ℓ_p-Spread Properties of Sparse Matrices
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Random subspaces X of ℝ^n of dimension proportional to n are, with high probability, well-spread with respect to the ℓ_p-norm (for p ∈ [1,2]). Namely, every nonzero x ∈ X is "robustly non-sparse" in the following sense: x is εx_p-far in ℓ_p-distance from all δ n-sparse vectors, for positive constants ε, δ bounded away from 0. This "ℓ_p-spread" property is the natural counterpart, for subspaces over the reals, of the minimum distance of linear codes over finite fields, and, for p = 2, corresponds to X being a Euclidean section of the ℓ_1 unit ball. Explicit ℓ_p-spread subspaces of dimension Ω(n), however, are not known except for p=1. The construction for p=1, as well as the best known constructions for p ∈ (1,2] (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-spread subspaces and ℓ_p-RIP matrices for 1 ≤ p < p_0, where 1 < p_0 < 2 is an absolute constant.
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