Improvements to k-means clustering

Working with huge amount of data and learning from it by extracting useful information is one of the prime challenges of the Internet era. Machine learning algorithms; provide an automatic and easy way to accomplish such tasks. These algorithms are classified into supervised, unsupervised, semi-supervised algorithms. Some of the most used algorithms belong to the class of unsupervised learning as training becomes a challenge for many practical applications. Machine learning algorithms for which the classes of input are unknown are called unsupervised algorithms. The k-means algorithm is one of the simplest and predominantly used algorithms for unsupervised learning procedure clustering. The k-means algorithm works grouping similar data based on some measure. The k in k-means denotes the number of such groups available. This study starts from the standard k-means algorithm and goes through some of the algorithmic improvements suggested in the machine learning and data mining literature. Traditional k-means algorithm is an iterative refinement algorithm with an assignment step and an update step. The distances from each of the clusters centroids are calculated and iteratively refined. The computational complexity of k –means algorithm mainly arises from the distance calculation or the so called nearest –neighbor query. Certain formulation of the k-means calculations are NP hard. For a dataset with dimension and values the computational complexity is for a single iteration. Some of the k-means clustering procedures take several iterations and still fail to produce an acceptable clustering error. Several authors have studied different ways of reducing the complexity of k-means algorithm in the literature. Particularly, this study focuses mainly on the algorithmic improvements suggested in the related works of Hamerly and Elkan. These algorithmic improvements are chosen to be studied as they are generic in nature and could be applied for many available datasets. The improved algorithms due to Elkan and Hamerly are applied to various available datasets and their computation performances are presented

Bridge between worlds: relating position and disposition in the mathematical field

Using ethnographic observations and interview based research I document the production of research mathematics in four European research institutes, interviewing 45 mathematicians from three areas of pure mathematics: topology, algebraic geometry and differential geometry. I use Bourdieu's notions of habitus, field and practice to explore how mathematicians come to perceive and interact with abstract mathematical spaces and constructions. Perception of mathematical reality, I explain, depends upon enculturation within a mathematical discipline. This process of socialisation involves positioning an individual within a field of production. Within a field mathematicians acquire certain structured sets of dispositions which constitute habitus, and these habitus then provide both perspectives and perceptual lenses through which to construe mathematical objects and spaces. I describe how mathematical perception is built up through interactions within three domains of experience: physical spaces, conceptual spaces and discourse spaces. These domains share analogous structuring schemas, which are related through Lakoff and Johnson's notions of metaphorical mappings and image schemas. Such schemas are mobilised during problem solving and proof construction, in order to guide mathematicians' intuitions; and are utilised during communicative acts, in order to create common ground and common reference frames. However, different structuring principles are utilised according to the contexts in which the act of knowledge production or communication take place. The degree of formality, privacy or competitiveness of environments affects the presentation of mathematicians' selves and ideas. Goffman's concept of interaction frame, front-stage and backstage are therefore used to explain how certain positions in the field shape dispositions, and lead to the realisation of different structuring schemas or scripts. I use Sewell's qualifications of Bourdieu's theories to explore the multiplicity of schemas present within mathematicians' habitus, and detail how they are given expression through craftwork and bricolage. I argue that mathematicians' perception of mathematical phenomena are dependent upon their positions and relations. I develop the notion of social space, providing definitions of such spaces and how they are generated, how positions are determined, and how individuals reposition within space through acquisition of capital