Provable Subspace Identification Under Post-Nonlinear Mixtures
Provable Subspace Identification Under Post-Nonlinear Mixtures
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DOI：
10.48550/arxiv.2210.07532
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发表时间：
2022-10
期刊：
影响因子：
--
通讯作者：
Qi Lyu;Xiao Fu
中科院分区：
文献类型：
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作者：
Qi Lyu;Xiao Fu
无监督混合学习（UML）旨在以盲的方式识别线性或非线性混合潜在成分。众所周知，UML是具有挑战性的：即使学习线性混合也需要高度重要的分析工具，例如，独立组件分析或非负矩阵分解。在这项工作中，重新审视了后非线性（PNL）混合模型——其中未知的单元非线性函数被施加到线性混合物上。PNL模型广泛应用于脑信号分类、语音分离、遥感、因果发现等领域。为了识别和去除未知的非线性函数，现有的作品通常在潜在成分上假设不同的性质（例如，统计独立性或概率单纯形结构）。这项工作表明，在精心设计的UML标准下，与底层混合系统相关的非平凡零空间的存在足以保证识别/去除未知的非线性。与之前的工作相比，我们的发现在很大程度上放宽了获得PNL可识别性的条件，因此可能有利于那些没有关于潜在成分的强结构信息的应用。一个有限样本的分析提供了表征的性能在现实设置所提出的方法。为了实现所提出的学习准则，提出了一种块坐标下降算法。一系列数值实验证实了我们的理论主张。
Unsupervised mixture learning (UML) aims at identifying linearly or nonlinearly mixed latent components in a blind manner. UML is known to be challenging: Even learning linear mixtures requires highly nontrivial analytical tools, e.g., independent component analysis or nonnegative matrix factorization. In this work, the post-nonlinear (PNL) mixture model -- where unknown element-wise nonlinear functions are imposed onto a linear mixture -- is revisited. The PNL model is widely employed in different fields ranging from brain signal classification, speech separation, remote sensing, to causal discovery. To identify and remove the unknown nonlinear functions, existing works often assume different properties on the latent components (e.g., statistical independence or probability-simplex structures). This work shows that under a carefully designed UML criterion, the existence of a nontrivial null space associated with the underlying mixing system suffices to guarantee identification/removal of the unknown nonlinearity. Compared to prior works, our finding largely relaxes the conditions of attaining PNL identifiability, and thus may benefit applications where no strong structural information on the latent components is known. A finite-sample analysis is offered to characterize the performance of the proposed approach under realistic settings. To implement the proposed learning criterion, a block coordinate descent algorithm is proposed. A series of numerical experiments corroborate our theoretical claims.