Ranking by Norm is an evaluation method used to assess learned prerequisite chains. It ranks all topics by increasing norm, which is defined as the distance between a concept $$u$$ and the origin of the Poincaré vector space. The method then evaluates performance by verifying how many annotated prerequisite relationships are satisfied or included within this rank.

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Improved Concept Embeddings for Learning Prerequisite Chains: Evaluation 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

The Mean Rank is the mean difference between the annotated ranks of the children of a node and the ranks of the same children obtained from the embedding (label vs. result). The lower the Mean Rank, the more correctly the embedding can determine what topics have a certain topic as their prerequisite.

Improved Concept Embeddings for Learning Prerequisite Chains: Mean Rank

For a single node/concept $u$ in the co-occurrence graph $G$, the Average Precision (AP) is calculated in two steps.
1. Constructed a list $labels = \{l_1, l_2, ..., l_n\}$, where $l_i = 1$ if $i$ is a child of $u$ and 0 otherwise, and calculated a list $distances = −\{d_1, d_2, ..., d_n\}$, where $d_i$ is the distance between $u$ and $i$. 
2. The AP algorithm considers every possible distance threshold to classify the nodes as children of $u$ in the graph $G$ , and for each threshold $j$ returns the recall $R_j$ and the precision $P_j$ . Sorted $R_j$ and calculated
$$AP = \sum_{i=1}^n (R_i − R_{i−1})P_i$$
The Mean Average Precision (mAP) is
$$mAP= \frac{1}{|G|}\sum_{u∈G} AP_u$$

A higher mAP means a more efficient embedding when reconstructing the prerequisite relationships of the annotated graph.

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