This work focuses on the problem of unraveling nonlinearly mixed latent components in an unsupervised manner. The latent components are assumed to reside in the probability simplex, and are transformed by an unknown post-nonlinear mixing system. This problem finds various applications in signal and data analytics, e.g., nonlinear hyperspectral unmixing, image embedding, and nonlinear clustering. Linear mixture learning problems are already ill-posed, as identifiability of the target latent components is hard to establish in general. With unknown nonlinearity involved, the problem is even more challenging. Prior work offered a function equation-based formulation for provable latent component identification. However, the identifiability conditions are somewhat stringent and unrealistic. In addition, the identifiability analysis is based on the infinite sample (i.e., population) case, while the understanding for practical finite sample cases has been elusive. Moreover, the algorithm in the prior work trades model expressiveness with computational convenience, which often hinders the learning performance. Our contribution is threefold. First, new identifiability conditions are derived under largely relaxed assumptions. Second, comprehensive sample complexity results are presented—which are the first of the kind. Third, a constrained autoencoder-based algorithmic framework is proposed for implementation, which effectively circumvents the challenges in the existing algorithm. Synthetic and real experiments corroborate our theoretical analyses.

中文翻译：

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通过自动编码器保证可识别性的单纯形结构后非线性混合学习


这项工作的重点是以无监督的方式解开非线性混合潜在成分的问题。假设潜在成分驻留在概率单纯形中，并通过未知的后非线性混合系统进行变换。该问题在信号和数据分析中有着多种应用，例如非线性高光谱分解、图像嵌入和非线性聚类。线性混合学习问题已经不适定，因为目标潜在组件的可识别性通常很难建立。由于涉及未知的非线性，该问题更具挑战性。先前的工作提供了一种基于函数方程的公式，用于可证明的潜在成分识别。然而，可识别性条件有些严格且不现实。此外，可识别性分析是基于无限样本（即总体）情况，而对实际有限样本情况的理解却难以捉摸。此外，现有工作中的算法以模型表达性和计算便利性为代价，这往往会阻碍学习性能。我们的贡献是三重的。首先，新的可识别性条件是在很大程度上宽松的假设下得出的。其次，给出了综合样本复杂性结果——这是此类结果中的第一个。第三，提出了一种基于约束自动编码器的算法框架进行实现，有效规避了现有算法中的挑战。综合和真实实验证实了我们的理论分析。