稀疏近似很难
Sparse Approximation is Hard
摘要：
给定由$M \times N$矩阵（$Φ\in \mathbb{R}^{M \times N}$）和目标信号$y \in \mathbb{R}^M$表示的冗余字典$Φ$，\emph{sparse approximation problem}要求使用至多$k$原子的线性组合来找到$y$的近似表示。通过对稀疏逼近问题的解的质量进行不同的度量，给出了稀疏逼近问题的一个新的复杂性理论困难性结果。有人认为，从算法的角度来看，这个问题是更有意义的，如果它要求最大限度地提高规范的目标信号的投影到所选择的原子列向量表示。然后，在一个合理的复杂性理论假设下，利用这个新测度建立了一个乘法不可逼近性结果。这个结果反过来又暗示了标准测度问题的加法不可逼近性。具体地，如果是$ZPP \neq NP$，则提供$k$ -稀疏向量$x$的所有多项式时间算法应当满足用于$1/4(1-1/e) > c \geq 0$的$$ {\|y-Φx\|}_2^2 \geq (1-c){\|y-Φx^*\|}_2^2 + c {\|y\|}_2^2, $$ \noindent，其中$x^*$是最优$k$ -稀疏解。该结果提供了$y-Φx^* = 0$案例的硬度的量化，揭示了关于问题的内在结构的更多细节。
英文摘要：
Given a redundant dictionary $Φ$, represented by an $M \times N$ matrix ($Φ\in \mathbb{R}^{M \times N}$) and a target signal $y \in \mathbb{R}^M$, the \emph{sparse approximation problem} asks to find an approximate representation of $y$ using a linear combination of at most $k$ atoms. In this paper, a new complexity theoretic hardness result for sparse approximation problem is presented via considering a different measure of quality for the solution. It is argued that, from an algorithmic standpoint, the problem is more meaningful if it asks to maximize the norm of the target signal's projection onto the selected atoms which are represented by column vectors. Then, a multiplicative inapproximability result is established with this new measure, under a reasonable complexity theoretic assumption. This result in turn implies additive inapproximability for the problem with the standard measure. Specifically, if $ZPP \neq NP$, all polynomial time algorithms which provide a $k$-sparse vector $x$ should satisfy $$ {\|y-Φx\|}_2^2 \geq (1-c){\|y-Φx^*\|}_2^2 + c {\|y\|}_2^2, $$ \noindent for $1/4(1-1/e) > c \geq 0$ where $x^*$ is the optimal $k$-sparse solution. This result provides a quantification of the hardness for the case $y-Φx^* = 0$, revealing more details about the inherent structure of the problem.
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