When Are Overcomplete Topic Models Identifiable? Uniqueness of Tensor Tucker Decompositions with Structured Sparsity
Overcomplete latent representations have been very popular for unsupervised feature learning in recent years. In this paper, we specify which overcomplete models can be identified given observable moments of a certain order. We consider probabilistic admixture or topic models in the overcomplete regime, where the number of latent topics can greatly exceed the size of the observed word vocabulary. While general overcomplete topic models are not identifiable, we establish generic identifiability under a constraint, referred to as topic persistence. Our sufficient conditions for identifiability involve a novel set of "higher order" expansion conditions on the topic-word matrix or the population structure of the model. This set of higher-order expansion conditions allow for overcomplete models, and require the existence of a perfect matching from latent topics to higher order observed words. We establish that random structured topic models are identifiable w.h.p. in the overcomplete regime. Our identifiability results allow for general (non-degenerate) distributions for modeling the topic proportions, and thus, we can handle arbitrarily correlated topics in our framework. Our identifiability results imply uniqueness of a class of tensor decompositions with structured sparsity which is contained in the class of Tucker decompositions, but is more general than the Candecomp/Parafac (CP) decomposition.
The paper is concerned with identifiability of latent 
      variable topic models, characterizing when the latent structure is 
      identifiable from observable moments in these models. The authors 
      introduce a multi-view type of model called the n-persistent topic model 
      where each set of n words (in sequence) share the same topic and study how 
      n plays a role in identifiability. The first theorem is a 
      deterministic result providing sufficient conditions for identifying the 
      population structure (or the topic distributions) from the (2rn)-th 
      observable moment. The three conditions involve non-degeneracy of the 
      prior distribution over topic probability, an non-overlapping condition on 
      the sparsity pattern of the columns of the population structure matrix, 
      and a condition relating the kruskal rank of the population matrix to the 
      sparsity of each column. The paper then analyzes population 
      structures with random sparsity pattern and characterizes parameter 
      settings (sparsity, degree of overcompleteness) under which these models 
      are w.h.p. identifiable. Here, the authors show that if the number of 
      topics is O(p^n) where p is the vocabulary size and n is the persistence 
      parameter, and the support of each topic distribution is O(p^{1/n}), then 
      these models are w.h.p. identifiable. While the paper deals with 
      matrix representations throughout, they comment that working with the 
      tensor representations would yield uniqueness conditions for a range of 
      tensor decompositions (depending on the persistence parameter). This is an 
      interesting result which is presented as an afterthought in the paper but 
      should be emphasized. Working with the tensor representations may also 
      lend to readability. The results of the paper are novel and 
      interesting. The paper, however, suffers in terms of readability; some 
      more intuition about the conditions and the main theorem would 
      significantly improve the clarity of the paper. For example, condition 2 
      and 3 together imply that ideal matrices have high degree of sparsity 
      while still containing a n-perfect matching but it's not obvious why these 
      matrices are the ideal ones. Moreover, the awkward matrix notation makes 
      the paper significantly harder to read. Questions: 1. For 
      condition 1 to be satisfied, it is essential that the topic proportions is 
      a random quantity, but this is not always assumed in analysis for latent 
      variable models (if h is fixed then M_{2r}(h) is always rank one). Why is 
      this condition required? Previous analysis [9] seem to not require a prior 
      distribution on the topic proportions while still guaranteeing 
      identifiability.