The probabilistic Latent Semantic Indexing model assumes that the expectation of the corpus matrix is low-rank and can be written as the product of a topic-word matrix and a word-document matrix. In this paper, we study the estimation of the topic-word matrix under the additional assumption that the ordered entries of its columns rapidly decay to zero. This sparsity assumption is motivated by the empirical observation that the word frequencies in a text often adhere to Zipf's law. We introduce a new spectral procedure for estimating the topic-word matrix that thresholds words based on their corpus frequencies, and show that its $\ell_1$-error rate under our sparsity assumption depends on the vocabulary size $p$ only via a logarithmic term. Our error bound is valid for all parameter regimes and in particular for the setting where $p$ is extremely large; this high-dimensional setting is commonly encountered but has not been adequately addressed in prior literature. Furthermore, our procedure also accommodates datasets that violate the separability assumption, which is necessary for most prior approaches in topic modeling. Experiments with synthetic data confirm that our procedure is computationally fast and allows for consistent estimation of the topic-word matrix in a wide variety of parameter regimes. Our procedure also performs well relative to well-established methods when applied to a large corpus of research paper abstracts, as well as the analysis of single-cell and microbiome data where the same statistical model is relevant but the parameter regimes are vastly different.

翻译：概率潜在语义索引模型假设语料矩阵的期望是低秩的，且可写作主题-词矩阵与词-文档矩阵的乘积。本文在额外假设主题-词矩阵各列的有序条目快速衰减至零的条件下，研究该矩阵的估计问题。这种稀疏性假设源于文本中词频常遵循齐普夫定律的经验观察。我们提出一种新的谱方法用于估计主题-词矩阵，该方法根据语料词频对词汇进行阈值化处理，并证明在我们提出的稀疏性假设下，其$\ell_1$误差率仅通过对数项依赖于词汇量大小$p$。该误差界适用于所有参数区间，尤其适用于$p$极大的高维场景——这类常见设置在前人文献中尚未得到充分解决。此外，我们的方法还能处理违反可分离性假设的数据集，而该假设是大多数现有主题建模方法所必需的。合成数据实验表明，本方法计算高效，可在多种参数区间内实现主题-词矩阵的一致估计。当应用于大规模研究论文摘要语料库，以及具有相同统计模型但参数区间迥异的单细胞和微生物组数据分析时，本方法相比现有成熟方法也表现出更优性能。