In compressed sensing, the restricted isometry property (RIP) on $M \times N$ sensing matrices (where $M < N$) guarantees efficient reconstruction of sparse vectors. A matrix has the $(s,δ)$-$\mathsf{RIP}$ property if behaves as a $δ$-approximate isometry on $s$-sparse vectors. It is well known that an $M\times N$ matrix with i.i.d. $\mathcal{N}(0,1/M)$ entries is $(s,δ)$-$\mathsf{RIP}$ with high probability as long as $s\lesssim δ^2 M/\log N$. On the other hand, most prior works aiming to deterministically construct $(s,δ)$-$\mathsf{RIP}$ matrices have failed when $s \gg \sqrt{M}$. An alternative way to find an RIP matrix could be to draw a random gaussian matrix and certify that it is indeed RIP. However, there is evidence that this certification task is computationally hard when $s \gg \sqrt{M}$, both in the worst case and the average case.
In this paper, we investigate the exact average-case time complexity of certifying the RIP property for $M\times N$ matrices with i.i.d. $\mathcal{N}(0,1/M)$ entries, in the "possible but hard" regime $\sqrt{M} \ll s\lesssim M/\log N$. Based on analysis of the low-degree likelihood ratio, we give rigorous evidence that subexponential runtime $N^{\tildeΩ(s^2/M)}$ is required, demonstrating a smooth tradeoff between the maximum tolerated sparsity and the required computational power. This lower bound is essentially tight, matching the runtime of an existing algorithm due to Koiran and Zouzias. Our hardness result allows $δ$ to take any constant value in $(0,1)$, which captures the relevant regime for compressed sensing. This improves upon the existing average-case hardness result of Wang, Berthet, and Plan, which is limited to $δ= o(1)$.
On the conditions used to prove oracle results for the Lasso
在对设计矩阵的各种不同假设下，建立了线性模型中Lasso的Oracle不等式和变量选择性质。在本文中，我们展示了不同的条件和概念是如何相互关联的。限制特征值条件（Bickel等人，2009）或稍微弱一点的相容性条件（van de Geer，2007）对于预言结果是足够的。我们认为，这两个条件允许一个相当普遍的一类设计矩阵。因此，Lasso用于预测和估计的最优性适用于比相干性（Bunea et al，2007 b，c）或限制等距（Candes and Tao，2005）假设更一般的情况。
Near-Ideal Behavior of Compressed Sensing Algorithms
在最近的一篇论文中，它表明，LASSO算法表现出“接近理想的行为”，在以下意义上：假设$y = Az + η$，其中$A$满足限制等距属性（RIP）与一个足够小的常数，和$\Vert η\Vert_2 \leq ε$。然后最小化服从$\Vert y - Az \Vert_2 \leq ε$的$\Vert z \Vert_1$导致估计$\hat{x}$，其误差$\Vert \hat{x} - x \Vert_2$由通用常数乘以由知道$x$的非零分量的位置的“oracle”获得的误差来限定。在优化领域，LASSO算法已经在几个方向上推广，例如组LASSO，稀疏组LASSO，没有或具有树结构重叠组，以及最近的排序LASSO。在本文中，它表明，{ \it任何算法\/}表现出接近理想的行为，在上述意义上，只提供（i）用于定义稀疏指数的规范是“可分解的”，（ii）在努力执行稀疏性最小化的惩罚规范是“$γ$ -可分解的”，和（iii）一个“压缩性条件”在一组限制等距属性是满足的。具体地，组LASSO和稀疏组LASSO（在组中具有一些允许的重叠）以及排序的$\ell_1$范数最小化都表现出接近理想的行为。显式的残余误差的界限推导出包含以前已知的结果作为特殊情况。
