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SNT-Rank: Kronecker Products and Euclidean Distance Matrices

This paper advances the theory of symmetric nonnegative matrix trifactorizations by deriving sharper upper bounds for the SNT-rank of Euclidean distance matrices, establishing new relationships between rank and SNT-rank, proving the submultiplicativity of SNT-rank under Kronecker products, and partially resolving conjectures regarding the multiplicativity of nonnegative rank.

Original authors: Bharat Pratap Chauhan, Projesh Nath Choudhury

Published 2026-07-30

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CC BY 4.0

Original authors: Bharat Pratap Chauhan, Projesh Nath Choudhury

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

Imagine you are a detective trying to solve a mystery using only a limited set of Lego bricks. In the world of mathematics, specifically a field called linear algebra, these "bricks" are numbers arranged in grids called matrices. Usually, mathematicians are happy to use any kind of brick—positive, negative, or zero—to build their structures. But sometimes, nature or data only gives us positive bricks (think of them as "nonnegative" numbers, like counts of people or amounts of money). When you are forced to build a complex shape using only positive bricks, the job becomes much harder. You might need way more bricks than if you were allowed to use negative ones. This is the heart of "Nonnegative Matrix Factorization": finding the smallest number of positive building blocks needed to reconstruct a specific pattern.

Now, imagine the pattern you are trying to build has a special rule: it must look the same if you flip it over (symmetry). This happens often in real life, like in the distances between cities on a map or the relationships between friends in a social network. A new type of puzzle has recently emerged called the "Symmetric Nonnegative Trifactorization." Instead of just stacking two layers of bricks, this puzzle asks you to build the shape using three layers: a left layer, a middle layer, and a right layer that is a mirror of the left. The goal is to find the smallest possible size for that middle layer. This size is called the "SNT-rank." The smaller the number, the more efficient your construction. Why does this matter? Because in fields like machine learning and data analysis, finding the most efficient way to compress and understand data can save massive amounts of computer power and reveal hidden patterns that were previously invisible.

In this paper, the authors Bharat Pratap Chauhan and Projesh Nath Choudhury tackle two main challenges regarding this SNT-rank puzzle. First, they look at a specific, tricky type of data called "Euclidean distance matrices." These are grids that show the squared distances between a list of points, like the distances between the numbers 1, 2, 3, and so on. Previous researchers had guessed how many bricks (the SNT-rank) were needed to build these shapes, but the authors found a way to build them with even fewer bricks than anyone thought possible. They proved that for a list of $n$ numbers, you never need more than $2 \lceil \log_2 n \rceil$ bricks. For example, if you have 16 numbers, you only need 8 bricks, which is a significant improvement over previous estimates.

Second, the authors investigate what happens when you combine two of these puzzles together using a mathematical operation called the "Kronecker product." You can think of this as taking two small Lego models and merging them into one giant, complex model. A long-standing question in the field was whether the number of bricks needed for the giant model is simply the product of the bricks needed for the two small ones. The authors show that this isn't always true for every possible puzzle, but they prove it is true under specific conditions, such as when one of the original models is very simple (rank 1) or when the models are small enough (3x3 or smaller). They also partially solve a conjecture about whether the number of bricks for a combined model is always at least as big as the product of the original ranks. By establishing these rules, the paper provides a clearer map for mathematicians and data scientists, showing them exactly when they can predict the complexity of a combined system and when they need to be more careful.

Technical Summary of "SNT-RANK: KRONECKER PRODUCTS AND EUCLIDEAN DISTANCE MATRICES"

Problem Statement
This paper investigates the Symmetric Nonnegative Trifactorization (SN-Trifactorization) of symmetric nonnegative matrices. Introduced by Bukovšek–Šmigoc, an SN-Trifactorization of a matrix $A \in S^+_n$ takes the form $A = BCB^T$ , where $B \in \mathbb{R}^{n \times k}_+$ and $C \in S^+_k$ . The SNT-rank, denoted $st_+(A)$ , is defined as the smallest integer $k$ for which such a factorization exists. While the SNT-rank is known to be bounded by the matrix dimension and related to the classical rank ($rk$), nonnegative rank ( $rk_+$ ), and completely positive rank ($cp$), determining its exact value remains a difficult problem, particularly for Euclidean distance matrices (EDMs). The paper addresses three primary challenges:

Deriving sharper upper bounds for the SNT-rank of specific families of EDMs, specifically $A(1, \dots, n)$ .
Establishing the behavior of the SNT-rank under the Kronecker product.
Investigating the multiplicativity of the nonnegative rank under the Kronecker product, specifically addressing conjectures regarding $rk_+(A_1 \otimes A_2)$ .

