Bo Kang

Nan Li, Bo Kang, Jefrey Lijffijt et al.

In settings such as e-recruitment and online dating, recommendation involves distributing limited opportunities, calling for novel approaches to quantify and enforce fairness. We introduce \emph{inferiority}, a novel (un)fairness measure quantifying a user's competitive disadvantage for their recommended items. Inferiority complements \emph{envy}, a fairness notion measuring preference for others' recommendations. We combine inferiority and envy with \emph{utility}, an accuracy-related measure of aggregated relevancy scores. Since these measures are non-differentiable, we reformulate them using a probabilistic interpretation of recommender systems, yielding differentiable versions. We combine these loss functions in a multi-objective optimization problem called \texttt{FEIR} (Fairness through Envy and Inferiority Reduction), applied as post-processing for standard recommender systems. Experiments on synthetic and real-world data demonstrate that our approach improves trade-offs between inferiority, envy, and utility compared to naive recommendations and the baseline methods.

Kai Puolamäki, Emilia Oikarinen, Bo Kang et al.

Visual exploration of high-dimensional real-valued datasets is a fundamental task in exploratory data analysis (EDA). Existing methods use predefined criteria to choose the representation of data. There is a lack of methods that (i) elicit from the user what she has learned from the data and (ii) show patterns that she does not know yet. We construct a theoretical model where identified patterns can be input as knowledge to the system. The knowledge syntax here is intuitive, such as "this set of points forms a cluster", and requires no knowledge of maths. This background knowledge is used to find a Maximum Entropy distribution of the data, after which the system provides the user data projections in which the data and the Maximum Entropy distribution differ the most, hence showing the user aspects of the data that are maximally informative given the user's current knowledge. We provide an open source EDA system with tailored interactive visualizations to demonstrate these concepts. We study the performance of the system and present use cases on both synthetic and real data. We find that the model and the prototype system allow the user to learn information efficiently from various data sources and the system works sufficiently fast in practice. We conclude that the information theoretic approach to exploratory data analysis where patterns observed by a user are formalized as constraints provides a principled, intuitive, and efficient basis for constructing an EDA system.

Xander Vankwikelberge, Bo Kang, Edith Heiter et al.

Dimensionality reduction and clustering techniques are frequently used to analyze complex data sets, but their results are often not easy to interpret. We consider how to support users in interpreting apparent cluster structure on scatter plots where the axes are not directly interpretable, such as when the data is projected onto a two-dimensional space using a dimensionality-reduction method. Specifically, we propose a new method to compute an interpretable clustering automatically, where the explanation is in the original high-dimensional space and the clustering is coherent in the low-dimensional projection. It provides a tunable balance between the complexity and the amount of information provided, through the use of information theory. We study the computational complexity of this problem and introduce restrictions on the search space of solutions to arrive at an efficient, tunable, greedy optimization algorithm. This algorithm is furthermore implemented in an interactive tool called ExClus. Experiments on several data sets highlight that ExClus can provide informative and easy-to-understand patterns, and they expose where the algorithm is efficient and where there is room for improvement considering tunability and scalability.

Robin Vandaele, Bo Kang, Jefrey Lijffijt et al.

Unsupervised feature learning often finds low-dimensional embeddings that capture the structure of complex data. For tasks for which prior expert topological knowledge is available, incorporating this into the learned representation may lead to higher quality embeddings. For example, this may help one to embed the data into a given number of clusters, or to accommodate for noise that prevents one from deriving the distribution of the data over the model directly, which can then be learned more effectively. However, a general tool for integrating different prior topological knowledge into embeddings is lacking. Although differentiable topology layers have been recently developed that can (re)shape embeddings into prespecified topological models, they have two important limitations for representation learning, which we address in this paper. First, the currently suggested topological losses fail to represent simple models such as clusters and flares in a natural manner. Second, these losses neglect all original structural (such as neighborhood) information in the data that is useful for learning. We overcome these limitations by introducing a new set of topological losses, and proposing their usage as a way for topologically regularizing data embeddings to naturally represent a prespecified model. We include thorough experiments on synthetic and real data that highlight the usefulness and versatility of this approach, with applications ranging from modeling high-dimensional single-cell data, to graph embedding.

Robin Vandaele, Bo Kang, Tijl De Bie et al.

Distances between data points are widely used in machine learning applications. Yet, when corrupted by noise, these distances -- and thus the models based upon them -- may lose their usefulness in high dimensions. Indeed, the small marginal effects of the noise may then accumulate quickly, shifting empirical closest and furthest neighbors away from the ground truth. In this paper, we exactly characterize such effects in noisy high-dimensional data using an asymptotic probabilistic expression. Previously, it has been argued that neighborhood queries become meaningless and unstable when distance concentration occurs, which means that there is a poor relative discrimination between the furthest and closest neighbors in the data. However, we conclude that this is not necessarily the case when we decompose the data in a ground truth -- which we aim to recover -- and noise component. More specifically, we derive that under particular conditions, empirical neighborhood relations affected by noise are still likely to be truthful even when distance concentration occurs. We also include thorough empirical verification of our results, as well as interesting experiments in which our derived 'phase shift' where neighbors become random or not turns out to be identical to the phase shift where common dimensionality reduction methods perform poorly or well for recovering low-dimensional reconstructions of high-dimensional data with dense noise.

