We derive a robust Euclidean embedding procedure based on semidefinite programming that may be used in place of the popular classical multidimensional scaling (cMDS) algorithm. We motivate this algorithm by arguing that cMDS is not particularly robust and has several other deficiencies. Generalpurpose semidefinite programming solvers are too memory intensive for medium to large sized applications, so we also describe a fast subgradient-based implementation of the robust algorithm. Additionally, since cMDS is often used for dimensionality reduction, we provide an in-depth look at reducing dimensionality with embedding procedures. In particular, we show that it is NP-hard to find optimal low-dimensional embeddings under a variety of cost functions.

We present a data structure enabling efficient nearest neighbor (NN) retrieval for bregman divergences. The family of bregman divergences includes many popular dissimilarity measures including KL-divergence (relative entropy), Mahalanobis distance, and ItakuraSaito divergence. These divergences present a challenge for efficient NN retrieval because they are not, in general, metrics, for which most NN data structures are designed. The data structure introduced in this work shares the same basic structure as the popular metric ball tree, but employs convexity properties of bregman divergences in place of the triangle inequality. Experiments demonstrate speedups over brute-force search of up to several orders of magnitude.

We develop methods for accelerating metric similarity search that are effective on modern hardware. Our algorithms factor into easily parallelizable components, making them simple to deploy and efficient on multicore CPUs and GPUs. Despite the simple structure of our algorithms, their search performance is provably sublinear in the size of the database, with a factor dependent only on its intrinsic dimensionality. We demonstrate that our methods provide substantial speedups on a range of datasets and hardware platforms. In particular, we present results on a 48-core server machine, on graphics hardware, and on a multicore desktop.

Due to recent advances in genotyping technologies, mapping phenotypes to single loci in the genome has become a standard technique in statistical genetics. However, one-locus mapping fails to explain much of the phenotypic variance in complex traits. Here, we present GLIDE, which maps phenotypes to pairs of genetic loci and systematically searches for the epistatic interactions expected to reveal part of this missing heritability. GLIDE makes use of the computational power of consumer-grade graphics cards to detect such interactions via linear regression. This enabled us to conduct a systematic two-locus mapping study on seven disease data sets from the Wellcome Trust Case Control Consortium and on in-house hippocampal volume data in 6 h per data set, while current single CPU-based approaches require more than a year’s time to complete the same task.