Projection onto the Cosparse Set is NP-Hard

TL;DR

This paper proves that projecting onto the set of k-cosparse vectors is strongly NP-hard, highlighting a significant computational challenge in sparse optimization unlike the easy projection onto k-sparse vectors.

Contribution

It establishes the NP-hardness of the projection onto the k-cosparse set, even for matrices with simple ternary or bipolar coefficients, contrasting with the easy sparse projection.

Findings

Projection onto k-cosparse vectors is NP-hard.

NP-hardness holds even for simple matrix coefficients.

Projection onto k-sparse vectors is computationally trivial.

Abstract

The computational complexity of a problem arising in the context of sparse optimization is considered, namely, the projection onto the set of $k$-cosparse vectors w.r.t. some given matrix $\Omeg$. It is shown that this projection problem is (strongly) \NP-hard, even in the special cases in which the matrix $\Omeg$ contains only ternary or bipolar coefficients. Interestingly, this is in contrast to the projection onto the set of $k$-sparse vectors, which is trivially solved by keeping only the $k$ largest coefficients.