Beyond Trace Reconstruction: Population Recovery from the Deletion Channel

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 deletion channel, in which each bit b is independently 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 trace reconstruction problem, which is a challenging problem that has received much recent attention. In this work we give algorithms and lower bounds for population recovery under the deletion channel when the population size is some value ℓ > 1. As our main sample complexity upper bound, we show that for any population size ℓ = o(log n / log log n), a population of ℓ strings from {o,1}^n can be learned under deletion channel noise using 2^n^1/2+o(1) samples. On the lower bound side, we show that at least n^Ω (ℓ) samples are required to perform population recovery under the deletion channel when the population size is ℓ, for all ℓ ≤ n^1/2-ε. Our upper bounds are obtained via a robust multivariate generalization of a polynomial-based analysis, due to Krasikov and Roddity [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 ℓ=1 case). Our lower bounds build on moment-matching results of Roos[Roos:00] and Daskalakis and Papadimitriou[DP15].