Scholarlink

<p>We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of giving a description of the correctable erasure patterns in a maximally recoverable tensor code with the problem of describing bipartite rigid graphs or low-rank completable matrix patterns. Additionally, we relate dependencies among symmetric products of generic vectors to graph rigidity and symmetric matrix completion. With an eye toward applications to computer science, we study the dependency of these matroids on the characteristic by giving new combinatorial descriptions in several cases, including the first description of the correctable patterns in an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((m, n, a=2, b=2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> maximally recoverable tensor code.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Rigidity Matroids and Linear Algebraic Matroids with Applications to Matrix Completion and Tensor Codes

  • 期刊论文

  • 发表日期：

    2026年3月20日

查看全文
  • Joshua Brakensiek,
  • Manik Dhar,
  • Jiyang Gao,
  • Sivakanth Gopi,
  • Matt Larson

摘要

We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of giving a description of the correctable erasure patterns in a maximally recoverable tensor code with the problem of describing bipartite rigid graphs or low-rank completable matrix patterns. Additionally, we relate dependencies among symmetric products of generic vectors to graph rigidity and symmetric matrix completion. With an eye toward applications to computer science, we study the dependency of these matroids on the characteristic by giving new combinatorial descriptions in several cases, including the first description of the correctable patterns in an \((m, n, a=2, b=2)\) ( m , n , a = 2 , b = 2 ) maximally recoverable tensor code.