TY - GEN
T1 - ℓp-Spread and Restricted Isometry Properties of Sparse Random Matrices
AU - Guruswami, Venkatesan
AU - Manohar, Peter
AU - Mosheiff, Jonathan
N1 - Funding Information:
Funding Venkatesan Guruswami: Supported in part by NSF grants CCF-1908125 and CCF-2210823, and a Simons Investigator Award. Peter Manohar: Supported in part by an ARCS Scholarship, NSF Graduate Research Fellowship (under grant numbers DGE1745016 and DGE2140739), and NSF CCF-1814603. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Jonathan Mosheiff : Supported in part by NSF CCF-1814603.
Publisher Copyright:
© Venkatesan Guruswami, Peter Manohar, and Jonathan Mosheiff
PY - 2022/7/1
Y1 - 2022/7/1
N2 - Random subspaces X of ℝn of dimension proportional to n are, with high probability, well-spread with respect to the ℓ2-norm. Namely, every nonzero x ∈ X is “robustly non-sparse” in the following sense: x is ε ∥x∥2-far in ℓ2-distance from all δn-sparse vectors, for positive constants ε, δ bounded away from 0. This “ℓ2-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 ℓ1 unit ball. Explicit ℓ2-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 ℓ2-norm, and in particular are not ℓ2-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 ℓp-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 ℓp 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 ℓ1 norm [6]. Instantiating this with suitable explicit expanders, we obtain the first explicit constructions of ℓp-RIP matrices for 1 ≤ p < p0, where 1 < p0 < 2 is an absolute constant.
AB - Random subspaces X of ℝn of dimension proportional to n are, with high probability, well-spread with respect to the ℓ2-norm. Namely, every nonzero x ∈ X is “robustly non-sparse” in the following sense: x is ε ∥x∥2-far in ℓ2-distance from all δn-sparse vectors, for positive constants ε, δ bounded away from 0. This “ℓ2-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 ℓ1 unit ball. Explicit ℓ2-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 ℓ2-norm, and in particular are not ℓ2-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 ℓp-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 ℓp 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 ℓ1 norm [6]. Instantiating this with suitable explicit expanders, we obtain the first explicit constructions of ℓp-RIP matrices for 1 ≤ p < p0, where 1 < p0 < 2 is an absolute constant.
KW - Euclidean Sections
KW - Restricted Isometry Property
KW - Sparse Matrices
KW - Spread Subspaces
UR - http://www.scopus.com/inward/record.url?scp=85130762072&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.CCC.2022.7
DO - 10.4230/LIPIcs.CCC.2022.7
M3 - Conference contribution
AN - SCOPUS:85130762072
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 37th Computational Complexity Conference, CCC 2022
A2 - Lovett, Shachar
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 37th Computational Complexity Conference, CCC 2022
Y2 - 20 July 2022 through 23 July 2022
ER -