Topic Modeling is an approach used for automatic comprehension and
classification of data in a variety of settings, and perhaps the canonical
application is in uncovering thematic structure in a corpus of documents. A
number of foundational works both in machine learning and in theory have
suggested a probabilistic model for documents, whereby documents arise as a
convex combination of (i.e. distribution on) a small number of topic vectors,
each topic vector being a distribution on words (i.e. a vector of
word-frequencies). Similar models have since been used in a variety of
application areas; the Latent Dirichlet Allocation or LDA model of Blei et al.
is especially popular.
Theoretical studies of topic modeling focus on learning the model's
parameters assuming the data is actually generated from it. Existing approaches
for the most part rely on Singular Value Decomposition(SVD), and consequently
have one of two limitations: these works need to either assume that each
document contains only one topic, or else can only recover the span of the
topic vectors instead of the topic vectors themselves.
This paper formally justifies Nonnegative Matrix Factorization(NMF) as a main
tool in this context, which is an analog of SVD where all vectors are
nonnegative. Using this tool we give the first polynomial-time algorithm for
learning topic models without the above two limitations. The algorithm uses a
fairly mild assumption about the underlying topic matrix called separability,
which is usually found to hold in real-life data. A compelling feature of our
algorithm is that it generalizes to models that incorporate topic-topic
correlations, such as the Correlated Topic Model and the Pachinko Allocation
Model.
We hope that this paper will motivate further theoretical results that use
NMF as a replacement for SVD - just as NMF has come to replace SVD in many
applications.

该论文从理论上证明了利用非负矩阵分解（NMF）作为主要工具来实现主题模型学习算法的可行性，具有学习效率高，不需要数据满足某些限制条件等优点，可用于包含主题 - 主题相关性的模型。

学习主题模型 —— 走出奇异值分解

Learning Topic Models - Going beyond SVD

Topic models, such as latent Dirichlet allocation (LDA), can be useful tools
for the statistical analysis of document collections and other discrete data.
The LDA model assumes that the words of each document arise from a mixture of
topics, each of which is a distribution over the vocabulary. A limitation of
LDA is the inability to model topic correlation even though, for example, a
document about genetics is more likely to also be about disease than X-ray
astronomy. This limitation stems from the use of the Dirichlet distribution to
model the variability among the topic proportions. In this paper we develop the
correlated topic model (CTM), where the topic proportions exhibit correlation
via the logistic normal distribution [J. Roy. Statist. Soc. Ser. B 44 (1982)
139--177]. We derive a fast variational inference algorithm for approximate
posterior inference in this model, which is complicated by the fact that the
logistic normal is not conjugate to the multinomial. We apply the CTM to the
articles from Science published from 1990--1999, a data set that comprises 57M
words. The CTM gives a better fit of the data than LDA, and we demonstrate its
use as an exploratory tool of large document collections.

本文提出了相关话题模型（CTM），通过逻辑正态分布，展现留言比例的相关性，我们还推导出了适用于该模型的快速变分推理算法，并将其应用于 Science 1990-1999 年间发表的文章，数据集包括 57M 词，证明 CTM 比 LDA 拟合效果更好，并将其作为大型文本集成探索性工具。