Sparse Recovery Using the Discrete Cosine Transform

Abstract

This note considers the problem of sparse recovery in ${\mathbb {R}}^{n}$ from linear measurements associated with a discrete cosine transform. The main theorem shows that an s-sparse vector in ${\mathbb {R}}^{n}$ can be recovered from the first 2s coefficients of its discrete cosine transform. This theorem is a real-valued analog of a result in Foucart and Rauhut (A mathematical introduction to compressive sensing, applied and numerical harmonic analysis, Birkhäuser/Springer, New York, 2013) concerned with sparse recovery in ${\mathbb {C}}^{n}$ based on linear measurements via the discrete Fourier transform.