Trace reconstruction from local statistical queries
Xi Chen, Anindya De, Chin Ho Lee, Rocco A. Servedio
TL;DR
This work analyzes trace reconstruction under a deletion channel through the lens of local Statistical Query (SQ) algorithms, introducing the notion of ℓ-local SQ and subword queries. It proves near-tight bounds: for worst-case reconstruction, an $ ilde{O}(n^{1/5})$-local SQ algorithm with subexponentially small tolerance exists, and any such algorithm must confront exponentially small tolerances at that locality; for average-case reconstruction, an $O( ext{log }n)$-local SQ algorithm achieves inverse-polynomial tolerance, while any $O( ext{log }n)$-local SQ method encounters similar lower bounds. The approach blends complex-polynomial techniques (via deletion-channel polynomials and BEK-type lemmas) with reductions to subword statistics, and extends smoothed-analysis ideas to the SQ model to handle average-case behavior. These results place strong, near-optimal restrictions on what local SQ methods can achieve for trace reconstruction and illuminate the interplay between locality, tolerance, and information content in deletion channels, with potential implications for restricted-query models in related reconstruction tasks.
Abstract
The goal of trace reconstruction is to reconstruct an unknown $n$-bit string $x$ given only independent random traces of $x$, where a random trace of $x$ is obtained by passing $x$ through a deletion channel. A Statistical Query (SQ) algorithm for trace reconstruction is an algorithm which can only access statistical information about the distribution of random traces of $x$ rather than individual traces themselves. Such an algorithm is said to be $\ell$-local if each of its statistical queries corresponds to an $\ell$-junta function over some block of $\ell$ consecutive bits in the trace. Since several -- but not all -- known algorithms for trace reconstruction fall under the local statistical query paradigm, it is interesting to understand the abilities and limitations of local SQ algorithms for trace reconstruction. In this paper we establish nearly-matching upper and lower bounds on local Statistical Query algorithms for both worst-case and average-case trace reconstruction. For the worst-case problem, we show that there is an $\tilde{O}(n^{1/5})$-local SQ algorithm that makes all its queries with tolerance $τ\geq 2^{-\tilde{O}(n^{1/5})}$, and also that any $\tilde{O}(n^{1/5})$-local SQ algorithm must make some query with tolerance $τ\leq 2^{-\tildeΩ(n^{1/5})}$. For the average-case problem, we show that there is an $O(\log n)$-local SQ algorithm that makes all its queries with tolerance $τ\geq 1/\mathrm{poly}(n)$, and also that any $O(\log n)$-local SQ algorithm must make some query with tolerance $τ\leq 1/\mathrm{poly}(n).$