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Given data drawn from an unknown distribution, $D$, to what extent is itpossible to ``amplify'' this dataset and output an even larger set of samplesthat appear to have been drawn from $D$? We formalize this question as follows:an $(n,m)$ $\text{amplification procedure}$ takes as input $n$ independentdraws from an unknown distribution $D$, and outputs a set of $m > n$``samples''. An amplification procedure is valid if no algorithm candistinguish the set of $m$ samples produced by the amplifier from a set of $m$independent draws from $D$, with probability greater than $2/3$. Perhapssurprisingly, in many settings, a valid amplification procedure exists, evenwhen the size of the input dataset, $n$, is significantly less than what wouldbe necessary to learn $D$ to non-trivial accuracy. Specifically we consider twofundamental settings: the case where $D$ is an arbitrary discrete distributionsupported on $\le k$ elements, and the case where $D$ is a $d$-dimensionalGaussian with unknown mean, and fixed covariance. In the first case, we showthat an $\left(n, n + \Theta(\frac{n}{\sqrt{k}})\right)$ amplifier exists. Inparticular, 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 needquadratically more data, $n=\Theta(k)$, to learn $D$ up to small constant totalvariation distance. In the Gaussian case, we show that an$\left(n,n+\Theta(\frac{n}{\sqrt{d}} )\right)$ amplifier exists, even thoughlearning the distribution to small constant total variation distance requires$\Theta(d)$ samples. In both the discrete and Gaussian settings, we show thatthese results are tight, to constant factors. Beyond these results, weformalize a number of curious directions for future research along this vein.

Sample Amplification: Increasing Dataset Size even when Learning is Impossible