# Information Theoretic Limits for Linear Prediction with Graph-Structured Sparsity

## TL;DR

This paper establishes the fundamental limits on the number of samples needed for sparse vector recovery in noisy linear prediction problems with graph-structured sparsity, using information theory.

## Contribution

It proves that the sample complexity bounds for the weighted graph model are both necessary and sufficient, advancing understanding of sample limits in structured sparsity recovery.

## Key findings

- Sample complexity bounds are tight for the weighted graph model.
- Information-theoretic lower bounds match existing upper bounds.
- The analysis applies Fano's inequality to structured ensemble constructions.

## Abstract

We analyze the necessary number of samples for sparse vector recovery in a noisy linear prediction setup. This model includes problems such as linear regression and classification. We focus on structured graph models. In particular, we prove that sufficient number of samples for the weighted graph model proposed by Hegde and others is also necessary. We use the Fano's inequality on well constructed ensembles as our main tool in establishing information theoretic lower bounds.