Linear Regression in $$p$$-Adic Metric Spaces

Many real-world machine learning problems involve inherently hierarchical data, yet traditional approaches rely on Euclidean metrics that fail to capture the discrete, branching nature of hierarchical relationships. We present a theoretical foundation for machine learning in $$p$$ -adic metric spaces, which naturally respect hierarchical structure. Our main result proves that an $$n$$ -dimensional plane minimizing the $$p$$ -adic sum of distances to points in a dataset must pass through at least $$n+1$$ of those points – a striking contrast to Euclidean regression that highlights how $$p$$ -adic metrics better align with the discrete nature of hierarchical data. As a corollary, a polynomial of degree $$n$$ constructed to minimise the $$p$$ -adic sum of residuals will pass through at least $$n+1$$ points. As a further corollary, a polynomial of degree $$n$$ approximating a higher degree polynomial at a finite number of points will yield a difference polynomial that has distinct rational roots. We demonstrate the practical significance of this result through two applications in natural language processing: analyzing hierarchical taxonomies and modeling grammatical morphology. These results suggest that $$p$$ -adic metrics may be fundamental to properly handling hierarchical data structures in machine learning. In hierarchical data, interpolation between points often makes less sense than selecting actual observed points as representatives.