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Dense subgraph discovery (DSD) is a key primitive in graph mining that typically deals with extracting cliques and near-cliques. In this paper, we revisit the optimal quasi-clique (OQC) formulation for DSD and establish that it is NP--hard. In addition, we reveal the hitherto unknown property that OQC can be used to explore the entire spectrum of densest subgraphs of all distinct sizes by appropriately varying a single hyperparameter, thereby forging an intimate link with the classic densest-k-subgraph problem (DkS). We corroborate these findings on real-world graphs by applying the simple greedy algorithm for OQC with improved hyperparameter tuning, to quickly generate high-quality approximations of the size-density frontier. Our findings indicate that OQC not only extracts high quality (near)-cliques, but also large and loosely-connected subgraphs that exhibit well defined local community structure. The latter discovery is particularly intriguing, since OQC is not explicitly geared towards community detection. 

Canonical correlation analysis (CCA) is a classic statistical method for discovering latent co-variation that underpins two or more observed random vectors. Several extensions and variations of CCA have been proposed that have strengthened our capabilities in terms of revealing common random factors from multiview datasets. In this work, we first revisit the most recent deterministic extensions of deep CCA and highlight the strengths and limitations of these state-of-the-art methods. Some methods allow trivial solutions, while others can miss weak common factors. Others overload the problem by also seeking to reveal what is {\em not common} among the views -- i.e., the private components that are needed to fully reconstruct each view. The latter tends to overload the problem and its computational and sample complexities. Aiming to improve upon these limitations, we design a novel and efficient formulation that alleviates some of the current restrictions. The main idea is to model the private components as {\em conditionally} independent given the common ones, which enables the proposed compact formulation. In addition, we also provide a sufficient condition for identifying the common random factors. Judicious experiments with synthetic and real datasets showcase the validity of our claims and the effectiveness of the proposed approach. 

We introduce the triangle-densest-K-subgraph problem (TDKS) for undirected graphs: given a size parameter K, compute a subset of K vertices that maximizes the number of induced triangles. The problem corresponds to the simplest generalization of the edge based densest-K-subgraph problem (DKS) to the case of higher-order network motifs. We prove that TDKS is NP-hard and is not amenable to efficient approximation, in the worst-case. By judiciously exploiting the structure of the problem, we propose a relaxation algorithm for the purpose of obtaining high-quality, sub-optimal solutions. Our approach utilizes the fact that the cost function of TDKS is submodular to construct a convex relaxation for the problem based on the Lovász extension for submodular functions. We demonstrate that our approaches attain state-of-the-art performance on real-world graphs and can offer substantially improved exploration of the optimal density-size curve compared to sophisticated approximation baselines for DKS. We use document summarization to showcase why TDKS is a useful generalization of DKS. 

Node embedding is the task of extracting concise and informative representations of certain entities that are connected in a network. Various real-world networks include information about both node connectivity and certain node attributes, in the form of features or time-series data. Modern representation learning techniques employ both the connectivity and attribute information of the nodes to produce embeddings in an unsupervised manner. In this context, deriving embeddings that preserve the geometry of the network and the attribute vectors would be highly desirable, as they would reflect both the topological neighborhood structure and proximity in feature space. While this is fairly straightforward to maintain when only observing the connectivity or attribute information of the network, preserving the geometry of both types of information is challenging. A novel tensor factorization approach for node embedding in attributed networks is proposed in this paper, that preserves the distances of both the connections and the attributes. Furthermore, an effective and lightweight algorithm is developed to tackle the learning task and judicious experiments with multiple state-of-the-art baselines suggest that the proposed algorithm offers significant performance improvements in downstream tasks. 

Collaborative filtering (CF) methods are making an impact on our daily lives in a wide range of applications, including recommender systems and personalization. Latent factor methods, e.g., matrix factorization (MF), have been the state-of-the-art in CF, however they lack interpretability and do not provide a straightforward explanation for their predictions. Explainability is gaining momentum in recommender systems for accountability, and because a good explanation can swing an undecided user. Most recent explainable recommendation methods require auxiliary data such as review text or item content on top of item ratings. In this paper, we address the case where no additional data are available and propose augmenting the classical MF framework for CF with a prior that encodes each user's embedding as a sparse linear combination of item embeddings, and vice versa for each item embedding. Our XPL-CF approach automatically reveals these user-item relationships, which underpin the latent factors and explain how the resulting recommendations are formed. We showcase the effectiveness of XPL-CF on real data from various application domains. We also evaluate the explainability of the user-item relationship obtained from XPL-CF through numeric evaluation and case study examples. 

Knowledge graphs (KGs) are powerful tools that codify relational behaviour between entities in knowledge bases. KGs can simultaneously model many different types of subject-predicate-object and higher-order relations. As such, they offer a flexible modeling framework that has been applied to many areas, including biology and pharmacology – most recently, in the fight against COVID-19. The flexibility of KG modeling is both a blessing and a challenge from the learning point of view. In this paper we propose a novel coupled tensor-matrix framework for KG embedding. We leverage tensor factorization tools to learn concise representations of entities and relations in knowledge bases and employ these representations to perform drug repurposing for COVID-19. Our proposed framework is principled, elegant, and achieves 100% improvement over the best baseline in the COVID-19 drug repurposing task using a recently developed biological KG.