ℓ_p-Spread and Restricted Isometry Properties of Sparse Random Matrices

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 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 ℓ₁ norm [6]. Instantiating this with suitable explicit expanders, we obtain the first explicit constructions of ℓ_p-RIP matrices for 1 ≤ p < p₀, where 1 < p₀ < 2 is an absolute constant.