Common complex diseases are likely influenced by the interplay of hundreds, or even thousands, of genetic variants. Converging evidence shows that genetic variants with low marginal effects (LME) play an important role in disease development. Despite their potential significance, discovering LME genetic variants and assessing their joint association on high dimensional data (e.g., genome wide association studies) remain a great challenge. To facilitate joint association analysis among a large ensemble of LME genetic variants, we proposed a computationally efficient and powerful approach, which we call Trees Assembling Mann whitney (TAMW). Through simulation studies and an empirical data application, we found that TAMW outperformed multifactor dimensionality reduction (MDR) and the likelihood ratio based Mann whitney approach (LRMW) when the underlying complex disease involves multiple LME loci and their interactions. For instance, in a simulation with 20 interacting LME loci, TAMW attained a higher power (power=0.931) than both MDR (power=0.599) and LRMW (power=0.704). In an empirical study of 29 known Crohn's disease (CD) loci, TAMW also identified a stronger joint association with CD than those detected by MDR and LRMW. Finally, we applied TAMW to Wellcome Trust CD GWAS to conduct a genome wide analysis. The analysis of 459K single nucleotide polymorphisms was completed in 40 hours using parallel computing, and revealed a joint association predisposing to CD (p-value=2.763e-19). Further analysis of the newly discovered association suggested that 13 genes, such as ATG16L1 and LACC1, may play an important role in CD pathophysiological and etiological processes.

We introduce an adaptive output-sensitive Metropolis-Hastings algorithm for probabilistic models expressed as programs, Adaptive Lightweight Metropolis-Hastings (AdLMH). The algorithm extends Lightweight Metropolis-Hastings (LMH) by adjusting the probabilities of proposing random variables for modification to improve convergence of the program output. We show that AdLMH converges to the correct equilibrium distribution and compare convergence of AdLMH to that of LMH on several test problems to highlight different aspects of the adaptation scheme. We observe consistent improvement in convergence on the test problems.

We investigate the difference between using an $\ell_1$ penalty versus an $\ell_1$ constraint in generalized eigenvalue problems, such as principal component analysis and discriminant analysis. Our main finding is that an $\ell_1$ penalty may fail to provide very sparse solutions; a severe disadvantage for variable selection that can be remedied by using an $\ell_1$ constraint. Our claims are supported both by empirical evidence and theoretical analysis. Finally, we illustrate the advantages of an $\ell_1$ constraint in the context of discriminant analysis and principal component analysis.

Data-aware methods for dimensionality reduction and matrix decomposition aim to find low-dimensional structure in a collection of data. Classical approaches discover such structure by learning a basis that can efficiently express the collection. Recently, "self expression", the idea of using a small subset of data vectors to represent the full collection, has been developed as an alternative to learning. Here, we introduce a scalable method for computing sparse SElf-Expressive Decompositions (SEED). SEED is a greedy method that constructs a basis by sequentially selecting incoherent vectors from the dataset. After forming a basis from a subset of vectors in the dataset, SEED then computes a sparse representation of the dataset with respect to this basis. We develop sufficient conditions under which SEED exactly represents low rank matrices and vectors sampled from a unions of independent subspaces. We show how SEED can be used in applications ranging from matrix approximation and denoising to clustering, and apply it to numerous real-world datasets. Our results demonstrate that SEED is an attractive low-complexity alternative to other sparse matrix factorization approaches such as sparse PCA and self-expressive methods for clustering.

In modern scientific research, massive datasets with huge numbers of observations are frequently encountered. To facilitate the computational process, a divide-and-conquer scheme is often used for the analysis of big data. In such a strategy, a full dataset is first split into several manageable segments; the final output is then averaged from the individual outputs of the segments. Despite its popularity in practice, it remains largely unknown that whether such a distributive strategy provides valid theoretical inferences to the original data. In this paper, we address this fundamental issue for the distributed kernel regression (DKR), where the algorithmic feasibility is measured by the generalization performance of the resulting estimator. To justify DKR, a uniform convergence rate is needed for bounding the generalization error over the individual outputs, which brings new and challenging issues in the big data setup. Under mild conditions, we show that, with a proper number of segments, DKR leads to an estimator that is generalization consistent to the unknown regression function. The obtained results justify the method of DKR and shed light on the feasibility of using other distributed algorithms for processing big data. The promising preference of the method is supported by both simulation and real data examples.

