We give a short argument that yields a new lower bound on the number of subsampled rows from a bounded, orthonormal matrix necessary to form a matrix with the restricted isometry property. We show that a matrix formed by uniformly subsampling rows of an $N \times N$ Walsh matrix contains a $K$-sparse vector in the kernel, unless the number of subsampled rows is $\Omega(K \log K \log (N/K))$ -- our lower bound applies whenever $\min(K, N/K) > \log^C N$. Containing a sparse vector in the kernel precludes not only the restricted isometry property, but more generally the application of those matrices for uniform sparse recovery.

翻译：我们给出一个简短论证，为具有约束等距性的有界正交矩阵所需的子采样行数建立了新的下界。研究表明，若从$N \times N$的Walsh矩阵中均匀子取样行构成矩阵，当且仅当子采样行数达到$\Omega(K \log K \log (N/K))$时，该矩阵的零空间才不含$K$-稀疏向量——该下界在满足$\min(K, N/K) > \log^C N$时成立。零空间包含稀疏向量不仅会破坏约束等距性，更会阻碍此类矩阵在均匀稀疏恢复中的普适应用。