The (efficient and parsimonious) decomposition of higher-order tensors is a
fundamental problem with numerous applications in a variety of fields. Several
methods have been proposed in the literature to that end, with the Tucker and
PARAFAC decompositions being the most prominent ones. Inspired by the latter,
in this work we propose a multi-resolution low-rank tensor decomposition to
describe (approximate) a tensor in a hierarchical fashion. The central idea of
the decomposition is to recast the tensor into \emph{multiple}
lower-dimensional tensors to exploit the structure at different levels of
resolution. The method is first explained, an alternating least squares
algorithm is discussed, and preliminary simulations illustrating the potential
practical relevance are provided.

通过多分辨率低秩张量分解以层次化方式描述张量，这种方法能够利用不同层次分辨率上的结构，创造性地解决了高阶张量分解这一基础问题。

多分辨率低秩张量分解

A Multi-resolution Low-rank Tensor Decomposition

Encoding facts as representations of entities and binary relationships
between them, as learned by knowledge graph representation models, is useful
for various tasks, including predicting new facts, question answering, fact
checking and information retrieval. The focus of this thesis is on (i)
improving knowledge graph representation with the aim of tackling the link
prediction task; and (ii) devising a theory on how semantics can be captured in
the geometry of relation representations. Most knowledge graphs are very
incomplete and manually adding new information is costly, which drives the
development of methods which can automatically infer missing facts. The first
contribution of this thesis is HypER, a convolutional model which simplifies
and improves upon the link prediction performance of the existing convolutional
state-of-the-art model ConvE and can be mathematically explained in terms of
constrained tensor factorisation. The second contribution is TuckER, a
relatively straightforward linear model, which, at the time of its
introduction, obtained state-of-the-art link prediction performance across
standard datasets. The third contribution is MuRP, first multi-relational graph
representation model embedded in hyperbolic space. MuRP outperforms all
existing models and its Euclidean counterpart MuRE in link prediction on
hierarchical knowledge graph relations whilst requiring far fewer dimensions.
Despite the development of a large number of knowledge graph representation
models with gradually increasing predictive performance, relatively little is
known of the latent structure they learn. We generalise recent theoretical
understanding of how semantic relations of similarity, paraphrase and analogy
are encoded in the geometric interactions of word embeddings to how more
general relations, as found in knowledge graphs, can be encoded in their
representations.

本文提出了三种知识图谱表征模型，分别是 HypER、TuckER 和 MuRP，它们都能够自动预测缺失的关系，并且在性能方面不断取得进步。同时，该文还将最新的理论研究成果推广到了知识图谱的领域，探讨了如何在这些模型中编码不同种类的关系。

学习实体和关系的表征

Learning Representations of Entities and Relations

Knowledge graphs are structured representations of real world facts. However,
they typically contain only a small subset of all possible facts. Link
prediction is a task of inferring missing facts based on existing ones. We
propose TuckER, a relatively straightforward but powerful linear model based on
Tucker decomposition of the binary tensor representation of knowledge graph
triples. TuckER outperforms previous state-of-the-art models across standard
link prediction datasets, acting as a strong baseline for more elaborate
models. We show that TuckER is a fully expressive model, derive sufficient
bounds on its embedding dimensionalities and demonstrate that several
previously introduced linear models can be viewed as special cases of TuckER.

TuckER 是一种基于 Tucker 分解的二元张量表示的线性模型，在标准的链接预测数据集上表现优于其他最先进的模型，并作为更复杂模型的强有力基准，是一种完全表达模型，可以作为先前引入的一些线性模型的特例来看待。