Trace reconstruction of matrices and hypermatrices

TL;DR

This paper extends trace reconstruction to matrices and hypermatrices, improving bounds on the number of traces needed for high-probability reconstruction through dimension reduction and new theoretical results.

Contribution

It introduces a dimension reduction technique and a multivariate Littlewood-type result to significantly improve trace reconstruction bounds for matrices and hypermatrices.

Findings

Reconstructed matrices with $ ilde{O}(n^{3/7})$ traces.

Reconstructed hypermatrices with $ ilde{O}(n^{3/5})$ traces.

Breaks the exponential growth trend with increasing dimension.

Abstract

A \emph{trace} of a sequence is generated by deleting each bit of the sequence independently with a fixed probability. The well-studied \emph{trace reconstruction} problem asks how many traces are required to reconstruct an unknown binary sequence with high probability. In this paper, we study the multivariate 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 et al. 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, 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. This breaks the tendency to trivial $\exp(O(n))$ as the dimension $d$ grows.