  In compressed sensing, one wishes to acquire an approximately sparse
high-dimensional signal $x\in\mathbb{R}^n$ via $m\ll n$ noisy linear
measurements, then later approximately recover $x$ given only those measurement
outcomes. Various guarantees have been studied in terms of the notion of
approximation in recovery, and some isolated folklore results are known stating
that some forms of recovery are stronger than others, via black-box reductions.
In this note we provide a general theorem concerning the hierarchy of strengths
of various recovery guarantees. As a corollary of this theorem, by reducing
from well-known results in the compressed sensing literature, we obtain an
efficient $\ell_p/\ell_p$ scheme for any $0<p<1$ with the fewest number of
measurements currently known amongst efficient schemes, improving recent bounds
of [SomaY16].