Xi Chen, Bo Kang, Jefrey Lijffijt et al.

In today's networked society, many real-world problems can be formalized as predicting links in networks, such as Facebook friendship suggestions, e-commerce recommendations, and the prediction of scientific collaborations in citation networks. Increasingly often, link prediction problem is tackled by means of network embedding methods, owing to their state-of-the-art performance. However, these methods lack transparency when compared to simpler baselines, and as a result their robustness against adversarial attacks is a possible point of concern: could one or a few small adversarial modifications to the network have a large impact on the link prediction performance when using a network embedding model? Prior research has already investigated adversarial robustness for network embedding models, focused on classification at the node and graph level. Robustness with respect to the link prediction downstream task, on the other hand, has been explored much less. This paper contributes to filling this gap, by studying adversarial robustness of Conditional Network Embedding (CNE), a state-of-the-art probabilistic network embedding model, for link prediction. More specifically, given CNE and a network, we measure the sensitivity of the link predictions of the model to small adversarial perturbations of the network, namely changes of the link status of a node pair. Thus, our approach allows one to identify the links and non-links in the network that are most vulnerable to such perturbations, for further investigation by an analyst. We analyze the characteristics of the most and least sensitive perturbations, and empirically confirm that our approach not only succeeds in identifying the most vulnerable links and non-links, but also that it does so in a time-efficient manner thanks to an effective approximation.

Ahmad Mel, Bo Kang, Jefrey Lijffijt et al.

Real-world data often presents itself in the form of a network. Examples include social networks, citation networks, biological networks, and knowledge graphs. In their simplest form, networks represent real-life entities (e.g. people, papers, proteins, concepts) as nodes, and describe them in terms of their relations with other entities by means of edges between these nodes. This can be valuable for a range of purposes from the study of information diffusion to bibliographic analysis, bioinformatics research, and question-answering. The quality of networks is often problematic though, affecting downstream tasks. This paper focuses on the common problem where a node in the network in fact corresponds to multiple real-life entities. In particular, we introduce FONDUE, an algorithm based on network embedding for node disambiguation. Given a network, FONDUE identifies nodes that correspond to multiple entities, for subsequent splitting. Extensive experiments on twelve benchmark datasets demonstrate that FONDUE is substantially and uniformly more accurate for ambiguous node identification compared to the existing state-of-the-art, at a comparable computational cost, while less optimal for determining the best way to split ambiguous nodes.

Xi Chen, Bo Kang, Jefrey Lijffijt et al.

Many real-world problems can be formalized as predicting links in a partially observed network. Examples include Facebook friendship suggestions, consumer-product recommendations, and the identification of hidden interactions between actors in a crime network. Several link prediction algorithms, notably those recently introduced using network embedding, are capable of doing this by just relying on the observed part of the network. Often, the link status of a node pair can be queried, which can be used as additional information by the link prediction algorithm. Unfortunately, such queries can be expensive or time-consuming, mandating the careful consideration of which node pairs to query. In this paper we estimate the improvement in link prediction accuracy after querying any particular node pair, to use in an active learning setup. Specifically, we propose ALPINE (Active Link Prediction usIng Network Embedding), the first method to achieve this for link prediction based on network embedding. To this end, we generalized the notion of V-optimality from experimental design to this setting, as well as more basic active learning heuristics originally developed in standard classification settings. Empirical results on real data show that ALPINE is scalable, and boosts link prediction accuracy with far fewer queries.

Junning Deng, Bo Kang, Jefrey Lijffijt et al.

The connectivity structure of graphs is typically related to the attributes of the nodes. In social networks for example, the probability of a friendship between two people depends on their attributes, such as their age, address, and hobbies. The connectivity of a graph can thus possibly be understood in terms of patterns of the form 'the subgroup of individuals with properties X are often (or rarely) friends with individuals in another subgroup with properties Y'. Such rules present potentially actionable and generalizable insights into the graph. We present a method that finds pairs of node subgroups between which the edge density is interestingly high or low, using an information-theoretic definition of interestingness. This interestingness is quantified subjectively, to contrast with prior information an analyst may have about the graph. This view immediately enables iterative mining of such patterns. Our work generalizes prior work on dense subgraph mining (i.e. subgraphs induced by a single subgroup). Moreover, not only is the proposed method more general, we also demonstrate considerable practical advantages for the single subgroup special case.

Bo Kang, Darío García García, Jefrey Lijffijt et al.

