Title: ArrowFlow: Implementing Hierarchical Machine Learning Within Permutation Spaces

Abstract:

We present ArrowFlow, a novel machine learning framework that functions exclusively within the domain of permutations. The architecture’s fundamental components are ranking filters, which employ learned orderings to evaluate inputs based on Spearman’s footrule distance. These filters update via permutation-matrix accumulation, a mechanism grounded in displacement evidence that operates without gradients. By composing these units hierarchically—where the output ranking of one layer serves as the input for the subsequent layer—the system facilitates deep ordinal representation learning entirely free of floating-point parameters in its core operations.

The design is theoretically linked to Arrow’s impossibility theorem. We demonstrate that deliberate violations of social-choice fairness axioms, specifically context dependence, specialization, and symmetry breaking, function as effective inductive biases. These violations promote nonlinearity, sparsity, and stability within the model. Empirical evaluations cover UCI tabular benchmarks, MNIST, gene expression cancer classification using TCGA data, and preference datasets, with performance measured against baselines tuned via GridSearchCV. ArrowFlow outperforms all competing methods on the Iris dataset, achieving an error rate of 2.7% compared to the baseline’s 3.3%, and remains competitive across most other UCI datasets.

A single hyperparameter, the polynomial degree, governs the model’s behavior. A degree of 1 prioritizes robustness, resulting in 8–28% less performance degradation under noise, enhanced privacy preservation with a minimal 0.5 percentage point cost, and resilience to missing features. Increasing the degree shifts the trade-off toward higher accuracy on clean data. ArrowFlow is not intended to replace gradient-based approaches; rather, it serves as an existence proof that competitive classification can be achieved through a fundamentally distinct computational paradigm. This approach elevates ordinal structure to a primary role, offering inherent compatibility with integer-only and neuromorphic hardware.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC