Coordination and Discoordination in Linear Algebra, Linear Information Theory, and Coded Caching

TL;DR

This paper introduces new linear algebra theorems related to information theory, defines a measure called discoordination for subspace collections, and applies these results to derive improved bounds and schemes in coded caching problems.

Contribution

It develops foundational theorems on discoordination of subspaces and applies them to derive new bounds and caching schemes in information theory.

Findings

New formulas involving three subspaces of a vector space.

A lower bound of 6M+5R≥11 for N=3, K=3, linear schemes.

A new caching scheme achieving (M,R) = (1/2,5/3).

Abstract

In the first part of this paper we develop some theorems in linear algebra applicable to information theory when all random variables involved are linear functions of the individual bits of a source of independent bits. We say that a collection of subspaces of a vector space are "coordinated" if the vector space has a basis such that each subspace is spanned by its intersection with the basis. We measure the failure of a collection of subspaces to be coordinated by an invariant that we call the "discoordination" of the family. We develop some foundational results regarding discoordination. In particular, these results give a number of new formulas involving three subspaces of a vector space. We then apply a number of our results, along with a method of Tian to obtain some new lower bounds in a special case of the basic coded caching problem. In terms of the usual notation for these problems, we show that for $N=3$ documents and $K=3$ caches, we have $6M+5R\ge 11$ for a scheme that achieves the memory-rate pair $(M,R)$, assuming the scheme is linear. We also give a new caching scheme for $N=K=3$ that achieves the pair $(M,R) = (1/2,5/3)$.