A Unique Normal Form for Tensor Trains over Arbitrary Fields

2026-07-07

AI Summary

This work addresses the long-standing challenge that tensor trains lack a unique canonical form over arbitrary fields—including finite fields—hindering irreducibility testing and standardization. The paper presents the first construction of a unique canonical form for tensor trains valid over any field, combining algebraic structural analysis with a polynomial-time reduction algorithm. This framework enables direct extraction of the canonical representation, principal indices, and corresponding values from a full tensor. Beyond establishing a theoretical upper bound on compression size, the approach substantially extends the theoretical foundations and applicability of tensor networks beyond real or complex fields, demonstrating their efficacy as a universal formalism across diverse algebraic settings.

Abstract

Tensor trains (or Matrix-Product States) are a data structure used in many fields of computer science and physics. They were recently shown to generalise binary decision diagrams when used over the 2-element Galois field, prompting the question of their reducibility in such a context, when the standard approach, over real or complex number, is not amenable to finite fields. We provide here a unique normal form and associated polynomial-time reduction strategy for tensor trains over arbitrary fields. We also show how to directly extract a normal form out of a full tensor, how to get the leading index and value of a normal form, and an upper bound on the size of a fully-reduced tensor train relative to a naive storage of the full tensor. On the one hand, this work strengthens the use of tensor trains as a relevant formal tool. On the other hand, from the perspective of tensor networks, it extends the formalism to more general settings than the well-studied real and complex fields, and crucially provides the first tensor train form with the uniqueness property.

Problem

• Tensor Trains
• Normal Form
• Arbitrary Fields
• Uniqueness
• Finite Fields

Innovation

• tensor trains
• unique normal form
• arbitrary fields
• polynomial-time reduction
• uniqueness property