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Average-Case to (Shifted) Worst-Case Reduction for the Trace Reconstruction Problem

Author Ittai Rubinstein

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Ittai Rubinstein
  • Qedma Quantum Computing, Tel Aviv, Israel

Acknowledgements

We would like to thank Zachary Chase, Roni Con and Aviad Rubinstein for their helpful comments on previous versions of this paper. We would also like to thank Nina Holden, Robin Pemantle, Yuval Peres and Alex Zhai for their help in understanding their paper. {}

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Ittai Rubinstein. Average-Case to (Shifted) Worst-Case Reduction for the Trace Reconstruction Problem. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 102:1-102:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.102

Abstract

In the trace reconstruction problem, one is given many outputs (called traces) of a noise channel applied to the same input message x, and is asked to recover the input message. Common noise channels studied in the context of trace reconstruction include the deletion channel which deletes each bit w.p. δ, the insertion channel which inserts a G_j i.i.d. uniformly distributed bits before each bit of the input message (where G_j is i.i.d. geometrically distributed with parameter σ) and the symmetry channel which flips each bit of the input message i.i.d. w.p. γ. De et al. and Nazarov and Peres [De et al., 2017; Nazarov and Peres, 2017] showed that any string x can be reconstructed from exp(O(n^{1/3})) traces. Holden et al. [Holden et al., 2018] adapted the techniques used to prove this upper bound, to construct an algorithm for average-case trace reconstruction from the insertion-deletion channel with a sample complexity of exp(O(log^{1/3} n)). However, it is not clear how to apply their techniques more generally and in particular for the recent worst-case upper bound of exp(Õ(n^{1/5})) shown by Chase [Chase, 2021] for the deletion channel. We prove a general reduction from the average-case to smaller instances of a problem similar to worst-case and extend Chase’s upper-bound to this problem and to symmetry and insertion channels as well. Using this reduction and generalization of Chase’s bound, we introduce an algorithm for the average-case trace reconstruction from the symmetry-insertion-deletion channel with a sample complexity of exp(Õ(log^{1/5} n)).

Subject Classification

ACM Subject Classification
  • Theory of computation → Sample complexity and generalization bounds
Keywords
  • Trace Reconstruction
  • Synchronization Channels
  • Computational Learning Theory
  • Computational Biology

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