$p$-adic Principal Component Analysis

Tomoki Mihara

Abstract

We formulate a $p$-adic optimisation problem on matrix factorisation, and investigate a heuristic method for it analogous to PCA.

We formulate a

-adic optimisation problem on matrix factorisation, and investigate a heuristic method for it analogous to PCA.

Paper Structure (12 sections, 4 theorems, 14 equations, 9 figures, 18 tables, 9 algorithms)

This paper contains 12 sections, 4 theorems, 14 equations, 9 figures, 18 tables, 9 algorithms.

Proposition 1.1

Let $V$ be a normed $k$-vector space of dimension $\leq 1$. Then $V$ satisfies the condition (1)'.

Figure : Pre-computation of the ratio data $\vec{\rho}$ and the valuation data $\vec{\nu}$ for $\vec{v}_0 = (v_{0,d})_{d=0}^{D-1} \in \mathbb{N}_{p^E}^D$ and $\vec{v}_1 = (v_{1,d})_{d=0}^{D-1} \in \mathbb{N}_{p^E}^D$
Figure : Construction of a trie tree for ratio data $\vec{\rho} = (\rho_d)_{d=0}^{D-1} \in \mathbb{N}_{p^E}^D$ and valuation data $\vec{\nu} = (\nu_d)_{d=0}^{D-1} \in \mathbb{N}_{\leq E}^D$
Figure : Solving the optimisation problem for $\vec{v}_0 = (v_{0,d})_{d=0}^{D-1} \in \mathbb{N}_{p^E}^D$ and $\vec{v}_1 = (v_{1,d})_{d=0}^{D-1} \in \mathbb{N}_{p^E}^D$
Figure : Orthogonalisation of $\vec{v} \in \mathbb{N}_{< p^E}^D$ by $X = (\vec{x}_j)_{j \in J} \in (\mathbb{N}_{p^E}^D)^J$
Figure : Iterated orthogonalisation of $X = (\vec{x}_j)_{j \in J} \in (\mathbb{N}_{p^E}^D)^J$
...and 4 more figures