Density mode clustering is a nonparametric clustering method. The clusters are the basins of attraction of the modes of a density estimator. We study the risk of mode-based clustering. We show that the clustering risk over the cluster cores --- the regions where the density is high --- is very small even in high dimensions. And under a low noise condition, the overall cluster risk is small even beyond the cores, in high dimensions.

In biospectroscopy, suitably annotated and statistically independent samples (e. g. patients, batches, etc.) for classifier training and testing are scarce and costly. Learning curves show the model performance as function of the training sample size and can help to determine the sample size needed to train good classifiers. However, building a good model is actually not enough: the performance must also be proven. We discuss learning curves for typical small sample size situations with 5 - 25 independent samples per class. Although the classification models achieve acceptable performance, the learning curve can be completely masked by the random testing uncertainty due to the equally limited test sample size. In consequence, we determine test sample sizes necessary to achieve reasonable precision in the validation and find that 75 - 100 samples will usually be needed to test a good but not perfect classifier. Such a data set will then allow refined sample size planning on the basis of the achieved performance. We also demonstrate how to calculate necessary sample sizes in order to show the superiority of one classifier over another: this often requires hundreds of statistically independent test samples or is even theoretically impossible. We demonstrate our findings with a data set of ca. 2550 Raman spectra of single cells (five classes: erythrocytes, leukocytes and three tumour cell lines BT-20, MCF-7 and OCI-AML3) as well as by an extensive simulation that allows precise determination of the actual performance of the models in question.

We introduce and develop a novel approach to outlier detection based on adaptation of random subspace learning. Our proposed method handles both high-dimension low-sample size and traditional low-dimensional high-sample size datasets. Essentially, we avoid the computational bottleneck of techniques like minimum covariance determinant (MCD) by computing the needed determinants and associated measures in much lower dimensional subspaces. Both theoretical and computational development of our approach reveal that it is computationally more efficient than the regularized methods in high-dimensional low-sample size, and often competes favorably with existing methods as far as the percentage of correct outlier detection is concerned.

In this chapter we review the main literature related to kernel spectral clustering (KSC), an approach to clustering cast within a kernel-based optimization setting. KSC represents a least-squares support vector machine based formulation of spectral clustering described by a weighted kernel PCA objective. Just as in the classifier case, the binary clustering model is expressed by a hyperplane in a high dimensional space induced by a kernel. In addition, the multi-way clustering can be obtained by combining a set of binary decision functions via an Error Correcting Output Codes (ECOC) encoding scheme. Because of its model-based nature, the KSC method encompasses three main steps: training, validation, testing. In the validation stage model selection is performed to obtain tuning parameters, like the number of clusters present in the data. This is a major advantage compared to classical spectral clustering where the determination of the clustering parameters is unclear and relies on heuristics. Once a KSC model is trained on a small subset of the entire data, it is able to generalize well to unseen test points. Beyond the basic formulation, sparse KSC algorithms based on the Incomplete Cholesky Decomposition (ICD) and $L_0$, $L_1, L_0 + L_1$, Group Lasso regularization are reviewed. In that respect, we show how it is possible to handle large scale data. Also, two possible ways to perform hierarchical clustering and a soft clustering method are presented. Finally, real-world applications such as image segmentation, power load time-series clustering, document clustering and big data learning are considered.

Kernel matrices are popular in machine learning and scientific computing, but they are limited by their quadratic complexity in both construction and storage. It is well-known that as one varies the kernel parameter, e.g., the width parameter in radial basis function kernels, the kernel matrix changes from a smooth low-rank kernel to a diagonally-dominant and then fully-diagonal kernel. Low-rank approximation methods have been widely-studied, mostly in the first case, to reduce the memory storage and the cost of computing matrix-vector products. Here, we use ideas from scientific computing to propose an extension of these methods to situations where the matrix is not well-approximated by a low-rank matrix. In particular, we construct an efficient block low-rank approximation method---which we call the Block Basis Factorization---and we show that it has $\mathcal{O}(n)$ complexity in both time and memory. Our method works for a wide range of kernel parameters, extending the domain of applicability of low-rank approximation methods, and our empirical results demonstrate the stability (small standard deviation in error) and superiority over current state-of-art kernel approximation algorithms.