ℓ_p-Spread and Restricted Isometry Properties of Sparse Random Matrices [article]

Venkatesan Guruswami, Peter Manohar, Jonathan Mosheiff
2022 arXiv   pre-print
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 εx_2-far in ℓ_2-distance from all δ n-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
more » ... ion of the ℓ_1 unit ball. Explicit ℓ_2-spread subspaces of dimension Ω(n), however, are unknown, and the best known constructions (which achieve weaker spread properties), are analogs of low density parity check (LDPC) codes over the reals, i.e., they are kernels of sparse matrices. We study the spread properties of the kernels of sparse random matrices. Rather surprisingly, 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. On the other hand, for p < 2 we prove that such subspaces are ℓ_p-spread with high probability. Moreover, we show that a random sparse matrix has the stronger restricted isometry property (RIP) with respect to the ℓ_p norm, and 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 [BGI+08]. Instantiating this with 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.
arXiv:2108.13578v2 fatcat:rg5xhaa3xzbghiogjrarebgl4m