Abstract

Grade of membership (GoM) models are popular individual-level mixture models for multivariate categorical data. GoM allows each subject to have mixed memberships in multiple extreme latent profiles. Therefore, GoM models have a richer modeling capacity than latent class models that restrict each subject to belong to a single profile. The flexibility of GoM comes at the cost of more challenging identifiability and estimation problems. In this work, we propose a singular value decomposition (SVD)-based spectral approach to GoM analysis with multivariate binary responses. Our approach hinges on the observation that the expectation of the data matrix has a low-rank decomposition under a GoM model. For identifiability, we develop sufficient and almost necessary conditions for a notion of expectation identifiability. For estimation, we extract only a few leading singular vectors of the observed data matrix and exploit the simplex geometry of these vectors to estimate the mixed membership scores and other parameters. We also establish the consistency of our estimator in the double-asymptotic regime where both the number of subjects and the number of items grow to infinity. Our spectral method has a huge computational advantage over Bayesian or likelihood-based methods and is scalable to large-scale and high-dimensional data. Extensive simulation studies demonstrate the superior efficiency and accuracy of our method. We also illustrate our method by applying it to a personality test dataset.

中文翻译：

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一种可识别二元响应隶属度分析等级的谱法

摘要

会员等级 (GoM) 模型是适用于多元分类数据的流行个人级别混合模型。 GoM 允许每个主题在多个极端潜在配置文件中拥有混合成员身份。因此，GoM 模型比限制每个主题属于单个配置文件的潜在类模型具有更丰富的建模能力。 GoM 的灵活性是以更具挑战性的可识别性和估计问题为代价的。在这项工作中，我们提出了一种基于奇异值分解 (SVD) 的谱方法，用于具有多元二元响应的 GoM 分析。我们的方法取决于数据矩阵的期望在 GoM 模型下具有低秩分解的观察。对于可识别性，我们为期望可识别性的概念开发了充分且几乎必要的条件。为了估计，我们仅提取观察到的数据矩阵的几个前导奇异向量，并利用这些向量的单纯形几何来估计混合隶属度分数和其他参数。我们还在双渐近机制中建立了估计量的一致性，其中受试者的数量和项目的数量都增长到无穷大。我们的谱方法比贝叶斯或基于似然的方法具有巨大的计算优势，并且可扩展到大规模和高维数据。广泛的模拟研究证明了我们的方法的卓越效率和准确性。我们还通过将其应用于性格测试数据集来说明我们的方法。