Researchers have derived an explicit orthogonal basis - a set of matrix building blocks arranged so their pairwise overlap is zero - for the part of pairwise-comparison matrices that satisfies additive consistency. They then use it to write a closed-form projection, a finite formula for projecting an input matrix onto that subspace. The result gives a direct mathematical route from a matrix of comparisons to a consistent representation.
The work is a constructive methods study. It asks whether an explicit orthogonal tensor basis can be built for the additively consistent subspace and used for closed-form projections, including comparisons with Saaty and singular-value-decomposition, or SVD, projections. The analysis is symbolic and illustrated with mathematical examples; it reports no participant, animal or observational sample.
From a consistent basis to an orthogonal one
The construction begins with a consistent tensor basis, a structured collection of matrix pieces, with minimal support. The paper establishes that these minimal-support matrices form a basis of the additively consistent subspace. They provide the starting basis for the paper's later orthogonalization.
To make the basis orthogonal, the authors use the standard Frobenius inner product - a way to compare matrices by combining corresponding entries - and a Gram-Schmidt recurrence. The resulting construction is a closed-form orthogonal basis expressed through tensor-difference matrices.
That basis leads directly to an orthogonal projection: a finite sum of scalar products between the input matrix and the constructed basis elements. The authors describe this calculation as having polynomial computational complexity. But the paper does not provide runtime or numerical-error benchmarks, so its practical speed and numerical stability remain unmeasured.
Three ways to reach consistency
Alongside the orthogonal construction, the paper treats the multiplicative model for pairwise-comparison matrices. Its logarithmic consistent projection is represented as a product of factors depending on the input matrix, and the authors describe that representation as not requiring iterative methods. This gives another closed-form route to a consistent matrix, but the formula itself is not a performance ranking.
For readers comparing the approaches, the Saaty projection forms a consistent ratio matrix from the positive Perron eigenvector of the input matrix. The SVD projection forms a consistent ratio matrix from the leading left singular vector. These definitions set up the paper's algebraic comparison of the three projection routes.
The most striking result is an identity within the paper's stated domain. For every pairwise-comparison matrix in that domain, the SVD projection equals the logarithmic projection applied after the tensor projection. In other words, the paper links the SVD result to a sequence of two operations. It does not, however, use that identity to claim better decision results.
A second identity links SVD to the Saaty method. For every matrix in the same stated domain, the SVD projection also equals the Saaty eigenvector projection applied after the tensor projection. Taken together, the two statements describe SVD as a composite tensor-based construction with either a logarithmic or an eigenvector-based second step.
The paper stops short of declaring a winner. The authors state that no windowing method is superior to the others in all cases. That caution matters because the work is theoretical and illustrated with examples, with no new systematic benchmark used to rank the methods. The identities show how the constructions are related; they do not establish which one should be preferred in every application.
What the formulas leave open
The formula also has a defined mathematical scope. The explicit orthogonalization is formulated with the standard Frobenius inner product. The paper leaves behavior under other norms and general weighted inner products for further work, so the construction should not be read as a result for every possible way of measuring matrix distance.
Practical validation is similarly absent. The reported work includes no benchmark timings, numerical-error study or real-world application evaluation. It therefore gives no empirical estimate of decision quality, generalization or human outcomes. Its evidence is the algebraic derivation and illustrative examples.
The paper's contribution is therefore a set of explicit formulas and algebraic relationships for the matrix classes and inner products it studies. It does not show that any projection improves human decisions or real-world ranking accuracy, and it does not establish a causal effect because no intervention or empirical comparison was conducted.
The document is labeled arXiv:2608.25923v1 and dated 26 August 2026, marking it as an arXiv preprint. The authors state that data sharing is not applicable because no datasets were generated or analyzed. Its acknowledgment reports National Science Centre, Poland support for Konrad Kułakowski under grant VIRGO 2024/55/B/HS4/00860.
Paper data and sources
Original title: Efficient tensor bases for pairwise comparisons
Authors: Konrad Kułakowski, Ryszard Smarzewski
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text