What is the relation between topic modeling and document clustering?

Question 1

A topic is quite different from a cluster of docs, after all, a topic is not composed of docs.

However, these two techniques are indeed related. I believe Topic Modeling is a viable way of deciding how similar documents are, hence a viable way for document clustering.

In representing each document as a topic distribution (actually a vector), topic modeling techniques reduce the feature dimensionality from number of distinct words appeared (in a corpus) to the number of topics. Similarity between docs' Topic distributions can be calculated using Cosine metrics and many other metrics, which reflect the similarity of the docs themselves in terms of the topics/themes they cover. Based on this quantified similarity measure, many clustering algorithms can be applied to group the documents.

And in this sense, I think it is right to say that topic modeling is a technique to do document clustering.

Question 2

The relation between clustering and classification is very similar to the relation between topic modeling and multi-label classification.

In single-label multi-class classification we assign just one label per each document. And in clustering we put each document in just one group. The fact is that we can't define the clusters in advance as we define labels. If we ignore this fact, grouping and labeling are essentially the same thing.

However, in real world problems flat classification is not sufficient. Often documents are related to multiple categories/classes. Thus we leverage the multi-label classification. Now, we can see the topic modeling as the unsupervised version of multi-label classification as we can put each document under multiple groups/topics. Here again, I'm ignoring the fact that we can't decide what topics to use as labels in advance.