Dimensionality reduction and manifold learning methods such as t-Distributed Stochastic Neighbor Embedding (t-SNE) are routinely used to map high-dimensional data into a 2-dimensional space to visualize and explore the data. However, two dimensions are typically insufficient to capture all structure in the data, the salient structure is often already known, and it is not obvious how to extract the remaining information in a similarly effective manner. To fill this gap, we introduce \emph{conditional t-SNE} (ct-SNE), a generalization of t-SNE that discounts prior information from the embedding in the form of labels. To achieve this, we propose a conditioned version of the t-SNE objective, obtaining a single, integrated, and elegant method. ct-SNE has one extra parameter over t-SNE; we investigate its effects and show how to efficiently optimize the objective. Factoring out prior knowledge allows complementary structure to be captured in the embedding, providing new insights. Qualitative and quantitative empirical results on synthetic and (large) real data show ct-SNE is effective and achieves its goal.

Bo Kang, Jefrey Lijffijt, Tijl De Bie

Networks are powerful data structures, but are challenging to work with for conventional machine learning methods. Network Embedding (NE) methods attempt to resolve this by learning vector representations for the nodes, for subsequent use in downstream machine learning tasks. Link Prediction (LP) is one such downstream machine learning task that is an important use case and popular benchmark for NE methods. Unfortunately, while NE methods perform exceedingly well at this task, they are lacking in transparency as compared to simpler LP approaches. We introduce ExplaiNE, an approach to offer counterfactual explanations for NE-based LP methods, by identifying existing links in the network that explain the predicted links. ExplaiNE is applicable to a broad class of NE algorithms. An extensive empirical evaluation for the NE method `Conditional Network Embedding' in particular demonstrates its accuracy and scalability.

Bo Kang, Jefrey Lijffijt, Tijl De Bie

Network Embeddings (NEs) map the nodes of a given network into $d$-dimensional Euclidean space $\mathbb{R}^d$. Ideally, this mapping is such that `similar' nodes are mapped onto nearby points, such that the NE can be used for purposes such as link prediction (if `similar' means being `more likely to be connected') or classification (if `similar' means `being more likely to have the same label'). In recent years various methods for NE have been introduced, all following a similar strategy: defining a notion of similarity between nodes (typically some distance measure within the network), a distance measure in the embedding space, and a loss function that penalizes large distances for similar nodes and small distances for dissimilar nodes. A difficulty faced by existing methods is that certain networks are fundamentally hard to embed due to their structural properties: (approximate) multipartiteness, certain degree distributions, assortativity, etc. To overcome this, we introduce a conceptual innovation to the NE literature and propose to create \emph{Conditional Network Embeddings} (CNEs); embeddings that maximally add information with respect to given structural properties (e.g. node degrees, block densities, etc.). We use a simple Bayesian approach to achieve this, and propose a block stochastic gradient descent algorithm for fitting it efficiently. We demonstrate that CNEs are superior for link prediction and multi-label classification when compared to state-of-the-art methods, and this without adding significant mathematical or computational complexity. Finally, we illustrate the potential of CNE for network visualization.

Jefrey Lijffijt, Bo Kang, Wouter Duivesteijn et al.

Deriving insights from high-dimensional data is one of the core problems in data mining. The difficulty mainly stems from the fact that there are exponentially many variable combinations to potentially consider, and there are infinitely many if we consider weighted combinations, even for linear combinations. Hence, an obvious question is whether we can automate the search for interesting patterns and visualizations. In this paper, we consider the setting where a user wants to learn as efficiently as possible about real-valued attributes. For example, to understand the distribution of crime rates in different geographic areas in terms of other (numerical, ordinal and/or categorical) variables that describe the areas. We introduce a method to find subgroups in the data that are maximally informative (in the formal Information Theoretic sense) with respect to a single or set of real-valued target attributes. The subgroup descriptions are in terms of a succinct set of arbitrarily-typed other attributes. The approach is based on the Subjective Interestingness framework FORSIED to enable the use of prior knowledge when finding most informative non-redundant patterns, and hence the method also supports iterative data mining.

Tijl De Bie, Jefrey Lijffijt, Raul Santos-Rodriguez et al.

Methods for Projection Pursuit aim to facilitate the visual exploration of high-dimensional data by identifying interesting low-dimensional projections. A major challenge is the design of a suitable quality metric of projections, commonly referred to as the projection index, to be maximized by the Projection Pursuit algorithm. In this paper, we introduce a new information-theoretic strategy for tackling this problem, based on quantifying the amount of information the projection conveys to a user given their prior beliefs about the data. The resulting projection index is a subjective quantity, explicitly dependent on the intended user. As a useful illustration, we developed this idea for two particular kinds of prior beliefs. The first kind leads to PCA (Principal Component Analysis), shining new light on when PCA is (not) appropriate. The second kind leads to a novel projection index, the maximization of which can be regarded as a robust variant of PCA. We show how this projection index, though non-convex, can be effectively maximized using a modified power method as well as using a semidefinite programming relaxation. The usefulness of this new projection index is demonstrated in comparative empirical experiments against PCA and a popular Projection Pursuit method.