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Title: Tensor Decompositions for Temporal Knowledge Base Completion

Paper content:
Most algorithms for representation learning and link prediction in relational data have been designed for static data.
However, the data they are applied to usually evolves with time, such as friend graphs in social networks or user interactions with items in recommender systems.
This is also the case for knowledge bases, which contain facts such as (US, has president, B. Obama, [2009-2017]) that are valid only at certain points in time.
For the problem of link prediction under temporal constraints, i.e., answering queries of the form (US, has president, ?, 2012), we propose a solution inspired by the canonical decomposition of tensors of order 4.
We introduce new regularization schemes and present an extension of ComplEx that achieves state-of-the-art performance.
Additionally, we propose a new dataset for knowledge base completion constructed from Wikidata, larger than previous benchmarks by an order of magnitude, as a new reference for evaluating temporal and non-temporal link prediction methods.
Link prediction in relational data has been the subject of interest, given the widespread availability of such data and the breadth of its use in bioinformatics (Zitnik et al., 2018) , recommender systems (Koren et al., 2009) or Knowledge Base completion (Nickel et al., 2016a) .
Relational data is often temporal, for example, the action of buying an item or watching a movie is associated to a timestamp.
Some medicines might not have the same adverse side effects depending on the subject's age.
The task of temporal link prediction is to find missing links in graphs at precise points in time.
In this work, we study temporal link prediction through the lens of temporal knowledge base completion, which provides varied benchmarks both in terms of the underlying data they represent, but also in terms of scale.
A knowledge base is a set of facts (subject, predicate, object) about the world that are known to be true.
Link prediction in a knowledge base amounts to answer incomplete queries of the form (subject, predicate, ?) by providing an accurate ranking of potential objects.
In temporal knowledge bases, these facts have some temporal metadata attached.
For example, facts might only hold for a certain time interval, in which case they will be annotated as such.
Other facts might be event that happened at a certain point in time.
Temporal link prediction amounts to answering queries of the form (subject, predicate, ?, timestamp) .
For example, we expect the ranking of queries (USA, president, ?, timestamp) to vary with the timestamps.
As tensor factorization methods have proved successful for Knowledge Base Completion (Nickel et al., 2016a; Trouillon et al., 2016; Lacroix et al., 2018) , we express our Temporal Knowledge Base Completion problem as an order 4 tensor completion problem.
That is, timestamps are discretized and used to index a 4-th mode in the binary tensor holding (subject, predicate, object, timestamps) facts.
First, we introduce a ComplEx (Trouillon et al., 2016) decomposition of this order 4 tensor, and link it with previous work on temporal Knowledge Base completion.
This decomposition yields embeddings for each timestamps.
A natural prior is for these timestamps representation to evolve slowly over time.
We are able to introduce this prior as a regularizer for which the optimum is a variation on the nuclear p-norm.
In order to deal with heterogeneous temporal knowledge bases where a significant amount of relations might be non-temporal, we add a non-temporal component to our decomposition.
Experiments on available benchmarks show that our method outperforms the state of the art for similar number of parameters.
We run additional experiments for larger, regularized models and obtain improvements of up to 0.07 absolute Mean Reciprocal Rank (MRR).
Finally, we propose a dataset of 400k entities, based on Wikidata, with 7M train triples, of which 10% contain temporal validity information.
This dataset is larger than usual benchmarks in the Knowledge Base completion community and could help bridge the gap between the method designed and the envisaged web-scale applications.
Tensor methods have been successful for Knowledge Base completion.
In this work, we suggest an extension of these methods to Temporal Knowledge Bases.
Our methodology adapts well to the various form of these datasets : point-in-time, beginning and endings or intervals.
We show that our methods reach higher performances than the state of the art for similar number of parameters.
For several datasets, we also provide performances for higher dimensions.
We hope that the gap between low-dimensional and high-dimensional models can motivate further research in models that have increased expressivity at lower number of parameters per entity.
Finally, we propose a large scale temporal dataset which we believe represents the challenges of large scale temporal completion in knowledge bases.
We give performances of our methods for low-ranks on this dataset.
We believe that, given its scale, this dataset could also be an interesting addition to non-temporal knowledge base completion.
Then according to Kolda & Bader (2009) , unfolding along modes 3 and 4 leads to an order three tensor of decompositionX
Where • is the Khatri-Rao product (Smilde et al., 2005) , which is the column-wise Kronecker product :
Note that for a fourth mode of size L:
This justifies the regularizers used in Section 3.2.

assistant: text: We propose new tensor decompositions and associated regularizers to obtain state of the art performances on temporal knowledge base completion.