  Given data drawn from an unknown distribution, $D$, to what extent is it
possible to ``amplify'' this dataset and output an even larger set of samples
that appear to have been drawn from $D$? We formalize this question as follows:
an $(n,m)$ $\text{amplification procedure}$ takes as input $n$ independent
draws from an unknown distribution $D$, and outputs a set of $m > n$
``samples''. An amplification procedure is valid if no algorithm can
distinguish the set of $m$ samples produced by the amplifier from a set of $m$
independent draws from $D$, with probability greater than $2/3$. Perhaps
surprisingly, in many settings, a valid amplification procedure exists, even
when the size of the input dataset, $n$, is significantly less than what would
be necessary to learn $D$ to non-trivial accuracy. Specifically we consider two
fundamental settings: the case where $D$ is an arbitrary discrete distribution
supported on $\le k$ elements, and the case where $D$ is a $d$-dimensional
Gaussian with unknown mean, and fixed covariance. In the first case, we show
that an $\left(n, n + \Theta(\frac{n}{\sqrt{k}})\right)$ amplifier exists. In
particular, given $n=O(\sqrt{k})$ samples from $D$, one can output a set of
$m=n+1$ datapoints, whose total variation distance from the distribution of $m$
i.i.d. draws from $D$ is a small constant, despite the fact that one would need
quadratically more data, $n=\Theta(k)$, to learn $D$ up to small constant total
variation distance. In the Gaussian case, we show that an
$\left(n,n+\Theta(\frac{n}{\sqrt{d}} )\right)$ amplifier exists, even though
learning the distribution to small constant total variation distance requires
$\Theta(d)$ samples. In both the discrete and Gaussian settings, we show that
these results are tight, to constant factors. Beyond these results, we
formalize a number of curious directions for future research along this vein.
