Coarse-refinement dilemma: on generalization bounds for data clustering

• Autor(a) principal: Vaz, Yule
• Data de Publicação: 2020
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/55/55134/tde-18112020-130229/
• topic: Agrupamento de dados; Agrupamento hierarquico; Algorithmic stability; Data clustering; Estabilidade algorítmica; Hierarchical clustering; Homologia persistente; Mudança de conceito topológico; Persistent homology; Topological concept drift

Machine Learning (ML) is typically organized into two main paradigms: (i) the Supervised Machine Learning (SML) to identify patterns from pre-labeled data, in which a loss function is used to adapt the corresponding model; and, (ii) the Unsupervised Machine Learning (UML) to organize data points in the absence of labels, taking similarity relations among elements into account. SML relies on well-consolidated theoretical frameworks, such as the Statistical Learning Theory (SLT) and the Algorithmic Stability (AS) to define assumptions, properties and convergence guarantees, allowing the comparison of different methods and, consequently, their improvements. Complementary, UML has been supported by investigations on Data Clustering (DC) and Hierarchical Clustering (HC) in order to define properties and improve their characterizations. Specifically, Kleinberg stated richness, scale-invariance and partition consistency as the necessary properties to define the DC problem, proving they do not hold simultaneously, while Ackerman, Ben-David and Loker explored other properties such as locality, and Carlsson and Mémoli developed stability and consistency frameworks for HC from metric spaces. To bring an additional contribution to UML, we considered topological spaces to design more general theoretical results given: (i) the invariance on topological spaces, more precisely isomorphism of homology groups, guarantees the properties of scale-invariance, partition consistency and locality; and (ii) this same invariance is inherited along less general spaces, such as the metric, thus allowing a more abstract clustering representation. Taking such invariance into account, we demonstrated that over-refined topologies endowed by DC and HC models lead to non-consistency in terms of their associated homology groups and, on the other hand, over-coarsed topologies devise consistent but unrepresentative homology groups, a phenomenon that we referred to as the Coarse-Refinement Dilemma (CRD). We then formulated DC and HC problems by employing Carlsson and Zomorodians bidimensional persistent homology, with the first dimension corresponding to the HC levels and the second to the inclusion of new data, thus allowing a probabilistic study based on martingales process and subsequent formalization of generalization bounds. From such results, we contributed with the related work by: (i) defining lower and upper bounds for Carlsson and Mémolis metric consistency; (ii) showing that Kleinbergs richness axiom must be relaxed otherwise over-refined or over-coarsed clusterings could be obtained; and, finally, (iii) defining unexpected changes in consistent topologies using what we named as Topological Concept Drift (TCD). An extensive set of experiments was performed to analyze the CRD and the TCD, including a brief study of a real-world scenario involving text documents. Results corroborated the usefulness in representing DC and HC problems using topological spaces, in detecting topology changes and the existence of CRD.