双线性反问题的可辨识性缩放定律
Identifiability Scaling Laws in Bilinear Inverse Problems
摘要：
信号处理中的一些不适定反问题，如盲反卷积、矩阵分解、字典学习和盲声源分离等，都具有双线性反问题的共同特征，即观测模型是两个变量的函数，且在一个变量已知的条件下，观测是另一个变量的线性函数。对于这样的逆问题出现的一个关键问题是可识别性，即观察是否足以明确地确定生成观察的输入对。可识别性是无线通信中的盲均衡和机器学习中的数据采掘等应用的关键问题。在这里，一个统一的和灵活的方法来识别分析一般圆锥先验约束BIP，利用连接到低秩矩阵恢复通过提升。我们开发确定性的可识别性条件的输入信号，并检查其满足性在实践中的三类信号分布，即依赖，但不相关，独立的高斯和独立的伯努利。在每一种情况下，缩放定律开发的权衡概率的鲁棒可识别性与秩二零空间的复杂性。我们的方法的一个额外的吸引力是，秩二零空间可以部分或完全表征为许多感兴趣的双线性问题（例如盲反卷积）。我们目前的数值实验涉及的变化的盲反卷积问题，利用秩二零空间的表征，并证明了缩放定律提供良好的估计可识别性。
英文摘要：
A number of ill-posed inverse problems in signal processing, like blind deconvolution, matrix factorization, dictionary learning and blind source separation share the common characteristic of being bilinear inverse problems (BIPs), i.e. the observation model is a function of two variables and conditioned on one variable being known, the observation is a linear function of the other variable. A key issue that arises for such inverse problems is that of identifiability, i.e. whether the observation is sufficient to unambiguously determine the pair of inputs that generated the observation. Identifiability is a key concern for applications like blind equalization in wireless communications and data mining in machine learning. Herein, a unifying and flexible approach to identifiability analysis for general conic prior constrained BIPs is presented, exploiting a connection to low-rank matrix recovery via lifting. We develop deterministic identifiability conditions on the input signals and examine their satisfiability in practice for three classes of signal distributions, viz. dependent but uncorrelated, independent Gaussian, and independent Bernoulli. In each case, scaling laws are developed that trade-off probability of robust identifiability with the complexity of the rank two null space. An added appeal of our approach is that the rank two null space can be partly or fully characterized for many bilinear problems of interest (e.g. blind deconvolution). We present numerical experiments involving variations on the blind deconvolution problem that exploit a characterization of the rank two null space and demonstrate that the scaling laws offer good estimates of identifiability.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
