Sample-Measurement Tradeoff in Support Recovery Under a Subgaussian Prior

Data samples from
${\mathbb{R}}^ {d}$
with a common support of size
$k$
are accessed through
$m$
random linear projections (measurements) per sample. It is well-known that roughly
$k$
measurements from a single sample are sufficient to recover the support. In the multiple sample setting, do
$k$

overall
measurements still suffice when only
$m$
measurements
per sample
are allowed, with
$m < k$
? We answer this question in the negative by considering a generative model setting with independent samples drawn from a subgaussian prior. We show that
$n=\Theta ((k^{2}/ m^{2})\cdot \log k(d- k))$
samples are necessary and sufficient to recover the support exactly. In turn, this shows that
when

$m < k$
,

$k$

overall measurements are insufficient for support recovery
; instead we need about
$m$
measurements each from
$k^{2}/ m^{2}$
samples, and therefore
$k^{2}/ m$
overall measurements are necessary.