Bruno Ordozgoiti

Bruno Ordozgoiti, Antonis Matakos, Aristides Gionis

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.

Antonis Matakos, Bruno Ordozgoiti, Suhas Thejaswi

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.

Suhas Thejaswi, Ameet Gadekar, Bruno Ordozgoiti et al.

In this work, we study diversity-aware clustering problems where the data points are associated with multiple attributes resulting in intersecting groups. A clustering solution needs to ensure that the number of chosen cluster centers from each group should be within the range defined by a lower and upper bound threshold for each group, while simultaneously minimizing the clustering objective, which can be either $k$-median, $k$-means or $k$-supplier. We study the computational complexity of the proposed problems, offering insights into their NP-hardness, polynomial-time inapproximability, and fixed-parameter intractability. We present parameterized approximation algorithms with approximation ratios $1+ \frac{2}{e} + ε\approx 1.736$, $1+\frac{8}{e} + ε\approx 3.943$, and $5$ for diversity-aware $k$-median, diversity-aware $k$-means and diversity-aware $k$-supplier, respectively. Assuming Gap-ETH, the approximation ratios are tight for the diversity-aware $k$-median and diversity-aware $k$-means problems. Our results imply the same approximation factors for their respective fair variants with disjoint groups -- fair $k$-median, fair $k$-means, and fair $k$-supplier -- with lower bound requirements.

Bruno Ordozgoiti, Lluís A. Belanche Muñoz

Kernel functions are a powerful tool to enhance the $k$-means clustering algorithm via the kernel trick. It is known that the parameters of the chosen kernel function can have a dramatic impact on the result. In supervised settings, these can be tuned via cross-validation, but for clustering this is not straightforward and heuristics are usually employed. In this paper we study the impact of kernel parameters on kernel $k$-means. In particular, we derive a lower bound, tight up to constant factors, below which the parameter of the RBF kernel will render kernel $k$-means meaningless. We argue that grid search can be ineffective for hyperparameter search in this context and propose an alternative algorithm for this purpose. In addition, we offer an efficient implementation based on fast approximate exponentiation with provable quality guarantees. Our experimental results demonstrate the ability of our method to efficiently reveal a rich and useful set of hyperparameter values.

Han Xiao, Bruno Ordozgoiti, Aristides Gionis

Signed graphs have been used to model interactions in social net-works, 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. In this paper we formulate the problem of finding local polarized communities in signed graphs as a locally-biased eigen-problem. By viewing the eigenvector associated with the smallest eigenvalue of the Laplacian matrix as the solution of a constrained optimization problem, we are able to incorporate the local information as an additional constraint. In addition, we show that the locally-biased vector can be used to find communities with approximation guarantee with respect to a local analogue of the Cheeger constant on signed graphs. By exploiting the sparsity in the input graph, an indicator vector for the polarized communities can be found in time linear to the graph size. Our experiments on real-world networks validate the proposed algorithm and demonstrate its usefulness in finding local structures in this semi-supervised manner.

Bruno Ordozgoiti, Alberto Mozo, Jesús García López de Lacalle

The Column Subset Selection Problem provides a natural framework for unsupervised feature selection. Despite being a hard combinatorial optimization problem, there exist efficient algorithms that provide good approximations. The drawback of the problem formulation is that it incorporates no form of regularization, and is therefore very sensitive to noise when presented with scarce data. In this paper we propose a regularized formulation of this problem, and derive a correct greedy algorithm that is similar in efficiency to existing greedy methods for the unregularized problem. We study its adequacy for feature selection and propose suitable formulations. Additionally, we derive a lower bound for the error of the proposed problems. Through various numerical experiments on real and synthetic data, we demonstrate the significantly increased robustness and stability of our method, as well as the improved conditioning of its output, all while remaining efficient for practical use.

Udi Margolin, Alberto Mozo, Bruno Ordozgoiti et al.

5G networks are expected to be more dynamic and chaotic in their structure than current networks. With the advent of Network Function Virtualization (NFV), Network Functions (NF) will no longer be tightly coupled with the hardware they are running on, which poses new challenges in network management. Noisy neighbor is a term commonly used to describe situations in NFV infrastructure where an application experiences degradation in performance due to the fact that some of the resources it needs are occupied by other applications in the same cloud node. These situations cannot be easily identified using straightforward approaches, which calls for the use of sophisticated methods for NFV infrastructure management. In this paper we demonstrate how Machine Learning (ML) techniques can be used to identify such events. Through experiments using data collected at real NFV infrastructure, we show that standard models for automated classification can detect the noisy neighbor phenomenon with an accuracy of more than 90% in a simple scenario.