Sparsity and ℓ_p-Restricted Isometry
share
A matrix A is said to have the ℓ_p-Restricted Isometry Property (ℓ_p-RIP) if for all vectors x of up to some sparsity k, Ax_p is roughly proportional to x_p. It is known that with high probability, random dense m× n matrices (e.g., with i.i.d. ± 1 entries) are ℓ_2-RIP with k ≈ m/log n, and sparse random matrices are ℓ_p-RIP for p ∈ [1,2) when k, m = Θ(n). However, when m = Θ(n), sparse random matrices are known to not be ℓ_2-RIP with high probability. With this backdrop, we show that there are no sparse matrices with ± 1 entries that are ℓ_2-RIP. On the other hand, for p ≠ 2, we show that any ℓ_p-RIP matrix must be sparse. Thus, sparsity is incompatible with ℓ_2-RIP, but necessary for ℓ_p-RIP for p ≠ 2.
READ FULL TEXT