Perspectives on Latent Factor Indeterminacy and its Implications for Data Representation
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潜在因子不确定性的多重视角及其数据表征含义
- 研究问题
- 即使已知潜在维度并消除旋转不确定性，为何因子得分仍无法唯一恢复潜在来源，增加特征维度能否缓解这一问题？
- 方法与思路
- 作者把共同因子模型视为基础生成式表征模型，比较因子不确定性的多个数学视角，并联系变分自编码器的潜变量坍缩。
- 材料陈述的贡献
- 摘要称特征维度趋于无穷时可获得各方面的因子确定性，并据此提出特征数很大时近乎不依赖分布的估计思路。
- 阅读关注点
- 应核对旋转不确定性与因子得分不确定性的区别、维度增长结论的条件，以及总体确定性如何转化为样本估计。
- 阅读边界
- 这是 arXiv 预印本，同行评审与期刊接收未核验；摘要未提供高维极限条件，不能推断有限维或深层模型同样适用。
原文摘要
The common factor analytic model is related to Helmholtz and Boltzmann machines, can be conceived as a linear autoencoder, or can be thought of as a single-hidden-layer generative neural network. We thus consider it a basal generative representation learner that can be used as a minimal model for studying the foundational characteristics of (deep) generative model architectures. We focus on the fundamental problem of indeterminacy in latent factor projections. This indeterminacy implies that, even when the intrinsic dimension of the latent vector is known, regularity conditions are met, and rotational indeterminacy is resolved, an inherent indefiniteness in the retrieval of causative latent sources remains: they will be uncertain, distributionally deviant, and non-unique. This can have major implications for data representation but remains an elusive issue, even to practitioners and theorists well-versed in the factor model. Moreover, this classic psychometric problem is intricately related to the modern issue of latent variable collapse in the variational autoencoder framework for deep generative modeling. Here, we assess this indeterminacy from various perspectives and show how these are mathematically and conceptually related and we discuss subsequent implications for the Psychometrics, Statistics, and Artificial Intelligence communities. We show that one has latent factor determinacy across all its facets when the feature-dimension grows to infinity. This feeds into an essentially distribution-free estimation approach in the sample case when the number of features grows very large. We conclude, as these are emergent properties at scale, that the factor model is suited for representation learning of very-high-dimensional data.