稀疏向量双线性反演的凸规划
A convex program for bilinear inversion of sparse vectors
摘要：
本文研究了从两个向量的乘积中恢复两个向量$\boldsymbol{x}\in\mathbb{R}^L$和$\boldsymbol{w}\in\mathbb{R}^L$的双线性逆问题。我们考虑的情况下，$\boldsymbol{x}$和$\boldsymbol{w}$具有已知的标志，并稀疏相对于已知的字典的大小$K$和$N$，分别。这里，$K$和$N$可以大于、小于或等于$L$。我们介绍$\ell_1$ -BranchHull，这是一个凸规划中的自然参数空间，并不需要一个近似的解决方案或初始化，以说明或解决。我们研究的情况下，$\boldsymbol{x}$和$\boldsymbol{w}$是$S_1$ -和$S_2$ -稀疏相对于一个随机字典，并提出了一个恢复保证，只依赖于测量的数量为$L\geqΩ(S_1+S_2)\log^{2}(K+N)$。数值实验表明，该定理中的标度常数并不太大。这个问题的一个应用是在介电成像中的扫频失真去除任务，其中一个信号是非负反射率，而另一个信号存在于已知的子空间中，例如由主小波系数给出的子空间。我们还介绍了一个变种的$\ell_1$ -BranchHull容忍噪声和离群值的目的，并恢复分段常数信号的目的。我们提供了这些变体的ADMM实现，并表明它们可以从真实图像中提取分段常数行为。
英文摘要：
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. We study the case where $\boldsymbol{x}$ and $\boldsymbol{w}$ are $S_1$- and $S_2$-sparse with respect to a random dictionary and present a recovery guarantee that only depends on the number of measurements as $L\geqΩ(S_1+S_2)\log^{2}(K+N)$. Numerical experiments verify that the scaling constant in the theorem is not too large. One application of this problem is 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 subspace, for example that given by dominant wavelet coefficients. We also introduce a 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.
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