In Bipartite Correlation Clustering (BCC) we are given a complete bipartite graph $G$ with `+' and `-' edges, and we seek a vertex clustering that maximizes the number of agreements: the number of all `+' edges within clusters plus all `-' edges cut across clusters. BCC is known to be NP-hard. We present a novel approximation algorithm for $k$-BCC, a variant of BCC with an upper bound $k$ on the number of clusters. Our algorithm outputs a $k$-clustering that provably achieves a number of agreements within a multiplicative ${(1-\delta)}$-factor from the optimal, for any desired accuracy $\delta$. It relies on solving a combinatorially constrained bilinear maximization on the bi-adjacency matrix of $G$. It runs in time exponential in $k$ and $\delta^{-1}$, but linear in the size of the input. Further, we show that, in the (unconstrained) BCC setting, an ${(1-\delta)}$-approximation can be achieved by $O(\delta^{-1})$ clusters regardless of the size of the graph. In turn, our $k$-BCC algorithm implies an Efficient PTAS for the BCC objective of maximizing agreements.

Many modern datasets can be represented as graphs and hence spectral decompositions such as graph principal component analysis (PCA) can be useful. Distinct from previous graph decomposition approaches based on subspace projection of a single topological feature, e.g., the Fiedler vector of centered graph adjacency matrix (graph Laplacian), we propose spectral decomposition approaches to graph PCA and graph dictionary learning that integrate multiple features, including graph walk statistics, centrality measures and graph distances to reference nodes. In this paper we propose a new PCA method for single graph analysis, called multi-centrality graph PCA (MC-GPCA), and a new dictionary learning method for ensembles of graphs, called multi-centrality graph dictionary learning (MC-GDL), both based on spectral decomposition of multi-centrality matrices. As an application to cyber intrusion detection, MC-GPCA can be an effective indicator of anomalous connectivity pattern and MC-GDL can provide discriminative basis for attack classification.

Blind source separation (BSS), i.e., the decoupling of unknown signals that have been mixed in an unknown way, has been a topic of great interest in the signal processing community for the last decade, covering a wide range of applications in such diverse fields as digital communications, pattern recognition, biomedical engineering, and financial data analysis, among others. This course aims at an introduction to the BSS problem via an exposition of well-known and established as well as some more recent approaches to its solution. A unified way is followed in presenting the various results so as to more easily bring out their similarities/differences and emphasize their relative advantages/disadvantages. Only a representative sample of the existing knowledge on BSS will be included in this course. The interested readers are encouraged to consult the list of bibliographical references for more details on this exciting and always active research topic.

Generalised Degrees of Freedom (GDF), as defined by Ye (1998 JASA 93:120-131), represent the sensitivity of model fits to perturbations of the data. As such they can be computed for any statistical model, making it possible, in principle, to derive the number of parameters in machine-learning approaches. Defined originally for normally distributed data only, we here investigate the potential of this approach for Bernoulli-data. GDF-values for models of simulated and real data are compared to model complexity-estimates from cross-validation. Similarly, we computed GDF-based AICc for randomForest, neural networks and boosted regression trees and demonstrated its similarity to cross-validation. GDF-estimates for binary data were unstable and inconsistently sensitive to the number of data points perturbed simultaneously, while at the same time being extremely computer-intensive in their calculation. Repeated 10-fold cross-validation was more robust, based on fewer assumptions and faster to compute. Our findings suggest that the GDF-approach does not readily transfer to Bernoulli data and a wider range of regression approaches.

We apply a wild bootstrap method to the Lancaster three-variable interaction measure in order to detect factorisation of the joint distribution on three variables forming a stationary random process, for which the existing permutation bootstrap method fails. As in the i.i.d. case, the Lancaster test is found to outperform existing tests in cases for which two independent variables individually have a weak influence on a third, but that when considered jointly the influence is strong. The main contributions of this paper are twofold: first, we prove that the Lancaster statistic satisfies the conditions required to estimate the quantiles of the null distribution using the wild bootstrap; second, the manner in which this is proved is novel, simpler than existing methods, and can further be applied to other statistics.

