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Analysis of sparse recovery for Legendre expansions using envelope bound. (arXiv:1810.02926v1 [math.NA])
来源于:arXiv
We provide novel sufficient conditions for the uniform recovery of sparse
Legendre expansions using $\ell_1$ minimization, where the sampling points are
drawn according to orthogonalization (uniform) measure. So far, conditions of
the form $m \gtrsim \Theta^2 s \times \textit{log factors}$ have been relied on
to determine the minimum number of samples $m$ that guarantees successful
reconstruction of $s$-sparse vectors when the measurement matrix is associated
to an orthonormal system. However, in case of sparse Legendre expansions, the
uniform bound $\Theta$ of Legendre systems is so high that these conditions are
unable to provide meaningful guarantees. In this paper, we present an analysis
which employs the envelop bound of all Legendre polynomials instead, and prove
a new recovery guarantee for $s$-sparse Legendre expansions, $$ m \gtrsim {s^2}
\times \textit{log factors}, $$ which is independent of $\Theta$. Arguably,
this is the first recovery condition established for orthonormal 查看全文>>