  Let $Q$ be a relatively compact subset in a Hilbert space $V$. For a given
$\e>0$ let $N(\e,Q)$ be the minimal number of linear measurements, sufficient
to reconstruct any $x \in Q$ with the accuracy $\e$. We call $N(\e,Q)$ a
sampling $\e$-entropy of $Q$. Using Dimensionality Reduction, as provided by
the Johnson-Lindenstrauss lemma, we show that, in an appropriate probabilistic
setting, $N(\e,Q)$ is bounded from above by the Kolmogorov's $\e$-entropy
$H(\e,Q)$, defined as $H(\e,Q)=\log M(\e,Q)$, with $M(\e,Q)$ being the minimal
number of $\e$-balls covering $Q$. As the main application, we show that
piecewise smooth (piecewise analytic) functions in one and several variables
can be sampled with essentially the same accuracy rate as their regular
counterparts. For univariate piecewise $C^k$-smooth functions this result,
which settles the so-called Eckhoff conjecture, was recently established in
\cite{Bat} via a deterministic "algebraic reconstruction" algorithm.