We consider a composite convex minimization problem associated with regularized empirical risk minimization, which often arises in machine learning. We propose two new stochastic gradient methods that are based on stochastic dual averaging method with variance reduction. Our methods generate a sparser solution than the existing methods because we do not need to take the average of the history of the solutions. This is favorable in terms of both interpretability and generalization. Moreover, our methods have theoretical support for both a strongly and a non-strongly convex regularizer and achieve the best known convergence rates among existing nonaccelerated stochastic gradient methods.

Manifold Learning is a class of algorithms seeking a low-dimensional non-linear representation of high-dimensional data. Thus manifold learning algorithms are, at least in theory, most applicable to high-dimensional data and sample sizes to enable accurate estimation of the manifold. Despite this, most existing manifold learning implementations are not particularly scalable. Here we present a Python package that implements a variety of manifold learning algorithms in a modular and scalable fashion, using fast approximate neighbors searches and fast sparse eigendecompositions. The package incorporates theoretical advances in manifold learning, such as the unbiased Laplacian estimator and the estimation of the embedding distortion by the Riemannian metric method. In benchmarks, even on a single-core desktop computer, our code embeds millions of data points in minutes, and takes just 200 minutes to embed the main sample of galaxy spectra from the Sloan Digital Sky Survey --- consisting of 0.6 million samples in 3750-dimensions --- a task which has not previously been possible.

A variety of real-world tasks involve the classification of images into pre-determined categories. Designing image classification algorithms that exhibit robustness to acquisition noise and image distortions, particularly when the available training data are insufficient to learn accurate models, is a significant challenge. This dissertation explores the development of discriminative models for robust image classification that exploit underlying signal structure, via probabilistic graphical models and sparse signal representations. Probabilistic graphical models are widely used in many applications to approximate high-dimensional data in a reduced complexity set-up. Learning graphical structures to approximate probability distributions is an area of active research. Recent work has focused on learning graphs in a discriminative manner with the goal of minimizing classification error. In the first part of the dissertation, we develop a discriminative learning framework that exploits the complementary yet correlated information offered by multiple representations (or projections) of a given signal/image. Specifically, we propose a discriminative tree-based scheme for feature fusion by explicitly learning the conditional correlations among such multiple projections in an iterative manner. Experiments reveal the robustness of the resulting graphical model classifier to training insufficiency.

The Efficient Global Optimization (EGO) algorithm uses a conditional Gaus-sian Process (GP) to approximate an objective function known at a finite number of observation points and sequentially adds new points which maximize the Expected Improvement criterion according to the GP. The important factor that controls the efficiency of EGO is the GP covariance function (or kernel) which should be chosen according to the objective function. Traditionally, a pa-rameterized family of covariance functions is considered whose parameters are learned through statistical procedures such as maximum likelihood or cross-validation. However, it may be questioned whether statistical procedures for learning covariance functions are the most efficient for optimization as they target a global agreement between the GP and the observations which is not the ultimate goal of optimization. Furthermore, statistical learning procedures are computationally expensive. The main alternative to the statistical learning of the GP is self-adaptation, where the algorithm tunes the kernel parameters based on their contribution to objective function improvement. After questioning the possibility of self-adaptation for kriging based optimizers, this paper proposes a novel approach for tuning the length-scale of the GP in EGO: At each iteration, a small ensemble of kriging models structured by their length-scales is created. All of the models contribute to an iterate in an EGO-like fashion. Then, the set of models is densified around the model whose length-scale yielded the best iterate and further points are produced. Numerical experiments are provided which motivate the use of many length-scales. The tested implementation does not perform better than the classical EGO algorithm in a sequential context but show the potential of the approach for parallel implementations.

Various $\ell_1$-penalised estimation methods such as graphical lasso and CLIME are widely used for sparse precision matrix estimation. Many of these methods have been shown to be consistent under various quantitative assumptions about the underlying true covariance matrix. Intuitively, these conditions are related to situations where the penalty term will dominate the optimisation. In this paper, we explore the consistency of $\ell_1$-based methods for a class of sparse latent variable -like models, which are strongly motivated by several types of applications. We show that all $\ell_1$-based methods fail dramatically for models with nearly linear dependencies between the variables. We also study the consistency on models derived from real gene expression data and note that the assumptions needed for consistency never hold even for modest sized gene networks and $\ell_1$-based methods also become unreliable in practice for larger networks.