arXiv subjects

Aristides Gionis

Publications and source records attributed to Aristides Gionis.

At least 55 records · Page 3Linked to original sources

arXiv2022

Diversity-aware $k$-median : Clustering with fair center representation

We introduce a novel problem for diversity-aware clustering. We assume that the potential cluster centers belong to a set of groups defined by protected attributes, such as ethnicity, gender, etc. We then ask to find a minimum-cost clustering of the data into $k$ clusters so that a specified minimum number of cluster centers are chosen from each group. We thus require that all groups are represented in the clustering solution as cluster centers, according to specified requirements. More precisely, we are given a set of clients $C$, a set of facilities $\pazocal{F}$, a collection $\mathcal{F}=\{F_1,\dots,F_t\}$ of facility groups $F_i \subseteq \pazocal{F}$, budget $k$, and a set of lower-bound thresholds $R=\{r_1,\dots,r_t\}$, one for each group in $\mathcal{F}$. The \emph{diversity-aware $k$-median problem} asks to find a set $S$ of $k$ facilities in $\pazocal{F}$ such that $|S \cap F_i| \geq r_i$, that is, at least $r_i$ centers in $S$ are from group $F_i$, and the $k$-median cost $\sum_{c \in C} \min_{s \in S} d(c,s)$ is minimized. We show that in the general case where the facility groups may overlap, the diversity-aware $k$-median problem is \np-hard, fixed-parameter intractable, and inapproximable to any multiplicative factor. On the other hand, when the facility groups are disjoint, approximation algorithms can be obtained by reduction to the \emph{matroid median} and \emph{red-blue median} problems. Experimentally, we evaluate our approximation methods for the tractable cases, and present a relaxation-based heuristic for the theoretically intractable case, which can provide high-quality and efficient solutions for real-world datasets.

cs.DS↗

arXiv2022

Coresets remembered and items forgotten: submodular maximization with deletions

In recent years we have witnessed an increase on the development of methods for submodular optimization, which have been motivated by the wide applicability of submodular functions in real-world data-science problems. In this paper, we contribute to this line of work by considering the problem of robust submodular maximization against unexpected deletions, which may occur due to privacy issues or user preferences. Specifically, we consider the minimum number of items an algorithm has to remember, in order to achieve a non-trivial approximation guarantee against adversarial deletion of up to $d$ items. We refer to the set of items that an algorithm has to keep before adversarial deletions as a deletion-robust coreset. Our theoretical contributions are two-fold. First, we propose a single-pass streaming algorithm that yields a $(1-2ε)/(4p)$-approximation for maximizing a non-decreasing submodular function under a general p-matroid constraint and requires a coreset of size $k + d/ε$, where $k$ is the maximum size of a feasible solution. To the best of our knowledge, this is the first work to achieve an (asymptotically) optimal coreset, as no constant-factor approximation is possible with a coreset of size sublinear in $d$. Second, we devise an effective offline algorithm that guarantees stronger approximation ratios with a coreset of size $O(d \log(k)/ε)$. We also demonstrate the superior empirical performance of the proposed algorithms in real-life applications.

cs.DS↗

arXiv2022

Diameter Minimization by Shortcutting with Degree Constraints

We consider the problem of adding a fixed number of new edges to an undirected graph in order to minimize the diameter of the augmented graph, and under the constraint that the number of edges added for each vertex is bounded by an integer. The problem is motivated by network-design applications, where we want to minimize the worst case communication in the network without excessively increasing the degree of any single vertex, so as to avoid additional overload. We present three algorithms for this task, each with their own merits. The special case of a matching augmentation, when every vertex can be incident to at most one new edge, is of particular interest, for which we show an inapproximability result, and provide bounds on the smallest achievable diameter when these edges are added to a path. Finally, we empirically evaluate and compare our algorithms on several real-life networks of varying types.

cs.DS↗

arXiv2022

Regularized impurity reduction: Accurate decision trees with complexity guarantees

Decision trees are popular classification models, providing high accuracy and intuitive explanations. However, as the tree size grows the model interpretability deteriorates. Traditional tree-induction algorithms, such as C4.5 and CART, rely on impurity-reduction functions that promote the discriminative power of each split. Thus, although these traditional methods are accurate in practice, there has been no theoretical guarantee that they will produce small trees. In this paper, we justify the use of a general family of impurity functions, including the popular functions of entropy and Gini-index, in scenarios where small trees are desirable, by showing that a simple enhancement can equip them with complexity guarantees. We consider a general setting, where objects to be classified are drawn from an arbitrary probability distribution, classification can be binary or multi-class, and splitting tests are associated with non-uniform costs. As a measure of tree complexity, we adopt the expected cost to classify an object drawn from the input distribution, which, in the uniform-cost case, is the expected number of tests. We propose a tree-induction algorithm that gives a logarithmic approximation guarantee on the tree complexity. This approximation factor is tight up to a constant factor under mild assumptions. The algorithm recursively selects a test that maximizes a greedy criterion defined as a weighted sum of three components. The first two components encourage the selection of tests that improve the balance and the cost-efficiency of the tree, respectively, while the third impurity-reduction component encourages the selection of more discriminative tests. As shown in our empirical evaluation, compared to the original heuristics, the enhanced algorithms strike an excellent balance between predictive accuracy and tree complexity.

