The problem of column subset selection asks for a subset of columns from an input matrix such that the matrix can be reconstructed as accurately as possible within the span of the selected columns. A natural extension is to consider a setting where the matrix rows are partitioned into two groups, and the goal is to choose a subset of columns that minimizes the maximum reconstruction error of both groups, relative to their respective best rank-k approximation. Extending the known results of column subset selection to this fair setting is not straightforward: in certain scenarios it is unavoidable to choose columns separately for each group, resulting in double the expected column count. We propose a deterministic leverage-score sampling strategy for the fair setting and show that sampling a column subset of minimum size becomes NP-hard in the presence of two groups. Despite these negative results, we give an approximation algorithm that guarantees a solution within 1.5 times the optimal solution size. We also present practical heuristic algorithms based on rank-revealing QR factorization. Finally, we validate our methods through an extensive set of experiments using real-world data.

We study a variant of classical clustering formulations in the context of algorithmic fairness, known as diversity-aware clustering. In this variant we are given a collection of facility subsets, and a solution must contain at least a specified number of facilities from each subset while simultaneously minimizing the clustering objective (k-median or k-means). We investigate the fixed-parameter tractability of these problems and show several negative hardness and inapproximability results, even when we afford exponential running time with respect to some parameters.

When searching for information in a data collection, we are often interested not only in finding relevant items, but also in assembling a diverse set, so as to explore different concepts that are present in the data. This problem has been researched extensively. However, finding a set of items with minimal pairwise similarities can be computationally challenging, and most existing works striving for quality guarantees assume that item relatedness is measured by a distance function. Given the widespread use of similarity functions in many domains, we believe this to be an important gap in the literature. In this paper we study the problem of finding a diverse set of items, when item relatedness is measured by a similarity function. We formulate the diversification task using a flexible, broadly applicable minimization objective, consisting of the sum of pairwise similarities of the selected items and a relevance penalty term. To find good solutions we adopt a randomized rounding strategy, which is challenging to analyze because of the cardinality constraint present in our formulation. Even though this obstacle can be overcome using dependent rounding, we show that it is possible to obtain provably good solutions using an independent approach, which is faster, simpler to implement and completely parallelizable. Our analysis relies on a novel bound for the ratio of Poisson-Binomial densities, which is of independent interest and has potential implications for other combinatorial-optimization problems. We leverage this result to design an efficient randomized algorithm that provides a lower-order additive approximation guarantee. We validate our method using several benchmark datasets, and show that it consistently outperforms the greedy approaches that are commonly used in the literature.

Dimensionality reduction techniques can be employed to produce robust, cost-effective predictive models, and to enhance interpretability in exploratory data analysis. However, the models produced by many of these methods are formulated in terms of abstract factors or are too high-dimensional to facilitate insight and fit within low computational budgets.

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.

Signed graphs have been used to model interactions in social networks, which can be either positive (friendly) or negative (antagonistic). The model has been used to study polarization and other related phenomena in social networks, which can be harmful to the process of democratic deliberation in our society. An interesting and challenging task in this application domain is to detect polarized communities in signed graphs. A number of different methods have been proposed for this task. However, existing approaches aim at finding globally optimal solutions. Instead, in this paper we are interested in finding polarized communities that are related to a small set of seed nodes provided as input. Seed nodes may consist of two sets, which constitute the two sides of a polarized structure.

Signed networks contain edge annotations to indicate whether each interaction is friendly (positive edge) or antagonistic (negative edge). The model is simple but powerful and it can capture novel and interesting structural properties of real-world phenomena. The analysis of signed networks has many applications from modeling discussions in social media, to mining user reviews, and to recommending products in e-commerce sites. In this paper we consider the problem of discovering polarized communities in signed networks. In particular, we search for two communities (subsets of the network vertices) where within communities there are mostly positive edges while across communities there are mostly negative edges. We formulate this novel problem as a "discrete eigenvector'' problem, which we show to be NP-hard. We then develop two intuitive spectral algorithms: one deterministic, and one randomized with quality guarantee $\sqrtn $ (where n is the number of vertices in the graph), tight up to constant factors. We validate our algorithms against non-trivial baselines on real-world signed networks. Our experiments confirm that our algorithms produce higher quality solutions, are much faster and can scale to much larger networks than the baselines, and are able to detect ground-truth polarized communities.

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.

Dimensionality reduction is often a crucial step for the successful application of machine learning and data mining methods. One way to achieve said reduction is feature selection. Due to the impossibility of labelling many data sets, unsupervised approaches are frequently the only option. The column subset selection problem translates naturally to this purpose, and has received consider able attention over the last few years, as it provides simple linear models for data reconstruction. Existing methods, however, often achieve approximation errors that are far from the optimum. In this paper we present a novel algorithm for column subset selection that consistently outperforms state-of-the-art methods in approximation error. We present a series of key derivations that allow an efficient implementation, making it comparable in speed and in some cases faster than other algorithms. We also prove results that make it possible to deal with huge matrices, which has strong implications for other algorithms of this type in the big data field. We validate our claimsthrough experiments on a wide variety of well-known data sets.