2^{n^m}$ tensors obtained by choosing each entry $a_{i_1...i_m}$ independently as $\underline{a}_{i_1...i_m}$ or $\overline{a}_{i_1...i_m}$.

A tensor $\mathcal{A}\in T_{m,n}$ is a double B-tensor if for each $i\in[n]$, letting

$$\gamma_i^+(\mathcal{A}) = \max\{ 0,\, a_{i i_2...i_m} : (i_2...i_m)\neq(i...i)\},$$

the following hold:

• (a) $a_{i...i} > \gamma_i^+(\mathcal{A})$,
• (b) $$a_{i...i} - \gamma_i^+(\mathcal{A}) \geq \sum_{(i_2...i_m)\neq(i...i)}[\gamma_i^+(\mathcal{A}) - a_{i i_2...i_m}]$$,
• (c) for every $j\neq i$,

$$(a_{i...i} - \gamma_i^+(\mathcal{A}))\;(a_{j...j} - \gamma_j^+(\mathcal{A})) > \Bigl( \sum_{(i_2...i_m)\neq(i...i)}[\gamma_i^+(\mathcal{A})-a_{i i_2...i_m}] \Bigr) \Bigl( \sum_{(j_2...j_m)\neq(j...j)}[\gamma_j^+(\mathcal{A})-a_{j j_2...j_m}] \Bigr).$$

An interval double B-tensor is an interval tensor family $$\mathcal{A}^I=[\underline{\mathcal{A}},\overline{\mathcal{A}}]$$ in which every member tensor is a double B-tensor, equivalently, in which all extreme-point tensors are double B-tensors (Ye et al., 18 Jan 2026).

2. Necessary and Sufficient Conditions for Interval B- and Double B-Tensors

Determination of whether $\mathcal{A}^I$ is an interval double B-tensor reduces to checking certain inequalities on its bounding tensors. Write $\underline{\mathcal{A}}$ and $\overline{\mathcal{A}}$ for the lower- and upper-bounding tensors.

Diagonal dominance at zero-level:

• For every $i$,

$$\underline{a}_{i...i} > \max \left\{ 0,\, \overline{a}_{i i_2...i_m} : (i_2...i_m)\neq(i...i) \right\}.$$

Row-wise single-index conditions:

• For every $i$ and each off-diagonal multi-index $(i_2...i_m)\neq(i...i)$,
  • (b1) $\underline{a}<em>{i...i} - \overline{a}</em>{i i_2...i_m}</li> <li>\geq \max \left{</li> <li>0,\;</li> <li>\sum_{(k_2...k_m)\neq(i...i),(k_2...k_m)\neq(i_2...i_m)} \left(\overline{a}<em>{i i_2...i_m} - \underline{a}</em>{i k_2...k_m}\right)</li> <li>\right}$,
  • (b2) $$\underline{a}_{i...i} \geq \max\left\{0, -\sum_{(k_2...k_m)\neq(i...i)} \underline{a}_{i k_2...k_m} \right\}$$.

Pairwise row-product conditions:

• For all $i\neq j$ and any $(i_2...i_m)\in$ row $i$ and $(j_2...j_m)\in$ row $j$,
  • (c1) $(\underline{a}<em>{i...i}-\overline{a}</em>{i i_2...i_m})(\underline{a}<em>{j...j}-\overline{a}</em>{j j_2...j_m})&gt;</li> <li>[\text{max-sum over row } i] \times [\text{max-sum over row } j]$- (c2)$(\underline{a}<em>{i...i}-\overline{a}</em>{i i_2...i_m})\;\underline{a}_{j...j} &gt;</li> <li>[\text{max-sum over row } i]\;\times [\text{max-sum over row } j]$- (c3)$\underline{a}<em>{i...i}\; \underline{a}</em>{j...j} &gt;</li> <li>[\text{max-sum over row } i]\; \times [\text{max-sum over row } j]$

These inequalities are necessary and sufficient for the interval double B-tensor property to hold (Ye et al., 18 Jan 2026).

For interval Z-tensors (all off-diagonals nonnegative), interval double B reduces to checking the double B-property only for $\underline{\mathcal{A}}$ (Ye et al., 18 Jan 2026).

