Research on knowledge graph (KG) construction, completion, inference, and applications has grown rapidly in recent years since it offers a powerful tool for modeling human knowledge in graph forms. Nodes in KGs denote entities and links represent relations between entities. The basic building blocks of KG are entity-relation triples in form of (subject, predicate, object) introduced by the Resource Description Framework (RDF). Learning representations for entities and relations in low dimensional vector spaces is one of the most active research topics in the field. Entity type offers a valuable piece of information to KG learning tasks. Better results in KG-related tasks have been achieved with the help of entity type. For example, TKRL [1] uses a hierarchical type encoder for KG completion by incorporating entity type information. AutoETER [2] adopts a similar approach but encodes the type information with projection matrices. Based on DistMult [3] and ComplEx [4] embedding, [5] propose an improved factorization model without explicit type supervision.

Many works have proven that existing neighbor-averaging Graph Neural Networks cannot efficiently catch structure information, such GNNs cannot even catch degree features in some cases. The reason is intuitive: as the neighbor-averaging GNNs can only combine neighbor's feature vectors for every node, if the neighbor's feature vectors contains no structure information, the hop-wise neighbor-averaging GNNs can only catch degree information at best([1];[2];[3]). So, as an intuitive idea, injecting structure information into feature vectors may improve the performance of GNNs. Numerous works have shown that injecting structure, distance, position or spatial information can significantly improve performance of neighbor-averaging GNNs([4];[5];[6];[7];[8];[9];[10]). However, existing works have their problems. Some of them has very high computation complexity which can not apply to large-scale graph(MotifNet[4]). Some of them simply concatenate structure information with intrinsic feature vector (ID-GNN[6]; P-GNN[8]; DE-GNN[9]), which may confuse the signals of different feature. For example, in ogbn-arxiv dataset, the intrinsic feature is semantic embedding of headline or abstract, which provides total different signal with structure information. Some of them are graph-level-task oriented and only deal with small graph(Graphormer[7]; SubGNN[10]).

Vector space models are to consider the relationship between data that are represented by vectors. It is popular in information retrieval systems but also useful for other purposes. Generally, this allows us to compare the similarity of two vectors from a geometric perspective. In this tutorial, we will see what is a vector space model and what it can do. A Gentle Introduction to Vector Space Models Photo by liamfletch, some rights reserved.

Convolutions are fundamental elements in deep learning architectures. Here, we present a theoretical framework for combining extrinsic and intrinsic approaches to manifold convolution through isometric embeddings into tori. In this way, we define a convolution operator for a manifold of arbitrary topology and dimension. We also explain geometric and topological conditions that make some local definitions of convolutions which rely on translating filters along geodesic paths on a manifold, computationally intractable. A result of Alan Turing from 1938 underscores the need for such a toric isometric embedding approach to achieve a global definition of convolution on computable, finite metric space approximations to a smooth manifold.

Graph neural networks (GNNs) have recently emerged as a dominant paradigm for machine learning with graphs. Research on GNNs has mainly focused on the family of message passing neural networks (MPNNs). Similar to the Weisfeiler-Leman (WL) test of isomorphism, these models follow an iterative neighborhood aggregation procedure to update vertex representations, and they next compute graph representations by aggregating the representations of the vertices. Although very successful, MPNNs have been studied intensively in the past few years. Thus, there is a need for novel architectures which will allow research in the field to break away from MPNNs. In this paper, we propose a new graph neural network model, so-called $\pi$-GNN which learns a "soft" permutation (i.e., doubly stochastic) matrix for each graph, and thus projects all graphs into a common vector space. The learned matrices impose a "soft" ordering on the vertices of the input graphs, and based on this ordering, the adjacency matrices are mapped into vectors. These vectors can be fed into fully-connected or convolutional layers to deal with supervised learning tasks. In case of large graphs, to make the model more efficient in terms of running time and memory, we further relax the doubly stochastic matrices to row stochastic matrices. We empirically evaluate the model on graph classification and graph regression datasets and show that it achieves performance competitive with state-of-the-art models.

