Bilinear Compressed Sensing Under Known Signs via Convex Programming
Bilinear Compressed Sensing Under Known Signs via Convex Programming
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DOI：
10.1109/tsp.2020.3017929
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发表时间：
2019-06
影响因子：
5.4
通讯作者：
A. Aghasi;Ali Ahmed;Paul Hand;Babhru Joshi
中科院分区：
文献类型：
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作者：
A. Aghasi;Ali Ahmed;Paul Hand;Babhru Joshi
我们考虑从入口积恢复两个向量的双线性反问题，即从入口积恢复两个向量，即{R}L$中的$\bold符号{x}和L$中的$\bold符号{w}。我们考虑这样的情形，其中$\boldSymbol{x}$和$\boldSymbol{w}$具有已知符号，并且分别相对于大小为$K$和$N$的已知字典稀疏。这里，$K$和$N$可以大于、小于或等于$L$。我们引入了$\ell_1$-BranchHull，它是在自然参数空间中提出的一个凸规划，它不需要近似解或初始化即可表述或求解。在假设$-BOLDARM{x}$和$\BOLDARM{w}$满足可比有效稀疏条件，且关于随机字典是$S_1$-稀疏和$S_2$-稀疏的情况下，我们给出了噪声情况下的恢复保证。证明了当测量次数满足$L((S_1+S_2)\log^{2}(K+N))$时，$-BranchHull对小而密集的噪声具有很强的稳健性。数值实验表明，该定理中的标度常数并不太大。我们还引入了$-BranchHull的变体，用于容忍噪声和野值，以及恢复分段恒定信号的目的。我们提供了这些变体的ADMM实现，并展示了它们可以从真实图像中提取分段常量行为。
We consider the bilinear inverse problem of recovering two vectors, $\boldsymbol{x}\in \mathbb {R}^L$ and $\boldsymbol{w}\in \mathbb {R}^L$, from their entrywise product. We consider the case where $\boldsymbol{x}$ and $\boldsymbol{w}$ have known signs and are sparse with respect to known dictionaries of size $K$ and $N$, respectively. Here, $K$ and $N$ may be larger than, smaller than, or equal to $L$. We introduce $\ell _1$-BranchHull, which is a convex program posed in the natural parameter space and does not require an approximate solution or initialization in order to be stated or solved. Under the assumptions that $\boldsymbol{x}$ and $\boldsymbol{w}$ satisfy a comparable-effective-sparsity condition and are $S_1$- and $S_2$-sparse with respect to a random dictionary, we present a recovery guarantee in a noisy case. We show that $\ell _1$-BranchHull is robust to small dense noise with high probability if the number of measurements satisfy $L\geq \Omega ((S_1+S_2)\log ^{2}(K+N))$. Numerical experiments show that the scaling constant in the theorem is not too large. We also introduce variants of $\ell _1$-BranchHull for the purposes of tolerating noise and outliers, and for the purpose of recovering piecewise constant signals. We provide an ADMM implementation of these variants and show they can extract piecewise constant behavior from real images.