The author tested 15, 50, and 100 epochs for Poincaré embeddings and chose 100 as the best number to train the model, with epochs of burn-in = 10.

Improved Concept Embeddings for Learning Prerequisite Chains: Epochs

- Learning rate = 0.2
- Number of negative examples = 80 
- Number of epochs = 100
- Epochs of burn-in = 10 
- Number of dimensions = 2, 3, 5, 10, 50, 100

University of Michigan - Ann Arbor

- Poincaré embeddings (Nickel and Kiela, 2017) is a method that better fit for data with latent hierarchies than other embedding models using the Euclidean vector space.
- Dhingra et al. (2018) introduced a method to obtain Poincaré word embeddings from natural language text.
- This paper combines the two methods above to obtain prerequisite relations between concepts from two datasets in the domain of Natural Language Processing. 

Improved Concept Embeddings for Learning Prerequisite Chains: Methods

Löffel, E. P. (2019). Improved Concept Embeddings for Learning Prerequisite Chains

Paper: https://pdfs.semanticscholar.org/6073/0900a80fcee29e9059fd293a1fa38742abf3.pdf

Github: https://github.com/eloffel/improved_embeddings

Improved Concept Embeddings for Learning Prerequisite Chains: Reference

Improved Concept Embeddings for Learning Prerequisite Chains: Datasets

- The embedding models using the Euclidean vector space are unable to capture hierarchical relationships, which is not optimized for embedding structures like graphs and trees. Nickel and Kiela (2017) proposed a model that embeds concepts into a hyperbolic space, which can be thought of as a continuous version of trees structure and therefore provides a better fit to embed data with latent hierarchies.
- Poincaré ball model represents the entire hyperbolic space within the border of an n-dimensional sphere. This model, referred to as Poincaré Embeddings, generates the embeddings having as its input a weighted directed graph $G = \{(b, a)\}$, in which each node represents a concept and a directed edge between a and b means that a is a hypernym of b, in other words, b is a prerequisite concept of a. 

Improved Concept Embeddings for Learning Prerequisite Chains: Poincaré embeddings

To generate Poincaré embeddings from a natural language text corpus, the author converted the text to a list of directed edges in the format $u,v,weight$, where $v$ is a hypernym of $u$, as input to the Poincaré embeddings algorithm, using the method described by Dhingra et al. (2018).  They constructed a co-occurrence graph $G = \{(u, v)\}$ that includes all pairs of words that occur within a fixed window of each other. Each edge $(u, v)$ in the graph has a weight $w = f^c$, where $f$ is the frequency of the co-occurrence pairs in the corpus and $c$ is a downsampling constant. 

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