Embedding Knowledge Graphs in Degenerate Clifford Algebras

Clifford algebras are a natural generalization of the real numbers, the complex numbers, and the quaternions. So far, solely Clifford algebras of the form $Cl_{p,q}$ (i.e., algebras without nilpotent base vectors) have been studied in the context of knowledge graph embeddings. We propose to consider nilpotent base vectors with a nilpotency index of two. In these spaces, denoted $Cl_{p,q,r}$, allows generalizing over approaches based on dual numbers (which cannot be modelled using $Cl_{p,q}$) and capturing patterns that emanate from the absence of higher-order interactions between real and complex parts of entity embeddings. We design two new models for the discovery of the parameters $p$, $q$, and $r$. The first model uses a greedy search to optimize $p$, $q$, and $r$. The second predicts $(p, q,r)$ based on an embedding of the input knowledge graph computed using neural networks. The results of our evaluation on seven benchmark datasets suggest that nilpotent vectors can help capture embeddings better. Our comparison against the state of the art suggests that our approach generalizes better than other approaches on all datasets w.r.t. the MRR it achieves on validation data. We also show that a greedy search suffices to discover values of $p$, $q$ and $r$ that are close to optimal.

A Computational Exploration of Emerging Methods of Variable Importance Estimation

Estimating the importance of variables is an essential task in modern machine learning. This help to evaluate the goodness of a feature in a given model. Several techniques for estimating the importance of variables have been developed during the last decade. In this paper, we proposed a computational and theoretical exploration of the emerging methods of variable importance estimation, namely: Least Absolute Shrinkage and Selection Operator (LASSO), Support Vector Machine (SVM), the Predictive Error Function (PERF), Random Forest (RF), and Extreme Gradient Boosting (XGBOOST) that were tested on different kinds of real-life and simulated data. All these methods can handle both regression and classification tasks seamlessly but all fail when it comes to dealing with data containing missing values. The implementation has shown that PERF has the best performance in the case of highly correlated data closely followed by RF. PERF and XGBOOST are "data-hungry" methods, they had the worst performance on small data sizes but they are the fastest when it comes to the execution time. SVM is the most appropriate when many redundant features are in the dataset. A surplus with the PERF is its natural cut-off at zero helping to separate positive and negative scores with all positive scores indicating essential and significant features while the negatives score indicates useless features. RF and LASSO are very versatile in a way that they can be used in almost all situations despite they are not giving the best results.