证明约束等距性的计算复杂性
Computational Complexity of Certifying Restricted Isometry Property
摘要：
给定具有$n$行、数$k<n$和$0<δ< 1$的矩阵$A$，如果对于任何向量$x \in \mathbb{R}^n$，至多具有$k$非零坐标，则$A$是$(k,δ)$-RIP（受限等距性质），$$(1-δ) \|x\|_2 \leq \|A x\|_2 \leq (1+δ)\|x\|_2$$在许多应用中，诸如压缩感测和稀疏恢复，期望构造具有大$k$和小$δ$的RIP矩阵。鉴于随机构造在生成有用的RIP矩阵中的功效，证明矩阵的RIP参数的问题变得重要。在本文中，我们证明了它是很难近似的RIP参数矩阵假设的小集合扩展假设。具体地说，我们证明，对于任何任意大的常数$C>0$和任何任意小的常数$0<δ<1$，存在一些$k$，使得给定一个矩阵$M$，它是SSE-难以区分以下两种情况：-（高度RIP）$M$是$(k,δ)$ -RIP。- （远离RIP）$M$不是$(k/C, 1-δ)$ -RIP。大多数关于RIP认证硬度主题的先前结果仅适用于$δ=o(1)$认证时。在实践中，理解证明具有接近于$\sqrt{2}-1$的$δ$的矩阵的复杂性是有意义的，因为对于许多实际应用来说，具有$δ= \sqrt{2}-1$的矩阵就足够了。我们的硬度结果适用于任何恒定的$δ$。具体地说，我们的结果证明，即使$δ$确实非常小，即矩阵实际上是\emph{strongly RIP}，证明矩阵表现出\emph{weak RIP}本身是SSE困难的。为了证明硬度的结果，我们证明了一个变种的奇格尔不等式的稀疏向量。
英文摘要：
Given a matrix $A$ with $n$ rows, a number $k<n$, and $0<δ< 1$, $A$ is $(k,δ)$-RIP (Restricted Isometry Property) if, for any vector $x \in \mathbb{R}^n$, with at most $k$ non-zero co-ordinates, $$(1-δ) \|x\|_2 \leq \|A x\|_2 \leq (1+δ)\|x\|_2$$ In many applications, such as compressed sensing and sparse recovery, it is desirable to construct RIP matrices with a large $k$ and a small $δ$. Given the efficacy of random constructions in generating useful RIP matrices, the problem of certifying the RIP parameters of a matrix has become important. In this paper, we prove that it is hard to approximate the RIP parameters of a matrix assuming the Small-Set-Expansion-Hypothesis. Specifically, we prove that for any arbitrarily large constant $C>0$ and any arbitrarily small constant $0<δ<1$, there exists some $k$ such that given a matrix $M$, it is SSE-Hard to distinguish the following two cases: - (Highly RIP) $M$ is $(k,δ)$-RIP. - (Far away from RIP) $M$ is not $(k/C, 1-δ)$-RIP. Most of the previous results on the topic of hardness of RIP certification only hold for certification when $δ=o(1)$. In practice, it is of interest to understand the complexity of certifying a matrix with $δ$ being close to $\sqrt{2}-1$, as it suffices for many real applications to have matrices with $δ= \sqrt{2}-1$. Our hardness result holds for any constant $δ$. Specifically, our result proves that even if $δ$ is indeed very small, i.e. the matrix is in fact \emph{strongly RIP}, certifying that the matrix exhibits \emph{weak RIP} itself is SSE-Hard. In order to prove the hardness result, we prove a variant of the Cheeger's Inequality for sparse vectors.
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