Learning Edge Representations via Low-Rank Asymmetric Projections

We propose a new method for embedding graphs while preserving directed edge information. Learning such continuous-space vector representations (or embeddings) of nodes in a graph is an important first step for using network information (from social networks, user-item graphs, knowledge bases, etc.) in many machine learning tasks. Unlike previous work, we (1) explicitly model an edge as a function of node embeddings, and we (2) propose a novel objective, the "graph likelihood", which contrasts information from sampled random walks with non-existent edges. Individually, both of these contributions improve the learned representations, especially when there are memory constraints on the total size of the embeddings. When combined, our contributions enable us to significantly improve the state-of-the-art by learning more concise representations that better preserve the graph structure. We evaluate our method on a variety of link-prediction task including social networks, collaboration networks, and protein interactions, showing that our proposed method learn representations with error reductions of up to 76% and 55%, on directed and undirected graphs. In addition, we show that the representations learned by our method are quite space efficient, producing embeddings which have higher structure-preserving accuracy but are 10 times smaller

Learning Neural Graph Representations in Non-Euclidean Geometries

The success of Deep Learning methods is heavily dependent on the choice of the data representation. For that reason, much of the actual effort goes into Representation Learning, which seeks to design preprocessing pipelines and data transformations that can support effective learning algorithms. The aim of Representation Learning is to facilitate the task of extracting useful information for classifiers and other predictor models. In this regard, graphs arise as a convenient data structure that serves as an intermediary representation in a wide range of problems. The predominant approach to work with graphs has been to embed them in an Euclidean space, due to the power and simplicity of this geometry. Nevertheless, data in many domains exhibit non-Euclidean features, making embeddings into Riemannian manifolds with a richer structure necessary. The choice of a metric space where to embed the data imposes a geometric inductive bias, with a direct impact on the performance of the models. This thesis is about learning neural graph representations in non-Euclidean geometries and showcasing their applicability in different downstream tasks. We introduce a toolkit formed by different graph metrics with the goal of characterizing the topology of the data. In that way, we can choose a suitable target embedding space aligned to the shape of the dataset. By virtue of the geometric inductive bias provided by the structure of the non-Euclidean manifolds, neural models can achieve higher performances with a reduced parameter footprint. As a first step, we study graphs with hierarchical structures. We develop different techniques to derive hierarchical graphs from large label inventories. Noticing the capacity of hyperbolic spaces to represent tree-like arrangements, we incorporate this information into an NLP model through hyperbolic graph embeddings and showcase the higher performance that they enable. Second, we tackle the question of how to learn hierarchical representations suited for different downstream tasks. We introduce a model that jointly learns task-specific graph embeddings from a label inventory and performs classification in hyperbolic space. The model achieves state-of-the-art results on very fine-grained labels, with a remarkable reduction of the parameter size. Next, we move to matrix manifolds to work on graphs with diverse structures and properties. We propose a general framework to implement the mathematical tools required to learn graph embeddings on symmetric spaces. These spaces are of particular interest given that they have a compound geometry that simultaneously contains Euclidean as well as hyperbolic subspaces, allowing them to automatically adapt to dissimilar features in the graph. We demonstrate a concrete implementation of the framework on Siegel spaces, showcasing their versatility on different tasks. Finally, we focus on multi-relational graphs. We devise the means to translate Euclidean and hyperbolic multi-relational graph embedding models into the space of symmetric positive definite (SPD) matrices. To do so we develop gyrocalculus in this geometry and integrate it with the aforementioned framework

Improving Heterogeneous Graph Learning with Weighted Mixed-Curvature Product Manifold

In graph representation learning, it is important that the complex geometric structure of the input graph, e.g. hidden relations among nodes, is well captured in embedding space. However, standard Euclidean embedding spaces have a limited capacity in representing graphs of varying structures. A promising candidate for the faithful embedding of data with varying structure is product manifolds of component spaces of different geometries (spherical, hyperbolic, or euclidean). In this paper, we take a closer look at the structure of product manifold embedding spaces and argue that each component space in a product contributes differently to expressing structures in the input graph, hence should be weighted accordingly. This is different from previous works which consider the roles of different components equally. We then propose WEIGHTED-PM, a data-driven method for learning embedding of heterogeneous graphs in weighted product manifolds. Our method utilizes the topological information of the input graph to automatically determine the weight of each component in product spaces. Extensive experiments on synthetic and real-world graph datasets demonstrate that WEIGHTED-PM is capable of learning better graph representations with lower geometric distortion from input data, and performs better on multiple downstream tasks, such as word similarity learning, top-$k$ recommendation, and knowledge graph embedding

Distributed PCP Theorems for Hardness of Approximation in P

We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment $x \in \{0,1\}^n$ to a CNF formula $\varphi$ is shared between two parties, where Alice knows $x_1, \dots, x_{n/2}$, Bob knows $x_{n/2+1},\dots,x_n$, and both parties know $\varphi$. The goal is to have Alice and Bob jointly write a PCP that $x$ satisfies $\varphi$, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic variant, where the players are helped by Merlin, a third party who knows all of $x$. Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that there are no truly-subquadratic approximation algorithms for Bichromatic Maximum Inner Product over {0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first two problems we obtain nearly-polynomial factors of $2^{(\log n)^{1-o(1)}}$; only $(1+o(1))$-factor lower bounds (under SETH) were known before

Off the Beaten Path: Let's Replace Term-Based Retrieval with k-NN Search

Retrieval pipelines commonly rely on a term-based search to obtain candidate records, which are subsequently re-ranked. Some candidates are missed by this approach, e.g., due to a vocabulary mismatch. We address this issue by replacing the term-based search with a generic k-NN retrieval algorithm, where a similarity function can take into account subtle term associations. While an exact brute-force k-NN search using this similarity function is slow, we demonstrate that an approximate algorithm can be nearly two orders of magnitude faster at the expense of only a small loss in accuracy. A retrieval pipeline using an approximate k-NN search can be more effective and efficient than the term-based pipeline. This opens up new possibilities for designing effective retrieval pipelines. Our software (including data-generating code) and derivative data based on the Stack Overflow collection is available online