On similarity prediction and pairwise clustering

We consider the problem of clustering a finite set of items from pairwise similarity information. Unlike what is done in the literature on this subject, we do so in a passive learning setting, and with no specific constraints on the cluster shapes other than their size. We investigate the problem in different settings: i. an online setting, where we provide a tight characterization of the prediction complexity in the mistake bound model, and ii. a standard stochastic batch setting, where we give tight upper and lower bounds on the achievable generalization error. Prediction performance is measured both in terms of the ability to recover the similarity function encoding the hidden clustering and in terms of how well we classify each item within the set. The proposed algorithms are time efficient

EpiAlign: an alignment-based bioinformatic tool for comparing chromatin state sequences.

The availability of genome-wide epigenomic datasets enables in-depth studies of epigenetic modifications and their relationships with chromatin structures and gene expression. Various alignment tools have been developed to align nucleotide or protein sequences in order to identify structurally similar regions. However, there are currently no alignment methods specifically designed for comparing multi-track epigenomic signals and detecting common patterns that may explain functional or evolutionary similarities. We propose a new local alignment algorithm, EpiAlign, designed to compare chromatin state sequences learned from multi-track epigenomic signals and to identify locally aligned chromatin regions. EpiAlign is a dynamic programming algorithm that novelly incorporates varying lengths and frequencies of chromatin states. We demonstrate the efficacy of EpiAlign through extensive simulations and studies on the real data from the NIH Roadmap Epigenomics project. EpiAlign is able to extract recurrent chromatin state patterns along a single epigenome, and many of these patterns carry cell-type-specific characteristics. EpiAlign can also detect common chromatin state patterns across multiple epigenomes, and it will serve as a useful tool to group and distinguish epigenomic samples based on genome-wide or local chromatin state patterns

Randomized Dimensionality Reduction for k-means Clustering

We study the topic of dimensionality reduction for $k$-means clustering. Dimensionality reduction encompasses the union of two approaches: \emph{feature selection} and \emph{feature extraction}. A feature selection based algorithm for $k$-means clustering selects a small subset of the input features and then applies $k$-means clustering on the selected features. A feature extraction based algorithm for $k$-means clustering constructs a small set of new artificial features and then applies $k$-means clustering on the constructed features. Despite the significance of $k$-means clustering as well as the wealth of heuristic methods addressing it, provably accurate feature selection methods for $k$-means clustering are not known. On the other hand, two provably accurate feature extraction methods for $k$-means clustering are known in the literature; one is based on random projections and the other is based on the singular value decomposition (SVD). This paper makes further progress towards a better understanding of dimensionality reduction for $k$-means clustering. Namely, we present the first provably accurate feature selection method for $k$-means clustering and, in addition, we present two feature extraction methods. The first feature extraction method is based on random projections and it improves upon the existing results in terms of time complexity and number of features needed to be extracted. The second feature extraction method is based on fast approximate SVD factorizations and it also improves upon the existing results in terms of time complexity. The proposed algorithms are randomized and provide constant-factor approximation guarantees with respect to the optimal $k$-means objective value.Comment: IEEE Transactions on Information Theory, to appea