The Layered Structure of Tensor Estimation and its Mutual Information

TL;DR

This paper introduces a recursive interpolation method to derive mutual information formulas for rank-one tensor estimation of any order, simplifying previous approaches and revealing a layered structure.

Contribution

It develops a new interpolation technique that simplifies the derivation of mutual information formulas for tensor estimation and generalizes to any tensor order.

Findings

Derived mutual information formulas for order 2 and 3 tensor problems.

Established a recursive method to obtain mutual information for higher-order tensors.

Simplified previous proofs using a new interpolation approach.

Abstract

We consider rank-one non-symmetric tensor estimation and derive simple formulas for the mutual information. We start by the order 2 problem, namely matrix factorization. We treat it completely in a simpler fashion than previous proofs using a new type of interpolation method developed in [1]. We then show how to harness the structure in "layers" of tensor estimation in order to obtain a formula for the mutual information for the order 3 problem from the knowledge of the formula for the order 2 problem, still using the same kind of interpolation. Our proof technique straightforwardly generalizes and allows to rigorously obtain the mutual information at any order in a recursive way.