NP-hardness of p-adic linear regression

TL;DR

Problem: determine the computational complexity of finding β that minimises the p-adic regression loss L(β). Approach: provide a polynomial-time reduction from Max Cut to 2-adic regression using a regularisation gadget, proving NP-hardness for p=2 when n is unrestricted. Contributions: hardness result complements existing fixed-dimension polynomial-time methods and clarifies the impact of the non-Archimedean ultrametric on the loss landscape. Significance: reveals a sharp contrast with Euclidean least-squares regression and guides future work on approximations and tractable regimes in p-adic learning.

Abstract

$p$-adic linear regression is the problem of finding coefficients $β$ that minimise $\sum_i |y_i - x_i^\topβ|_p$. We prove that computing an optimal solution is NP-hard via a polynomial-time reduction from Max Cut using a regularisation gadget.

Theorems & Definitions (12)

• Definition 1: $p$-adic linear regression
• Definition 2: Max Cut
• Theorem 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof