# Mutual information for low-rank even-order symmetric tensor estimation

## TL;DR

This paper derives a variational formula for the asymptotic mutual information in finite-rank symmetric tensor factorization of even order, extending adaptive interpolation methods to more complex tensor models.

## Contribution

It introduces a novel extension of the adaptive interpolation method for finite-rank, even-order symmetric tensors, advancing theoretical understanding of tensor estimation.

## Key findings

- Derived a single-letter variational expression for mutual information
- Extended adaptive interpolation to finite-rank, even-order tensors
- Identified limitations for odd-order tensor cases

## Abstract

We consider a statistical model for finite-rank symmetric tensor factorization and prove a single-letter variational expression for its asymptotic mutual information when the tensor is of even order. The proof applies the adaptive interpolation method originally invented for rank-one factorization. Here we show how to extend the adaptive interpolation to finite-rank and even-order tensors. This requires new nontrivial ideas with respect to the current analysis in the literature. We also underline where the proof falls short when dealing with odd-order tensors.