cs.LG↗

arXiv2022

Generalized Leverage Scores: Geometric Interpretation and Applications

In problems involving matrix computations, the concept of leverage has found a large number of applications. In particular, leverage scores, which relate the columns of a matrix to the subspaces spanned by its leading singular vectors, are helpful in revealing column subsets to approximately factorize a matrix with quality guarantees. As such, they provide a solid foundation for a variety of machine-learning methods. In this paper we extend the definition of leverage scores to relate the columns of a matrix to arbitrary subsets of singular vectors. We establish a precise connection between column and singular-vector subsets, by relating the concepts of leverage scores and principal angles between subspaces. We employ this result to design approximation algorithms with provable guarantees for two well-known problems: generalized column subset selection and sparse canonical correlation analysis. We run numerical experiments to provide further insight on the proposed methods. The novel bounds we derive improve our understanding of fundamental concepts in matrix approximations. In addition, our insights may serve as building blocks for further contributions.

cs.LG↗

arXiv2022

Ranking with submodular functions on a budget

Submodular maximization has been the backbone of many important machine-learning problems, and has applications to viral marketing, diversification, sensor placement, and more. However, the study of maximizing submodular functions has mainly been restricted in the context of selecting a set of items. On the other hand, many real-world applications require a solution that is a ranking over a set of items. The problem of ranking in the context of submodular function maximization has been considered before, but to a much lesser extent than item-selection formulations. In this paper, we explore a novel formulation for ranking items with submodular valuations and budget constraints. We refer to this problem as max-submodular ranking (MSR). In more detail, given a set of items and a set of non-decreasing submodular functions, where each function is associated with a budget, we aim to find a ranking of the set of items that maximizes the sum of values achieved by all functions under the budget constraints. For the MSR problem with cardinality- and knapsack-type budget constraints we propose practical algorithms with approximation guarantees. In addition, we perform an empirical evaluation, which demonstrates the superior performance of the proposed algorithms against strong baselines.

cs.DS↗

arXiv2022

Improved analysis of randomized SVD for top-eigenvector approximation

Computing the top eigenvectors of a matrix is a problem of fundamental interest to various fields. While the majority of the literature has focused on analyzing the reconstruction error of low-rank matrices associated with the retrieved eigenvectors, in many applications one is interested in finding one vector with high Rayleigh quotient. In this paper we study the problem of approximating the top-eigenvector. Given a symmetric matrix $\mathbf{A}$ with largest eigenvalue $λ_1$, our goal is to find a vector \hu that approximates the leading eigenvector $\mathbf{u}_1$ with high accuracy, as measured by the ratio $R(\hat{\mathbf{u}})=λ_1^{-1}{\hat{\mathbf{u}}^T\mathbf{A}\hat{\mathbf{u}}}/{\hat{\mathbf{u}}^T\hat{\mathbf{u}}}$. We present a novel analysis of the randomized SVD algorithm of \citet{halko2011finding} and derive tight bounds in many cases of interest. Notably, this is the first work that provides non-trivial bounds of $R(\hat{\mathbf{u}})$ for randomized SVD with any number of iterations. Our theoretical analysis is complemented with a thorough experimental study that confirms the efficiency and accuracy of the method.

cs.LG↗

arXiv2021

Maximizing the Diversity of Exposure in a Social Network

Social-media platforms have created new ways for citizens to stay informed and participate in public debates. However, to enable a healthy environment for information sharing, social deliberation, and opinion formation, citizens need to be exposed to sufficiently diverse viewpoints that challenge their assumptions, instead of being trapped inside filter bubbles. In this paper, we take a step in this direction and propose a novel approach to maximize the diversity of exposure in a social network. We formulate the problem in the context of information propagation, as a task of recommending a small number of news articles to selected users. We propose a realistic setting where we take into account content and user leanings, and the probability of further sharing an article. This setting allows us to capture the balance between maximizing the spread of information and ensuring the exposure of users to diverse viewpoints. The resulting problem can be cast as maximizing a monotone and submodular function subject to a matroid constraint on the allocation of articles to users. It is a challenging generalization of the influence maximization problem. Yet, we are able to devise scalable approximation algorithms by introducing a novel extension to the notion of random reverse-reachable sets. We experimentally demonstrate the efficiency and scalability of our algorithm on several real-world datasets.

