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Trace reconstruction considers the task of recovering an unknown string $x in {0,1}^n$ given a number of independent traces, i.e., subsequences of $x$ obtained by randomly and independently deleting every symbol of $x$ with some probability $p$. The information-theoretic limit of the number of traces needed to recover a string of length $n$ are still unknown. This limit is essentially the same as the number of traces needed to determine, given strings $x$ and $y$ and traces of one of them, which string is the source. The most studied class of algorithms for the worst-case version of the problem are mean-based algorithms. These are a restricted class of distinguishers that only use the mean value of each coordinate on the given samples. In this work we study limitations of mean-based algorithms on strings at small Hamming or edit distance. We show on the one hand that distinguishing strings that are nearby in Hamming distance is easy for such distinguishers. On the other hand, we show that distinguishing strings that are nearby in edit distance is hard for mean-based algorithms. Along the way we also describe a connection to the famous Prouhet-Tarry-Escott (PTE) problem, which shows a barrier to finding explicit hard-to-distinguish strings: namely such strings would imply explicit short solutions to the PTE problem, a well-known difficult problem in number theory. Our techniques rely on complex analysis arguments that involve careful trigonometric estimates, and algebraic techniques that include applications of Descartes rule of signs for polynomials over the reals.
Mean-based reconstruction is a fundamental, natural approach to worst-case trace reconstruction over channels with synchronization errors. It is known that $exp(O(n^{1/3}))$ traces are necessary and sufficient for mean-based worst-case trace reconstr
Bailey showed that the general pointwise forecasting for stationary and ergodic time series has a negative solution. However, it is known that for Markov chains the problem can be solved. Morvai showed that there is a stopping time sequence ${lambda_
Consider a collection of random variables attached to the vertices of a graph. The reconstruction problem requires to estimate one of them given `far away observations. Several theoretical results (and simple algorithms) are available when their join
Bures distance holds a special place among various distance measures due to its several distinguished features and finds applications in diverse problems in quantum information theory. It is related to fidelity and, among other things, it serves as a
We consider a broad class of Approximate Message Passing (AMP) algorithms defined as a Lipschitzian functional iteration in terms of an $ntimes n$ random symmetric matrix $A$. We establish universality in noise for this AMP in the $n$-limit and valid