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Reconstruction of hypermatrices from subhypermatrices

Xiande Zhang, Wenjie Zhong

Abstract

For a given $n$, what is the smallest number $k$ such that every sequence of length $n$ is determined by the multiset of all its $k$-subsequences? This is called the $k$-deck problem for sequence reconstruction, and has been generalized to the two-dimensional case -- reconstruction of $n\times n$-matrices from submatrices. Previous works show that the smallest $k$ is at most $O(n^\frac{1}{2})$ for sequences and at most $O(n^\frac{2}{3})$ for matrices. We study this $k$-deck problem for general dimension $d$ and prove that, the smallest $k$ is at most $O(n^\frac{d}{d+1})$ for reconstructing a $d$ dimensional hypermatrix of order $n$ from the multiset of all its subhypermatrices of order $k$.

Reconstruction of hypermatrices from subhypermatrices

Abstract

For a given , what is the smallest number such that every sequence of length is determined by the multiset of all its -subsequences? This is called the -deck problem for sequence reconstruction, and has been generalized to the two-dimensional case -- reconstruction of -matrices from submatrices. Previous works show that the smallest is at most for sequences and at most for matrices. We study this -deck problem for general dimension and prove that, the smallest is at most for reconstructing a dimensional hypermatrix of order from the multiset of all its subhypermatrices of order .
Paper Structure (18 sections, 28 theorems, 83 equations, 4 figures)

This paper contains 18 sections, 28 theorems, 83 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The polynomial method
  3. The nonsymmetric case
  4. The symmetric case
  5. Proof of \ref{['S_k reconstruction']}
  6. Construction sketch of our polynomial
  7. Notations
  8. Idea of the polynomial construction
  9. Construction of the polynomial
  10. Construction of $p_0=f_0\circ g_0$
  11. Lattice and Mahler basis
  12. Construction of $p_i=f_i\circ g_i,1\leq i\leq d-1$
  13. Proof of \ref{['d_polynomial']}
  14. Peak value
  15. Low degree
  16. ...and 3 more sections

Key Result

Theorem 1.1

For any fixed dimension $d\geq3$, when $n$ is sufficiently large, we have $\kappa_d(n)\leq d^{\frac{3}{2}d}n^\frac{d}{d+1}$ and $\kappa_d^\text{p}(n)\leq d^{\frac{3}{2}d}n^\frac{d}{d+1}$. That is, all $n^{\times d}$-hypermatrices can be reconstructed by their (principal) $k$-decks when $k=\Omega(n^\

Figures (4)

  • Figure 1: $P$ is tangent to $H$
  • Figure 2: Construct ${\mathbf{a}}_i$ in $\text{Span}({\mathbf{h}}_i,{\mathbf{a}}_0)$
  • Figure 3: For \ref{['DPS']}. Here $\partial B({\mathbf{o}},R)$ denotes the $d$-sphere of radius $R$.
  • Figure 4: \ref{['g_i(H)']}

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Lemma 2.1
  • Proof
  • Lemma 2.2
  • Proof
  • Lemma 2.3
  • Corollary 2.1
  • ...and 44 more