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Data samples from $\mathbb{R}^{d}$ with a common support of size $k$ areaccessed through $m$ random linear projections (measurements) per sample. It iswell-known that roughly $k$ measurements from a single sample are sufficient torecover the support. In the multiple sample setting, do $k$ overallmeasurements still suffice when only $m$ measurements per sample are allowed,with $m<k$? We answer this question in the negative by considering a generativemodel setting with independent samples drawn from a subgaussian prior. We showthat $n=\Theta((k^2/m^2)\cdot\log k(d-k))$ samples are necessary and sufficientto recover the support exactly. In turn, this shows that when $m<k$, $k$overall measurements are insufficient for support recovery; instead we needabout $m$ measurements each from $k^{2}/m^2$ samples, i.e., $k^{2}/m$ overallmeasurements are necessary.

Sample-Measurement Tradeoff in Support Recovery under a Subgaussian Prior