## Abstract

Random subspaces X of ℝ^{n} of dimension proportional to n are, with high probability, well-spread with respect to the ℓ_{2}-norm. Namely, every nonzero x ∈ X is “robustly non-sparse” in the following sense: x is ε ∥x∥_{2}-far in ℓ_{2}-distance from all δn-sparse vectors, for positive constants ε, δ bounded away from 0. This “ℓ_{2}-spread” property is the natural counterpart, for subspaces over the reals, of the minimum distance of linear codes over finite fields, and corresponds to X being a Euclidean section of the ℓ_{1} unit ball. Explicit ℓ_{2}-spread subspaces of dimension Ω(n), however, are unknown, and the best known explicit constructions (which achieve weaker spread properties), are analogs of low density parity check (LDPC) codes over the reals, i.e., they are kernels of certain sparse matrices. Motivated by this, we study the spread properties of the kernels of sparse random matrices. We prove that with high probability such subspaces contain vectors x that are o(1) · ∥x∥_{2}-close to o(n)-sparse with respect to the ℓ_{2}-norm, and in particular are not ℓ_{2}-spread. This is strikingly different from the case of random LDPC codes, whose distance is asymptotically almost as good as that of (dense) random linear codes. On the other hand, for p < 2 we prove that such subspaces are ℓ_{p}-spread with high probability. The spread property of sparse random matrices thus exhibits a threshold behavior at p = 2. Our proof for p < 2 moreover shows that a random sparse matrix has the stronger restricted isometry property (RIP) with respect to the ℓ_{p} norm, and in fact this follows solely from the unique expansion of a random biregular graph, yielding a somewhat unexpected generalization of a similar result for the ℓ_{1} norm [6]. Instantiating this with suitable explicit expanders, we obtain the first explicit constructions of ℓ_{p}-RIP matrices for 1 ≤ p < p_{0}, where 1 < p_{0} < 2 is an absolute constant.

Original language | English |
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Title of host publication | 37th Computational Complexity Conference, CCC 2022 |

Editors | Shachar Lovett |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772419 |

DOIs | |

State | Published - 1 Jul 2022 |

Externally published | Yes |

Event | 37th Computational Complexity Conference, CCC 2022 - Philadelphia, United States Duration: 20 Jul 2022 → 23 Jul 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 234 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 37th Computational Complexity Conference, CCC 2022 |
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Country/Territory | United States |

City | Philadelphia |

Period | 20/07/22 → 23/07/22 |

## Keywords

- Euclidean Sections
- Restricted Isometry Property
- Sparse Matrices
- Spread Subspaces

## ASJC Scopus subject areas

- Software