The Average-Case Time Complexity of Certifying the Restricted Isometry Property

In compressed sensing, the restricted isometry property (RIP) on $M \times N$ sensing matrices (where < ) guarantees efficient reconstruction of sparse vectors. A matrix has notation="LaTeX">$(s,\delta)$ - notation="LaTeX">$\mathsf {RIP}$ if behaves as a notation="LaTeX">$\delta $ -approximate notation="LaTeX">$s$ -sparse It is well known that an notation="LaTeX">$M\times with i.i.d. notation="LaTeX">$\mathcal {N}(0,1/M)$ entries high probability long notation="LaTeX">$s\lesssim \delta ^{2}~M/\log . On other hand, most prior works aiming to deterministically construct have failed when notation="LaTeX">$s \gg \sqrt {M}$ An alternative way find RIP could be draw random gaussian and certify it indeed RIP. However, there evidence this certification task computationally hard , both in worst case average case. paper, we investigate exact average-case time complexity certifying for entries, “possible but hard” regime notation="LaTeX">$\sqrt {M} \ll s\lesssim M/\log Based analysis low-degree likelihood ratio, give rigorous subexponential runtime notation="LaTeX">$N^{\tilde \Omega (s^{2}/M)}$ required, demonstrating smooth tradeoff between maximum tolerated sparsity required computational power. This lower bound essentially tight, matching existing algorithm due Koiran Zouzias. Our hardness result allows take any constant value (0, 1), which captures relevant sensing. improves upon Wang et al. limited = o(1)$