Local Guarantees in Graph Cuts and Clustering

Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Min $s-t$ Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization. Here, we are given a graph with edges labeled $+$ or $-$ and the goal is to produce a clustering that agrees with the labels as much as possible: $+$ edges within clusters and $-$ edges across clusters. The classical approach towards Correlation Clustering (and other graph cut problems) is to optimize a global objective. We depart from this and study local objectives: minimizing the maximum number of disagreements for edges incident on a single node, and the analogous max min agreements objective. This naturally gives rise to a family of basic min-max graph cut problems. A prototypical representative is Min Max $s-t$ Cut: find an $s-t$ cut minimizing the largest number of cut edges incident on any node. We present the following results: $(1)$ an $O(\sqrt{n})$-approximation for the problem of minimizing the maximum total weight of disagreement edges incident on any node (thus providing the first known approximation for the above family of min-max graph cut problems), $(2)$ a remarkably simple $7$-approximation for minimizing local disagreements in complete graphs (improving upon the previous best known approximation of $48$), and $(3)$ a $1/(2+\varepsilon)$-approximation for maximizing the minimum total weight of agreement edges incident on any node, hence improving upon the $1/(4+\varepsilon)$-approximation that follows from the study of approximate pure Nash equilibria in cut and party affiliation games

Bayesian Approaches For Modeling Variation

A core focus of statistics is determining how much of the variation in data may be attributed to the signal of interest, and how much to noise. When the sources of variation are many and complex, a Bayesian approach to data analysis offers a number of advantages. In this thesis, we propose and implement new Bayesian methods for modeling variation in two general settings. The first setting is high-dimensional linear regression where the unknown error variance is also of interest. Here, we show that a commonly used class of conjugate shrinkage priors can lead to underestimation of the error variance. We then extend the Spike-and-Slab Lasso (SSL, Rockova and George, 2018) to the unknown variance case, using an alternative, independent prior framework. This extended procedure outperforms both the fixed variance approach and alternative penalized likelihood methods on both simulated and real data. For the second setting, we move from univariate response data where the predictors are known, to multivariate response data in which potential predictors are unobserved. In this setting, we first consider the problem of biclustering, where a motivating example is to find subsets of genes which have similar expression in a subset of patients. For this task, we propose a new biclustering method called Spike-and-Slab Lasso Biclustering (SSLB). SSLB utilizes the SSL prior to find a doubly-sparse factorization of the data matrix via a fast EM algorithm. Applied to both a microarray dataset and a single-cell RNA-sequencing dataset, SSLB recovers biologically meaningful signal in the data. The second problem we consider in this setting is nonlinear factor analysis. The goal here is to find low-dimensional, unobserved ``factors\u27\u27 which drive the variation in the high-dimensional observed data in a potentially nonlinear fashion. For this purpose, we develop factor analysis BART (faBART), an MCMC algorithm which alternates sampling from the posterior of (a) the factors and (b) a functional approximation to the mapping from the factors to the data. The latter step utilizes Bayesian Additive Regression Trees (BART, Chipman et al., 2010). On a variety of simulation settings, we demonstrate that with only the observed data as the input, faBART is able to recover both the unobserved factors and the nonlinear mapping

Single-Pass Pivot Algorithm for Correlation Clustering. Keep it simple!

We show that a simple single-pass semi-streaming variant of the Pivot algorithm for Correlation Clustering gives a (3 + {\epsilon})-approximation using O(n/{\epsilon}) words of memory. This is a slight improvement over the recent results of Cambus, Kuhn, Lindy, Pai, and Uitto, who gave a (3 + {\epsilon})-approximation using O(n log n) words of memory, and Behnezhad, Charikar, Ma, and Tan, who gave a 5-approximation using O(n) words of memory. One of the main contributions of this paper is that both the algorithm and its analysis are very simple, and also the algorithm is easy to implement

User-Specific Bicluster-based Collaborative Filtering

Tese de mestrado, Ciência de Dados, Universidade de Lisboa, Faculdade de Ciências, 2020Collaborative Filtering is one of the most popular and successful approaches for Recommender Systems. However, some challenges limit the effectiveness of Collaborative Filtering approaches when dealing with recommendation data, mainly due to the vast amounts of data and their sparse nature. In order to improve the scalability and performance of Collaborative Filtering approaches, several authors proposed successful approaches combining Collaborative Filtering with clustering techniques. In this work, we study the effectiveness of biclustering, an advanced clustering technique that groups rows and columns simultaneously, in Collaborative Filtering. When applied to the classic U-I interaction matrices, biclustering considers the duality relations between users and items, creating clusters of users who are similar under a particular group of items. We propose USBCF, a novel biclustering-based Collaborative Filtering approach that creates user specific models to improve the scalability of traditional CF approaches. Using a realworld dataset, we conduct a set of experiments to objectively evaluate the performance of the proposed approach, comparing it against baseline and state-of-the-art Collaborative Filtering methods. Our results show that the proposed approach can successfully suppress the main limitation of the previously proposed state-of-the-art biclustering-based Collaborative Filtering (BBCF) since BBCF can only output predictions for a small subset of the system users and item (lack of coverage). Moreover, USBCF produces rating predictions with quality comparable to the state-of-the-art approaches

