\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} noise channel and for the erasure noise channel.
In this paper we initiate the study of population recovery under the \emph{deletion channel}, in which each bit $b$ 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 is of size 1 (corresponding to a trivial probability distribution supported on a single string) corresponds to the \emph{trace reconstruction} problem, which is a challenging problem that has received much recent attention (see e.g.~\cite{DOS17,NazarovPeres17,PeresZhai17,HPP18,HHP18}).
In this work we give algorithms and lower bounds for population recovery under the deletion channel when the population size is some value $\ell > 1$. As our main sample complexity upper bound, we show that for any population size $\ell = o(\log n / \log \log n)$, a population of $\ell$ strings from $\zo^n$ can be learned under deletion channel noise using $\smash{2^{n^{1/2 + o(1)}}}$ samples. On the lower bound side, we show that at least $n^{\Omega{(\ell)}}$ samples are required to perform population recovery under the deletion channel when the population size is $\ell$, 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{Roos:00} and Daskalakis and Papadimitriou~\cite{DP15}.