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Trace reconstruction of matrices and hypermatrices

Part of: Theory of computing Combinatorial probability Extremal combinatorics

Published online by Cambridge University Press:  10 March 2026

Wenjie Zhong  and
Xiande Zhang
Wenjie Zhong
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, China
Xiande Zhang*
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, China Hefei National Laboratory, University of Science and Technology of China, Hefei, China
*
Corresponding author: Xiande Zhang; Email: drzhangx@ustc.edu.cn
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Abstract

A trace of a sequence is generated by deleting each bit of the sequence independently with a fixed probability. The well-studied trace reconstruction problem asks how many traces are required to reconstruct an unknown binary sequence with high probability. In this paper, we study the multidimensional version of this problem for matrices and hypermatrices, where a trace is generated by deleting each row/column of the matrix or each slice of the hypermatrix independently with a constant probability. Previously, Krishnamurthy, Mazumdar, McGregor and Pal showed that $\exp (\widetilde {O}(n^{d/(d+2)}))$ traces suffice to reconstruct any unknown $n\times n$ matrix (for $d=2$) and any unknown $n^{\times d}$ hypermatrix. By developing a dimension reduction procedure and establishing a multivariate version of the Littlewood-type result that lower bounds sparse complex polynomials around $1$, we improve this upper bound by showing that $\exp (\widetilde {O}(n^{3/7}))$ traces suffice to reconstruct any unknown $n\times n$ matrix, and $\exp (\widetilde {O}(n^{3/5}))$ traces suffice to reconstruct any unknown $n^{\times d}$ hypermatrix. In contrast to the earlier bound, our new exponent is bounded away from $1$ even as $d$ becomes very large.

Keywords

Trace reconstructionmatriceshypermatricesLittlewood-type problems

MSC classification

Primary: 05D99: None of the above, but in this section
Secondary: 60C05: Combinatorial probability 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 35 , Issue 3 , May 2026 , pp. 356 - 374
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Copyright
© The Author(s), 2026. Published by Cambridge University Press

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