BranchHull：来自具有已知符号的信号的逐项乘积的凸双线性反演
BranchHull: Convex bilinear inversion from the entrywise product of signals with known signs
摘要：
本文研究了从$\mathbb{R}^L$中的两个向量$x$和$w$的项积恢复它们的双线性逆问题。对于向量具有已知符号且属于已知子空间的情况，我们引入凸程序BranchHull，其在自然参数空间中提出，不需要近似解或初始化以便陈述或求解。在结构假设下，$x$和$w$是已知的$K$和$N$维随机子空间的成员，我们给出了无噪声和噪声情况下的恢复保证.在无噪声的情况下，我们证明了BranchHull恢复$x$和$w$高达固有的缩放模糊度时，$L\ \gg\ 2(K+N)$的概率很高。分析提供了一个精确的上限的系数为样本的复杂性。在有噪声的情况下，我们表明，高概率的BranchHull是强大的小密集噪声时，$L = Ω(K+N)$。BranchHull是由电介质成像中的扫描失真去除任务激发的，其中一个信号是非负反射率，另一个信号存在于已知的小波子空间中。其他潜在的应用是盲解卷积和自校准。
英文摘要：
We consider the bilinear inverse problem of recovering two vectors, $x$ and $w$, in $\mathbb{R}^L$ from their entrywise product. For the case where the vectors have known signs and belong to known subspaces, we introduce the convex program BranchHull, which is posed in the natural parameter space that does not require an approximate solution or initialization in order to be stated or solved. Under the structural assumptions that $x$ and $w$ are members of known $K$ and $N$ dimensional random subspaces, we present a recovery guarantee for the noiseless case and a noisy case. In the noiseless case, we prove that the BranchHull recovers $x$ and $w$ up to the inherent scaling ambiguity with high probability when $L\ \gg\ 2(K+N)$. The analysis provides a precise upper bound on the coefficient for the sample complexity. In a noisy case, we show that with high probability the BranchHull is robust to small dense noise when $L = Ω(K+N)$. BranchHull is motivated by the sweep distortion removal task in dielectric imaging, where one of the signals is a nonnegative reflectivity, and the other signal lives in a known wavelet subspace. Additional potential applications are blind deconvolution and self-calibration.
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