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.

本文提出了一种用于一般锥形约束双线性逆问题可识别性分析的统一灵活的方法，利用通过 lift 方式与低秩矩阵恢复的联系。我们针对三类信号分布开发了确定性可识别性条件，并考虑了其在实践中的满足情况。同时，我们展示了利用秩二零空间特征进行盲反卷积问题变异的数值实验，并证明了我们的方法提供了可识别性的良好估计。