Methodology
The authors employ a combination of algebraic matrix theory, combinatorial arguments, and structural analysis of symmetric nonnegative matrices. Key methodological components include:

Structural Decomposition: The paper utilizes the properties of principal submatrices, direct sums, and congruence transformations (permutation and positive diagonal scaling) to relate the SNT-rank of complex matrices to simpler components.
Kronecker Product Analysis: The authors analyze the SN-Trifactorization of Kronecker products $A_1 \otimes A_2$ by constructing factorizations from the factors of $A_1$ and $A_2$ . They leverage the identity $(B_1 C_1 B_1^T) \otimes (B_2 C_2 B_2^T) = (B_1 \otimes B_2)(C_1 \otimes C_2)(B_1 \otimes B_2)^T$ .
Recurrence Relations for EDMs: For Euclidean distance matrices, the authors construct specific block structures. By rearranging indices and decomposing the matrix into block sums involving Kronecker products of small matrices (e.g., $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ and $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ ), they derive recurrence relations for the SNT-rank.
Rank Inequalities: The proofs frequently rely on established inequalities relating $rk$, $rk_+$ , and $st_+$ , as well as specific lemmas regarding the rank of matrices with zero diagonals and positive off-diagonal entries.

Key Contributions and Results

Improved Bounds for Euclidean Distance Matrices:
The paper improves the known upper bound for the SNT-rank of the specific EDM $A(1, \dots, n)$ . While Shitov previously established $st_+(A(1, \dots, n)) \le 4 \log_2 n + 4$ , the authors prove:
$st_+(A(1, \dots, n)) \le 2 \lceil \log_2 n \rceil$
This result is extended to broader classes of EDMs, including those with integer parameters $A(x_1, \dots, x_n)$ and rational parameters, providing bounds dependent on the range of the parameters.
Submultiplicativity of SNT-rank:
The authors establish that the SNT-rank is submultiplicative with respect to the Kronecker product. For any $A_1 \in S^+_{n_1}$ and $A_2 \in S^+_{n_2}$ :
$rk(A_1)rk(A_2) \le st_+(A_1 \otimes A_2) \le st_+(A_1)st_+(A_2)$
Furthermore, they prove that if $n_1, n_2 \le 3$ , the equality $st_+(A_1 \otimes A_2) = st_+(A_1)st_+(A_2)$ holds. They also show that if at least one matrix has rank 1, the SNT-rank of the product equals the product of the SNT-ranks.
Multiplicativity of Nonnegative Rank under Structural Assumptions:
Addressing the conjecture that $rk_+(A_1 \otimes A_2) = rk_+(A_1)rk_+(A_2)$ (which was disproven in general by Vandaele et al.), the paper proves the identity holds under specific conditions:
- If at least one of the matrices $A_1$ or $A_2$ has rank 1.
- If for each matrix $A_i$ , either the number of rows or columns is at most 3 (i.e., $m_i \le 3$ or $n_i \le 3$ ). In this case, the result relies on the fact that for matrices of dimension $\le 3$ , the nonnegative rank coincides with the classical rank.
Lower Bounds and Structural Properties:
The paper provides a lower bound for the SNT-rank of symmetric nonnegative matrices with zero diagonals and positive off-diagonal entries, showing $st_+(A) \ge \min \{k : n \le \binom{k}{\lfloor k/2 \rfloor}\}$ . This implies that for $n=4$ , $st_+(A) = 4$ . Additionally, the authors demonstrate that for matrices of order $n \le 3$ , the SNT-rank coincides exactly with the classical rank, a property that fails for $n \ge 4$ .

Significance
The paper contributes to the theoretical understanding of low-rank structures in symmetric nonnegative matrices. By tightening the bounds for the SNT-rank of Euclidean distance matrices, it refines the computational complexity landscape for these specific structures. The establishment of submultiplicativity for the SNT-rank under the Kronecker product provides a new tool for constructing symmetric nonnegative matrices with controlled SNT-ranks (e.g., matrices with SNT-rank at most 9). Finally, by resolving the multiplicativity of the nonnegative rank under specific dimensional or rank constraints, the work clarifies the boundaries of the conjecture posed in the Dagstuhl Seminar Report, offering precise conditions under which the nonnegative rank behaves multiplicatively.

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