Jean-Pierre Magnot

Jean-Pierre Magnot

We attack the problem of getting a strict ranking (i.e. a ranking without equally ranked items) of $n$ items from a pairwise comparisons matrix. Basic structures are described, a first heuristical approach based on a condition, the $\mathcal{R}-$condition, is proposed. Analyzing the limits of this ranking procedure, we finish with a minimization problem which can be applied to a wider class of pairwise comparisons matrices. If solved, it produces consistent pairwise comparisons that produce a strict ranking.

Jean-Pierre Magnot

Pairwise comparisons are widely used in decision analysis, preference modeling, and evaluation problems. In many practical situations, the observed comparison matrix is not reciprocal. This lack of reciprocity is often treated as a defect to be corrected immediately. In this article, we adopt a different point of view: part of the nonreciprocity may reflect a genuine variation in the evaluation scale, while another part is due to random perturbations. We introduce an additive model in which the unknown underlying comparison matrix is consistent but not necessarily reciprocal. The reciprocal component carries the global ranking information, whereas the symmetric component describes possible scale variation. Around this structured matrix, we add a random perturbation and show how to estimate the noise level, assess whether the scale variation remains moderate, and assign probabilities to admissible ranking regions in the sense of strict ranking by pairwise comparisons. We also compare this approach with the brutal projection onto reciprocal matrices, which suppresses all symmetric information at once. The Gaussian perturbation model is used here not because human decisions are exactly Gaussian, but because observed judgment errors often result from the accumulation of many small effects. In such a context, the central limit principle provides a natural heuristic justification for Gaussian noise. This makes it possible to derive explicit estimators and probability assessments while keeping the model interpretable for decision problems.

Jean-Pierre Magnot

We propose a geometric extension of restricted Boltzmann machines (RBMs) by allowing weights to take values in abstract groups such as \( \mathrm{GL}_n(\mathbb{R}) \), \( \mathrm{SU}(2) \), or even infinite-dimensional operator groups. This generalization enables the modeling of complex relational structures, including projective transformations, spinor dynamics, and functional symmetries, with direct applications to vision, language, and quantum learning. A central contribution of this work is the introduction of a \emph{contextuality index} based on group-valued holonomies computed along cycles in the RBM graph. This index quantifies the global inconsistency or "curvature" induced by local weights, generalizing classical notions of coherence, consistency, and geometric flatness. We establish links with sheaf-theoretic contextuality, gauge theory, and noncommutative geometry, and provide numerical and diagrammatic examples in both finite and infinite dimensions. This framework opens novel directions in AI, from curvature-aware learning architectures to topological regularization in uncertain or adversarial environments.