# Sparse Approximation is Provably Hard under Coherent Dictionaries

## TL;DR

This paper proves that sparse approximation remains computationally hard even under dictionaries with low coherence, bridging the gap between known NP-hardness results and positive algorithmic results under restrictive assumptions.

## Contribution

It demonstrates that the assumption of low coherence is essentially necessary for efficient sparse approximation algorithms, using a new multilayered PCP reduction.

## Key findings

- Sparse approximation is NP-hard under low coherence dictionaries.
- Low coherence assumption is asymptotically optimal for efficient algorithms.
- Introduces a new multilayered PCP tailored for coherence-based reductions.

## Abstract

It is well known that sparse approximation problem is \textsf{NP}-hard under general dictionaries. Several algorithms have been devised and analyzed in the past decade under various assumptions on the \emph{coherence} $\mu$ of the dictionary represented by an $M \times N$ matrix from which a subset of $k$ column vectors is selected. All these results assume $\mu=O(k^{-1})$. This article is an attempt to bridge the big gap between the negative result of \textsf{NP}-hardness under general dictionaries and the positive results under this restrictive assumption. In particular, it suggests that the aforementioned assumption might be asymptotically the best one can make to arrive at any efficient algorithmic result under well-known conjectures of complexity theory. In establishing the results, we make use of a new simple multilayered PCP which is tailored to give a matrix with small coherence combined with our reduction.