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High-dimensional data sets can be difficult to visualize and analyze, while data in low-dimensional space tend to be more accessible. In order to aid visualization of the underlying structure of a dataset, the dimension of the dataset is reduced. The simplest approach to accomplish this task of dimensionality reduction is by a random projection of the data. Even though this approach allows some degree of visualization of the underlying structure, it is possible to lose more interesting underlying structure within the data. In order to address this concern, various supervised and unsupervised linear dimensionality reduction algorithms have been designed, such as Principal Component Analysis and Linear Discriminant Analysis. These methods can be powerful, but often miss important non-linear structure in the data. In this thesis, manifold learning approaches to dimensionality reduction are developed. These approaches combine both linear and non-linear methods of dimension reduction.

Sequences of event intervals occur in several application domains, while their inherent complexity hinders scalable solutions to tasks such as clustering and classification. In this thesis, we propose a novel spectral embedding representation of event interval sequences that relies on bipartite graphs. More concretely, each event interval sequence is represented by a bipartite graph by following three main steps: (1) creating a hash table that can quickly convert a collection of event interval sequences into a bipartite graph representation, (2) creating and regularizing a bi-adjacency matrix corresponding to the bipartite graph, (3) defining a spectral embedding mapping on the bi-adjacency matrix. In addition, we show that substantial improvements can be achieved with regard to classification performance through pruning parameters that capture the nature of the relations formed by the event intervals. We demonstrate through extensive experimental evaluation on five real-world datasets that our approach can obtain runtime speedups of up to two orders of magnitude compared to other state-of-the-art methods and similar or better clustering and classification performance. / Sekvenser av händelsesintervall förekommer i flera applikationsdomäner, medan deras inneboende komplexitet hindrar skalbara lösningar på uppgifter som kluster och klassificering. I den här avhandlingen föreslår vi en ny spektral inbäddningsrepresentation av händelsens intervallsekvenser som förlitar sig på bipartitgrafer. Mer konkret representeras varje händelsesintervalsekvens av en bipartitgraf genom att följa tre huvudsteg: (1) skapa en hashtabell som snabbt kan konvertera en samling händelsintervalsekvenser till en bipartig grafrepresentation, (2) skapa och reglera en bi-adjacency-matris som motsvarar bipartitgrafen, (3) definiera en spektral inbäddning på bi-adjacensmatrisen. Dessutom visar vi att väsentliga förbättringar kan uppnås med avseende på klassificeringsprestanda genom beskärningsparametrar som fångar arten av relationerna som bildas av händelsesintervallen. Vi demonstrerar genom omfattande experimentell utvärdering på fem verkliga datasätt att vår strategi kan erhålla runtime-hastigheter på upp till två storlekar jämfört med andra modernaste metoder och liknande eller bättre kluster- och klassificerings- prestanda.