Biclustering electronic health records to unravel disease presentation patterns

Tese de mestrado, Ciência de Dados, Universidade de Lisboa, Faculdade de Ciências, 2019A Esclerose Lateral Amiotrófica (ELA) é uma doença neurodegenerativa heterogénea com padrões de apresentação altamente variáveis. Dada a natureza heterogénea dos doentes com ELA, aquando do diagnóstico os clínicos normalmente estimam a progressão da doença utilizando uma taxa de decaimento funcional, calculada com base na Escala Revista de Avaliação Funcional de ELA (ALSFRS-R). A utilização de modelos de Aprendizagem Automática que consigam lidar com este padrões complexos é necessária para compreender a doença, melhorar os cuidados aos doentes e a sua sobrevivência. Estes modelos devem ser explicáveis para que os clínicos possam tomar decisões informadas. Desta forma, o nosso objectivo é descobrir padrões de apresentação da doença, para isso propondo uma nova abordagem de Prospecção de Dados: Descoberta de Meta-atributos Discriminativos (DMD), que utiliza uma combinação de Biclustering, Classificação baseada em Biclustering e Prospecção de Regras de Associação para Classificação. Estes padrões (chamados de Meta-atributos) são compostos por subconjuntos de atributos discriminativos conjuntamente com os seus valores, permitindo assim distinguir e caracterizar subgrupos de doentes com padrões similares de apresentação da doença. Os Registos de Saúde Electrónicos (RSE) utilizados neste trabalho provêm do conjunto de dados JPND ONWebDUALS (ONTology-based Web Database for Understanding Amyotrophic Lateral Sclerosis), composto por questões standardizadas acerca de factores de risco, mutações genéticas, atributos clínicos ou informação de sobrevivência de uma coorte de doentes e controlos seguidos pelo consórcio ENCALS (European Network to Cure ALS), que inclui vários países europeus, incluindo Portugal. Nesta tese a metodologia proposta foi utilizada na parte portuguesa do conjunto de dados ONWebDUALS para encontrar padrões de apresentação da doença que: 1) distinguissem os doentes de ELA dos seus controlos e 2) caracterizassem grupos de doentes de ELA com diferentes taxas de progressão (categorizados em grupos Lentos, Neutros e Rápidos). Nenhum padrão coerente emergiu das experiências efectuadas para a primeira tarefa. Contudo, para a segunda tarefa os padrões encontrados para cada um dos três grupos de progressão foram reconhecidos e validados por clínicos especialistas em ELA, como sendo características relevantes de doentes com progressão Lenta, Neutra e Rápida. Estes resultados sugerem que a nossa abordagem genérica baseada em Biclustering tem potencial para identificar padrões de apresentação noutros problemas ou doenças semelhantes.Amyotrophic Lateral Sclerosis (ALS) is a heterogeneous neurodegenerative disease with a high variability of presentation patterns. Given the heterogeneous nature of ALS patients and targeting a better prognosis, clinicians usually estimate disease progression at diagnosis using the rate of decay computed from the Revised ALS Functional Rating Scale (ALSFRS-R). In this context, the use of Machine Learning models able to unravel the complexity of disease presentation patterns is paramount for disease understanding, targeting improved patient care and longer survival times. Furthermore, explainable models are vital, since clinicians must be able to understand the reasoning behind a given model’s result before making a decision that can impact a patient’s life. Therefore we aim at unravelling disease presentation patterns by proposing a new Data Mining approach called Discriminative Meta-features Discovery (DMD), which uses a combination of Biclustering, Biclustering-based Classification and Class Association Rule Mining. These patterns (called Metafeatures) are composed of discriminative subsets of features together with their values, allowing to distinguish and characterize subgroups of patients with similar disease presentation patterns. The Electronic Health Record (EHR) data used in this work comes from the JPND ONWebDUALS (ONTology-based Web Database for Understanding Amyotrophic Lateral Sclerosis) dataset, comprised of standardized questionnaire answers regarding risk factors, genetic mutations, clinical features and survival information from a cohort of patients and controls from ENCALS (European Network to Cure ALS), a consortium of diverse European countries, including Portugal. In this work the proposed methodology was used on the ONWebDUALS Portuguese EHR data to find disease presentation patterns that: 1) distinguish the ALS patients from their controls and 2) characterize groups of ALS patients with different progression rates (categorized into Slow, Neutral and Fast groups). No clear pattern emerged from the experiments performed for the first task. However, in the second task the patterns found for each of the three progression groups were recognized and validated by ALS expert clinicians, as being relevant characteristics of slow, neutral and fast progressing patients. These results suggest that our generic Biclustering approach is a promising way to unravel disease presentation patterns and could be applied to similar problems and other diseases

On Solving Selected Nonlinear Integer Programming Problems in Data Mining, Computational Biology, and Sustainability

This thesis consists of three essays concerning the use of optimization techniques to solve four problems in the fields of data mining, computational biology, and sustainable energy devices. To the best of our knowledge, the particular problems we discuss have not been previously addressed using optimization, which is a specific contribution of this dissertation. In particular, we analyze each of the problems to capture their underlying essence, subsequently demonstrating that each problem can be modeled as a nonlinear (mixed) integer program. We then discuss the design and implementation of solution techniques to locate optimal solutions to the aforementioned problems. Running throughout this dissertation is the theme of using mixed-integer programming techniques in conjunction with context-dependent algorithms to identify optimal and previously undiscovered underlying structure