  \emph{Population recovery} is the problem of learning an unknown distribution
over an unknown set of $n$-bit strings, given access to independent draws from
the distribution that have been independently corrupted according to some noise
channel. Recent work has intensively studied such problems both for the
bit-flip and erasure noise channels.
  We initiate the study of population recovery under the \emph{deletion
channel}, in which each bit is independently \emph{deleted} with some fixed
probability and the surviving bits are concatenated and transmitted. This is a
far more challenging noise model than bit-flip~noise or erasure noise; indeed,
even the simplest case in which the population size is 1 (corresponding to a
trivial distribution supported on a single string) corresponds to the
\emph{trace reconstruction} problem, a challenging problem that has received
much recent attention (see e.g.~\cite{DOS17,NP17,PZ17,HPP18,HHP18}).
  We give algorithms and lower bounds for population recovery under the
deletion channel when the population size is some $\ell>1$. As our main sample
complexity upper bound, we show that for any $\ell=o(\log n/\log \log n)$, a
population of $\ell$ strings from $\{0,1\}^n$ can be learned under deletion
channel noise using $\smash{2^{n^{1/2+o(1)}}}$ samples. On the lower bound
side, we show that $n^{\Omega(\ell)}$ samples are required to perform
population recovery under the deletion channel, for all $\ell \leq
n^{1/2-\epsilon}$.
  Our upper bounds are obtained via a robust multivariate generalization of a
polynomial-based analysis, due to Krasikov and Roddity \cite{KR97}, of how the
$k$-deck of a bit-string uniquely identifies the string; this is a very
different approach from recent algorithms for trace reconstruction (the
$\ell=1$ case). Our lower bounds build on moment-matching results of
Roos~\cite{Roo00} and Daskalakis and Papadimitriou~\cite{DP15}.