cs.SI↗

arXiv2021

Workload-Aware Materialization of Junction Trees

Bayesian networks are popular probabilistic models that capture the conditional dependencies among a set of variables. Inference in Bayesian networks is a fundamental task for answering probabilistic queries over a subset of variables in the data. However, exact inference in Bayesian networks is \NP-hard, which has prompted the development of many practical inference methods. In this paper, we focus on improving the performance of the junction-tree algorithm, a well-known method for exact inference in Bayesian networks. In particular, we seek to leverage information in the workload of probabilistic queries to obtain an optimal workload-aware materialization of junction trees, with the aim to accelerate the processing of inference queries. We devise an optimal pseudo-polynomial algorithm to tackle this problem and discuss approximation schemes. Compared to state-of-the-art approaches for efficient processing of inference queries via junction trees, our methods are the first to exploit the information provided in query workloads. Our experimentation on several real-world Bayesian networks confirms the effectiveness of our techniques in speeding-up query processing.

cs.DB↗

arXiv2021

Reconciliation k-median: Clustering with Non-Polarized Representatives

We propose a new variant of the k-median problem, where the objective function models not only the cost of assigning data points to cluster representatives, but also a penalty term for disagreement among the representatives. We motivate this novel problem by applications where we are interested in clustering data while avoiding selecting representatives that are too far from each other. For example, we may want to summarize a set of news sources, but avoid selecting ideologically-extreme articles in order to reduce polarization. To solve the proposed k-median formulation we adopt the local-search algorithm of Arya et al. We show that the algorithm provides a provable approximation guarantee, which becomes constant under an assumption on the minimum number of points for each cluster. We experimentally evaluate our problem formulation and proposed algorithm on datasets inspired by the motivating applications. In particular, we experiment with data extracted from Twitter, the US Congress voting records, and popular news sources. The results show that our objective can lead to choosing less polarized groups of representatives without significant loss in representation fidelity.

cs.DS↗

arXiv2021

Maximizing diversity over clustered data

Maximum diversity aims at selecting a diverse set of high-quality objects from a collection, which is a fundamental problem and has a wide range of applications, e.g., in Web search. Diversity under a uniform or partition matroid constraint naturally describes useful cardinality or budget requirements, and admits simple approximation algorithms. When applied to clustered data, however, popular algorithms such as picking objects iteratively and performing local search lose their approximation guarantees towards maximum intra-cluster diversity because they fail to optimize the objective in a global manner. We propose an algorithm that greedily adds a pair of objects instead of a singleton, and which attains a constant approximation factor over clustered data. We further extend the algorithm to the case of monotone and submodular quality function, and under a partition matroid constraint. We also devise a technique to make our algorithm scalable, and on the way we obtain a modification that gives better solutions in practice while maintaining the approximation guarantee in theory. Our algorithm achieves excellent performance, compared to strong baselines in a mix of synthetic and real-world datasets.

cs.DS↗

arXiv2021

Discovering Dense Correlated Subgraphs in Dynamic Networks

Given a dynamic network, where edges appear and disappear over time, we are interested in finding sets of edges that have similar temporal behavior and form a dense subgraph. Formally, we define the problem as the enumeration of the maximal subgraphs that satisfy specific density and similarity thresholds. To measure the similarity of the temporal behavior, we use the correlation between the binary time series that represent the activity of the edges. For the density, we study two variants based on the average degree. For these problem variants we enumerate the maximal subgraphs and compute a compact subset of subgraphs that have limited overlap. We propose an approximate algorithm that scales well with the size of the network, while achieving a high accuracy. We evaluate our framework on both real and synthetic datasets. The results of the synthetic data demonstrate the high accuracy of the approximation and show the scalability of the framework.

cs.SI↗

arXiv2021

Query the model: precomputations for efficient inference with Bayesian Networks

Variable Elimination is a fundamental algorithm for probabilistic inference over Bayesian networks. In this paper, we propose a novel materialization method for Variable Elimination, which can lead to significant efficiency gains when answering inference queries. We evaluate our technique using real-world Bayesian networks. Our results show that a modest amount of materialization can lead to significant improvements in the running time of queries. Furthermore, in comparison with junction tree methods that also rely on materialization, our approach achieves comparable efficiency during inference using significantly lighter materialization.

