Spectral clustering and its use in bioinformatics

We formulate a discrete optimization problem that leads to a simple and informative derivation of a widely used class of spectral clustering algorithms. Regarding the algorithms as attempting to bi-partition a weighted graph with N vertices, our derivation indicates that they are inherently tuned to tolerate all partitions into two non-empty sets, independently of the cardinality of the two sets. This approach also helps to explain the difference in behaviour observed between methods based on the unnormalized and normalized graph Laplacian. We also give a direct explanation of why Laplacian eigenvectors beyond the Fiedler vector may contain fine-detail information of relevance to clustering. We show numerical results on synthetic data to support the analysis. Further, we provide examples where normalized and unnormalized spectral clustering is applied to microarray data—here the graph summarizes similarity of gene activity across different tissue samples, and accurate clustering of samples is a key task in bioinformatics.     

Some implications of a cosmological phase transition

I discuss the reasons for believing that phase transitions occurred in the very early history of the universe, and the topological structures that may have been generated thereby-in particular, the strings and monopoles. The aim is mainly pedagogical, with special emphasis on unsolved problems.     

Zur Entwicklung von Gasen

Ohne Zusammenfassung