Related papers: Learning with the $p$-adics

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…

In this paper, we offer a brief introduction to the $p$-adic numbers and operations in the metric space defined under the $p$-adic norm. Specifically, we provide a clear description of the derivation of the $p$-adic number via the…

In this paper we study certain category of smooth modules for reductive $p$--adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a $\mathbb Q$--algebra. We prove some…

We present a new p-adic version of the Jackiw-Rebbi model. In the new model, the real numeric line is replaced by a p-adic line (the field of p-adic numbers Q_{p}), and the Dirac Hamiltonian is replaced by a non-local operator acting on…

Spatial networks are networks whose graph topology is constrained by their embedded spatial space. Understanding the coupled spatial-graph properties is crucial for extracting powerful representations from spatial networks. Therefore,…

Nondegenerate quadratic forms over $p$-adic fields are classified by their dimension, discriminant, and Hasse invariant. This paper uses these three invariants, elementary facts about $p$-adic fields and the theory of quadratic forms to…

In machine learning, data is usually represented in a (flat) Euclidean space where distances between points are along straight lines. Researchers have recently considered more exotic (non-Euclidean) Riemannian manifolds such as hyperbolic…

The mathematical basis of p-adic Higgs mechanism discussed in papers [email protected] 9410058-62 is considered in this paper. The basic properties of p-adic numbers, of their algebraic extensions and the so called canonical…

Deep learning methods capable of handling relational data have proliferated over the last years. In contrast to traditional relational learning methods that leverage first-order logic for representing such data, these deep learning methods…

The p-adic numbers have found applications in a wide range of diverse fields of research. In some applications the algebraic properties of p-adics enter as an indispensable ingredient of the theory. Another class of applications has to do…

The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data…

We introduce the notion of $p$-adic quantum bit ($p$-qubit) in the context of the $p$-adic quantum mechanics initiated and developed by Volovich and his followers. In this approach, physics takes place in three-dimensional $p$-adic space…

Learning useful representations is a key ingredient to the success of modern machine learning. Currently, representation learning mostly relies on embedding data into Euclidean space. However, recent work has shown that data in some domains…

In this article we introduce theory and algorithms for learning discrete representations that take on a lattice that is embedded in an Euclidean space. Lattice representations possess an interesting combination of properties: a) they can be…

Graph-structured data are widespread in real-world applications, such as social networks, recommender systems, knowledge graphs, chemical molecules etc. Despite the success of Euclidean space for graph-related learning tasks, its ability to…

We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear $p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric…

A method for approximating continuous functions $\mathbb{Z}_{p}^{n}\rightarrow\mathbb{Z}_{p}$ by a linear superposition of continuous functions $\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ is presented and a polynomial regression model is…

Graph is a universe data structure that is widely used to organize data in real-world. Various real-word networks like the transportation network, social and academic network can be represented by graphs. Recent years have witnessed the…

Motivated by the Langlands program in representation theory, number theory and geometry, the theory of representations of a reductive $p$-adic group over a coefficient ring different from the field of complex numbers has been widely…

Deep neural networks can empirically perform efficient hierarchical learning, in which the layers learn useful representations of the data. However, how they make use of the intermediate representations are not explained by recent theories…