cs.DB↗

arXiv2020

Finding path motifs in large temporal graphs using algebraic fingerprints

We study a family of pattern-detection problems in vertex-colored temporal graphs. In particular, given a vertex-colored temporal graph and a multiset of colors as a query, we search for temporal paths in the graph that contain the colors specified in the query. These types of problems have several applications, for example in recommending tours for tourists or detecting abnormal behavior in a network of financial transactions. For the family of pattern-detection problems we consider, we establish complexity results and design an algebraic-algorithmic framework based on constrained multilinear sieving. We demonstrate that our solution scales to massive graphs with up to a billion edges for a multiset query with five colors and up to hundred million edges for a multiset query with ten colors, despite the problems being NP-hard. Our implementation, which is publicly available, exhibits practical edge-linear scalability and is highly optimized. For instance, in a real-world graph dataset with more than six million edges and a multiset query with ten colors, we can extract an optimum solution in less than eight minutes on a Haswell desktop with four cores.

cs.DS↗

arXiv2020

Mining Dense Subgraphs with Similar Edges

When searching for interesting structures in graphs, it is often important to take into account not only the graph connectivity, but also the metadata available, such as node and edge labels, or temporal information. In this paper we are interested in settings where such metadata is used to define a similarity between edges. We consider the problem of finding subgraphs that are dense and whose edges are similar to each other with respect to a given similarity function. Depending on the application, this function can be, for example, the Jaccard similarity between the edge label sets, or the temporal correlation of the edge occurrences in a temporal graph. We formulate a Lagrangian relaxation-based optimization problem to search for dense subgraphs with high pairwise edge similarity. We design a novel algorithm to solve the problem through parametric MinCut, and provide an efficient search scheme to iterate through the values of the Lagrangian multipliers. Our study is complemented by an evaluation on real-world datasets, which demonstrates the usefulness and efficiency of the proposed approach.

cs.DS↗

arXiv2020

Diverse Rule Sets

While machine-learning models are flourishing and transforming many aspects of everyday life, the inability of humans to understand complex models poses difficulties for these models to be fully trusted and embraced. Thus, interpretability of models has been recognized as an equally important quality as their predictive power. In particular, rule-based systems are experiencing a renaissance owing to their intuitive if-then representation. However, simply being rule-based does not ensure interpretability. For example, overlapped rules spawn ambiguity and hinder interpretation. Here we propose a novel approach of inferring diverse rule sets, by optimizing small overlap among decision rules with a 2-approximation guarantee under the framework of Max-Sum diversification. We formulate the problem as maximizing a weighted sum of discriminative quality and diversity of a rule set. In order to overcome an exponential-size search space of association rules, we investigate several natural options for a small candidate set of high-quality rules, including frequent and accurate rules, and examine their hardness. Leveraging the special structure in our formulation, we then devise an efficient randomized algorithm, which samples rules that are highly discriminative and have small overlap. The proposed sampling algorithm analytically targets a distribution of rules that is tailored to our objective. We demonstrate the superior predictive power and interpretability of our model with a comprehensive empirical study against strong baselines.

cs.AI↗

arXiv2020

Explainable Classification of Brain Networks via Contrast Subgraphs

Mining human-brain networks to discover patterns that can be used to discriminate between healthy individuals and patients affected by some neurological disorder, is a fundamental task in neuroscience. Learning simple and interpretable models is as important as mere classification accuracy. In this paper we introduce a novel approach for classifying brain networks based on extracting contrast subgraphs, i.e., a set of vertices whose induced subgraphs are dense in one class of graphs and sparse in the other. We formally define the problem and present an algorithmic solution for extracting contrast subgraphs. We then apply our method to a brain-network dataset consisting of children affected by Autism Spectrum Disorder and children Typically Developed. Our analysis confirms the interestingness of the discovered patterns, which match background knowledge in the neuroscience literature. Further analysis on other classification tasks confirm the simplicity, soundness, and high explainability of our proposal, which also exhibits superior classification accuracy, to more complex state-of-the-art methods.

cs.SI↗

arXiv2020

Finding large balanced subgraphs in signed networks

Signed networks are graphs whose edges are labelled with either a positive or a negative sign, and can be used to capture nuances in interactions that are missed by their unsigned counterparts. The concept of balance in signed graph theory determines whether a network can be partitioned into two perfectly opposing subsets, and is therefore useful for modelling phenomena such as the existence of polarized communities in social networks. While determining whether a graph is balanced is easy, finding a large balanced subgraph is hard. The few heuristics available in the literature for this purpose are either ineffective or non-scalable. In this paper we propose an efficient algorithm for finding large balanced subgraphs in signed networks. The algorithm relies on signed spectral theory and a novel bound for perturbations of the graph Laplacian. In a wide variety of experiments on real-world data we show that our algorithm can find balanced subgraphs much larger than those detected by existing methods, and in addition, it is faster. We test its scalability on graphs of up to 34 million edges.

cs.SI↗