Question: In their approach, what is the only condition that the authors can derive tractable exact inference and marginal likelihood computation procedures?
Evidence: Data annotation is a major bottleneck for the application of supervised learning approaches to many problems. As a result, unsupervised methods that learn directly from unlabeled data are increasingly important. For tasks related to unsupervised syntactic analysis, discrete generative models have dominated in recent years – for example, for both part-of-speech (POS) induction BIBREF0 , BIBREF1 and unsupervised dependency parsing BIBREF2 , BIBREF3 , BIBREF4 . While similar models have had success on a range of unsupervised tasks, they have mostly ignored the apparent utility of continuous word representations evident from supervised NLP applications BIBREF5 , BIBREF6 . In this work, we focus on leveraging and explicitly representing continuous word embeddings within unsupervised models of syntactic structure.
Pre-trained word embeddings from massive unlabeled corpora offer a compact way of injecting a prior notion of word similarity into models that would otherwise treat words as discrete, isolated categories. However, the specific properties of language captured by any particular embedding scheme can be difficult to control, and, further, may not be ideally suited to the task at hand. For example, pre-trained skip-gram embeddings BIBREF7 with small context window size are found to capture the syntactic properties of language well BIBREF8 , BIBREF9 . However, if our goal is to separate syntactic categories, this embedding space is not ideal – POS categories correspond to overlapping interspersed regions in the embedding space, evident in Figure SECREF4 .
In our approach, we propose to learn a new latent embedding space as a projection of pre-trained embeddings (depicted in Figure SECREF5 ), while jointly learning latent syntactic structure – for example, POS categories or syntactic dependencies. To this end, we introduce a new generative model (shown in Figure FIGREF6 ) that first generates a latent syntactic representation (e.g. a dependency parse) from a discrete structured prior (which we also call the “syntax model”), then, conditioned on this representation, generates a sequence of latent embedding random variables corresponding to each word, and finally produces the observed (pre-trained) word embeddings by projecting these latent vectors through a parameterized non-linear function. The latent embeddings can be jointly learned with the structured syntax model in a completely unsupervised fashion.
By choosing an invertible neural network as our non-linear projector, and then parameterizing our model in terms of the projection's inverse, we are able to derive tractable exact inference and marginal likelihood computation procedures so long as inference is tractable in the underlying syntax model. In sec:learn-with-inv we show that this derivation corresponds to an alternate view of our approach whereby we jointly learn a mapping of observed word embeddings to a new embedding space that is more suitable for the syntax model, but include an additional Jacobian regularization term to prevent information loss.
Recent work has sought to take advantage of word embeddings in unsupervised generative models with alternate approaches BIBREF9 , BIBREF10 , BIBREF11 , BIBREF12 . BIBREF9 build an HMM with Gaussian emissions on observed word embeddings, but they do not attempt to learn new embeddings. BIBREF10 , BIBREF11 , and BIBREF12 extend HMM or dependency model with valence (DMV) BIBREF2 with multinomials that use word (or tag) embeddings in their parameterization. However, they do not represent the embeddings as latent variables.
In experiments, we instantiate our approach using both a Markov-structured syntax model and a tree-structured syntax model – specifically, the DMV. We evaluate on two tasks: part-of-speech (POS) induction and unsupervised dependency parsing without gold POS tags. Experimental results on the Penn Treebank BIBREF13 demonstrate that our approach improves the basic HMM and DMV by a large margin, leading to the state-of-the-art results on POS induction, and state-of-the-art results on unsupervised dependency parsing in the difficult training scenario where neither gold POS annotation nor punctuation-based constraints are available.

By choosing an invertible neural network as our non-linear projector, and then parameterizing our model in terms of the projection’s inverse, we are able to derive tractable exact inference and marginal likelihood computation procedures so long as inference is tractable in the underlying syntax model.
Answer: Tractable in the underlying syntax model.