TY - JOUR
T1 - A Gradient-Enhanced $ℓ_1$ Approach for the Recovery of Sparse Trigonometric Polynomials
AU - Zhiqiang Xu & Tao Zhou
JO - Communications in Computational Physics
VL - 1
SP - 286
EP - 308
PY - 2018
DA - 2018/03
SN - 24
DO - http://doi.org/10.4208/cicp.OA-2018-0006
UR - https://global-sci.org/intro/article_detail/cicp/10938.html
KW - Gradient-enhanced $ℓ_1$ minimization, compressed sensing, sparse Fourier expansions, restricted isometry property, mutual incoherence.
AB - <p style="text-align: justify;">In this paper, we discuss a gradient-enhanced $ℓ_1$ approach for the recovery
of sparse Fourier expansions. By $gradient$-$enhanced$ approaches we mean that the
directional derivatives along given vectors are utilized to improve the sparse approximations.
We first consider the case where both the function values and the directional
derivatives at sampling points are known. We show that, under some mild conditions,
the inclusion of the derivatives information can indeed decrease the coherence
of measurement matrix, and thus leads to the improved the sparse recovery conditions
of the $ℓ_1$ minimization. We also consider the case where either the function values or
the directional derivatives are known at the sampling points, in which we present a
sufficient condition under which the measurement matrix satisfies RIP, provided that
the samples are distributed according to the uniform measure. This result shows that
the derivatives information plays a similar role as that of the function values. Several
numerical examples are presented to support the theoretical statements. Potential applications
to function (Hermite-type) interpolations and uncertainty quantification are
also discussed.</p>