3. Special Cases and Simplifying Criteria

Certain structured tensor classes permit further simplification:

• Circulant interval tensors: If both $\underline{\mathcal{A}},\overline{\mathcal{A}}$ are circulant, interval double B- and interval B-tensor conditions coincide, and only the first-row inequalities suffice.
• Slice-extremal case: If, in each row, a unique off-diagonal attains the maximum upper bound, then it suffices to check only $2n$ specific extreme-point tensors.
• Even order symmetric interval tensors: If $m$ is even and both bounds are symmetric, every interval double B-tensor is automatically an interval P-tensor (Ye et al., 18 Jan 2026).

This reduction to checking finitely many (but possibly exponentially many) inequalities enables verifiable certification of the interval property.

4. Relationships to Other Structured Tensor Classes

Interval double B-tensors relate to other matrix and tensor classes as summarized below:

Tensor Class	Inclusion Relationship	Special Condition for Coincidence
Interval double B-tensor	$\subseteq$ interval B-tensor	Always
Interval double B-tensor	$\implies$ interval P-tensor (even-order, symmetric)	Even $m$, symmetric bounds
Interval double B-tensor	$\iff$ lower bound double B-tensor (for interval Z-tensor)	All off-diagonals nonnegative

In the broader tensor landscape, interval B-tensors extend B-matrix theory, while double B-tensors guarantee strong forms of positivity and eigenvalue localization akin to Gerschgorin and Cassini-type intervals for matrices (Ye et al., 18 Jan 2026, Jin et al., 2015).

5. Eigenvalue Localization and Interval B-Intervals

Direct implications of the interval double B-tensor inequalities include tight bounds on tensor eigenvalues:

• For even-order, real symmetric tensors, every $H$-eigenvalue of any member of the interval is guaranteed to lie in the appropriate double B-intervals. These sets are constructed as a union of one-index and two-index (Cassini-oval) pieces determined by the defining tensor entries (Jin et al., 2015).
• Formally, for a symmetric tensor $\mathcal{A} \in T(\mathbb{R}^n, m)$ of even order, the inclusion region $\Lambda(\mathcal{A})$ satisfies

$$\text{All real } H\text{-eigenvalues } \lambda \in \Lambda(\mathcal{A}).$$

• This interval can be strictly tighter than those produced by the Brauer-type or classical Gerschgorin approaches, especially after intersection.

A plausible implication is that robust spectral bounds applicable to all tensors in the interval family can be directly read off from the interval bounds on tensor entries (Ye et al., 18 Jan 2026).

6. Illustrative Example and Applications

Consider a $3$rd-order, $n=3$ interval tensor with only one uncertain entry:

• Diagonals fixed: $$\underline{a}_{iii} = \overline{a}_{iii} = 100,\ i=1,2,3$$.
• One off-diagonal uncertain: $0 \leq a_{112} \leq 1$.
• All other off-diagonals zero.

Checking the double B conditions at the two extreme points ($a_{112}=0$ and $a_{112}=1$) confirms the interval double B-tensor property.

Applications:

• Extension of interval-matrix theory to the tensor setting, generalizing criteria such as row-sum and dominance inequalities.
• Certification of robust positive definiteness (interval P-tensor property) in polynomial optimization, relevant in sums-of-squares hierarchies under uncertainty.
• Constraining the solution structure in complementarity problems with uncertain data; interval double B-tensors yield existence and uniqueness results.
• Rigorous inclusion for $H$- or $Z$-eigenvalues of all tensors in the family, facilitating robust spectral analysis (Ye et al., 18 Jan 2026).

7. Broader Significance and Research Directions

Interval double B-tensors provide a precise route to generalizing classical matrix interval results to high-order tensors. The theory supplies tools for analyzing stability, definiteness, and solution uniqueness in a range of optimization and equilibrium problems. Connections to interval Z-tensors and explicit reductions in special cases (such as circulant structure) highlight the flexibility of the approach.

A plausible implication is that further study of interval B- and double B-tensors will yield new advances in uncertainty quantification for multilinear algebraic systems, expanding applications in robust data analysis, control, and computational mathematics (Ye et al., 18 Jan 2026).