With the rapid development of data collection techniques, complex data objects that are not in the Euclidean space are frequently encountered in new statistical applications. Fr\'echet regression model (Peterson & M\"uller 2019) provides a promising framework for regression analysis with metric space-valued responses. In this paper, we introduce a flexible sufficient dimension reduction (SDR) method for Fr\'echet regression to achieve two purposes: to mitigate the curse of dimensionality caused by high-dimensional predictors, and to provide a tool for data visualization for Fr\'echet regression. Our approach is flexible enough to turn any existing SDR method for Euclidean (X,Y) into one for Euclidean X and metric space-valued Y. The basic idea is to first map the metric-space valued random object $Y$ to a real-valued random variable $f(Y)$ using a class of functions, and then perform classical SDR to the transformed data. If the class of functions is sufficiently rich, then we are guaranteed to uncover the Fr\'echet SDR space. We showed that such a class, which we call an ensemble, can be generated by a universal kernel. We established the consistency and asymptotic convergence rate of the proposed methods. The finite-sample performance of the proposed methods is illustrated through simulation studies for several commonly encountered metric spaces that include Wasserstein space, the space of symmetric positive definite matrices, and the sphere. We illustrated the data visualization aspect of our method by exploring the human mortality distribution data across countries and by studying the distribution of hematoma density.

Word embedding is the approach of learning word and their relative meanings from a corpus of text and representing the word as a dense vector. The word vector is the projection of the word into a continuous feature vector space, see Figure 1 (A) for clarity. Words that have similar meaning should be close together in the vector space as illustrated in see Figure 1 (B). Word2vec is one of the most popular words embedding in NLP. Word2vec has two types, Continuous Bag-of-Words Model (CBOW) and Continuous Skip-gram Model [3], the model architectures are shown in Figure 2. CBOW predicts the word according to the given context, where Skip-gram predicts the context according to the given word, which increases the computational complexity [3].

In cross-lingual language models, representations for many different languages live in the same space. Here, we investigate the linguistic and non-linguistic factors affecting sentence-level alignment in cross-lingual pretrained language models for 101 languages and 5,050 language pairs. Using BERT-based LaBSE and BiLSTM-based LASER as our models, and the Bible as our corpus, we compute a task-based measure of cross-lingual alignment in the form of bitext retrieval performance, as well as four intrinsic measures of vector space alignment and isomorphism. We then examine a range of linguistic, quasi-linguistic, and training-related features as potential predictors of these alignment metrics. The results of our analyses show that word order agreement and agreement in morphological complexity are two of the strongest linguistic predictors of cross-linguality. We also note in-family training data as a stronger predictor than language-specific training data across the board. We verify some of our linguistic findings by looking at the effect of morphological segmentation on English-Inuktitut alignment, in addition to examining the effect of word order agreement on isomorphism for 66 zero-shot language pairs from a different corpus. We make the data and code for our experiments publicly available.

We focus on the problem of domain adaptation when the goal is shifting the model towards the target distribution, rather than learning domain invariant representations. It has been shown that under the following two assumptions: (a) access to samples from intermediate distributions, and (b) samples being annotated with the amount of change from the source distribution, self-training can be successfully applied on gradually shifted samples to adapt the model toward the target distribution. We hypothesize having (a) is enough to enable iterative self-training to slowly adapt the model to the target distribution, by making use of an implicit curriculum. In the case where (a) does not hold, we observe that iterative self-training falls short. We propose GIFT, a method that creates virtual samples from intermediate distributions by interpolating representations of examples from source and target domains. We evaluate an iterative-self-training method on datasets with natural distribution shifts, and show that when applied on top of other domain adaptation methods, it improves the performance of the model on the target dataset. We run an analysis on a synthetic dataset to show that in the presence of (a) iterative-self-training naturally forms a curriculum of samples. Furthermore, we show that when (a) does not hold, GIFT performs better than iterative self-training.

A database describing such user-item interactions often takes the form of a matrix, where each entry describes the interaction between one user and one item. The overall Recommender systems are often designed based rating or purchasing pattern of a user can therefore be described on a collaborative filtering approach, where user by the corresponding row in such a matrix. However, preferences are predicted by modelling interactions since there are typically large numbers of users and items between users and items. Many common approaches in the database, and each user is usually only interested in to solve the collaborative filtering task a small subset of items, this user-item matrix is often large are based on learning representations of users and and sparse. It is therefore inefficient to define the similarity items, including simple matrix factorization, Gaussian between users in the high dimensional feature space defined process latent variable models, and neuralnetwork by all items. Instead, it is more advantageous to derive abstract based embeddings. While matrix factorization feature vectors that represent users and items, which approaches fail to model nonlinear relations, inspired a large variety of low-rank matrix decomposition neural networks can potentially capture such models such as non-negative matrix decomposition [Zhang complex relations with unprecedented predictive et al., 2006], biased matrix decomposition [Koren et al., power and are highly scalable. However, neither 2009] and non-parametric decomposition [Yu et al., 2009]. of them is able to model predictive uncertainties. These methods aim at learning low dimensional representations In contrast, Gaussian Process based models can for all users and items, allowing for the prediction of generate a predictive distribution, but cannot scale the unobserved interaction between a new pair of user and to large amounts of data.