[2604.04087] ArrowFlow: Hierarchical Machine Learning in the Space of Permutations

Computer Science > Machine Learning arXiv:2604.04087 (cs) [Submitted on 5 Apr 2026] Title:ArrowFlow: Hierarchical Machine Learning in the Space of Permutations Authors:Ozgur Yilmaz View a PDF of the paper titled ArrowFlow: Hierarchical Machine Learning in the Space of Permutations, by Ozgur Yilmaz View PDF HTML (experimental) Abstract:We introduce ArrowFlow, a machine learning architecture that operates entirely in the space of permutations. Its computational units are ranking filters, learned orderings that compare inputs via Spearman's footrule distance and update through permutation-matrix accumulation, a non-gradient rule rooted in displacement evidence. Layers compose hierarchically: each layer's output ranking becomes the next layer's input, enabling deep ordinal representation learning without any floating-point parameters in the core computation. We connect the architecture to Arrow's impossibility theorem, showing that violations of social-choice fairness axioms (context dependence, specialization, symmetry breaking) serve as inductive biases for nonlinearity, sparsity, and stability. Experiments span UCI tabular benchmarks, MNIST, gene expression cancer classification (TCGA), and preference data, all against GridSearchCV-tuned baselines. ArrowFlow beats all baselines on Iris (2.7% vs. 3.3%) and is competitive on most UCI datasets. A single parameter, polynomial degree, acts as a master switch: degree 1 yields noise robustness (8-28% less degradation), privacy pre...