Extracting the underlying low-dimensional space where high-dimensional signals often reside has long been at the center of numerous algorithms in the signal processing and machine learning literature during the past few decades. At the same time, working with incomplete (partly observed) large scale datasets has recently been commonplace for diverse reasons. This so called {\it big data era} we are currently living calls for devising online subspace learning algorithms that can suitably handle incomplete data. Their envisaged objective is to {\it recursively} estimate the unknown subspace by processing streaming data sequentially, thus reducing computational complexity, while obviating the need for storing the whole dataset in memory. In this paper, an online variational Bayes subspace learning algorithm from partial observations is presented. To account for the unawareness of the true rank of the subspace, commonly met in practice, low-rankness is explicitly imposed on the sought subspace data matrix by exploiting sparse Bayesian learning principles. Moreover, sparsity, {\it simultaneously} to low-rankness, is favored on the subspace matrix by the sophisticated hierarchical Bayesian scheme that is adopted. In doing so, the proposed algorithm becomes adept in dealing with applications whereby the underlying subspace may be also sparse, as, e.g., in sparse dictionary learning problems. As shown, the new subspace tracking scheme outperforms its state-of-the-art counterparts in terms of estimation accuracy, in a variety of experiments conducted on simulated and real data.

The conjugate gradient (CG) method is an efficient iterative method for solving large-scale strongly convex quadratic programming (QP). In this paper we propose some generalized CG (GCG) methods for solving the $\ell_1$-regularized (possibly not strongly) convex QP that terminate at an optimal solution in a finite number of iterations. At each iteration, our methods first identify a face of an orthant and then either perform an exact line search along the direction of the negative projected minimum-norm subgradient of the objective function or execute a CG subroutine that conducts a sequence of CG iterations until a CG iterate crosses the boundary of this face or an approximate minimizer of over this face or a subface is found. We determine which type of step should be taken by comparing the magnitude of some components of the minimum-norm subgradient of the objective function to that of its rest components. Our analysis on finite convergence of these methods makes use of an error bound result and some key properties of the aforementioned exact line search and the CG subroutine. We also show that the proposed methods are capable of finding an approximate solution of the problem by allowing some inexactness on the execution of the CG subroutine. The overall arithmetic operation cost of our GCG methods for finding an $\epsilon$-optimal solution depends on $\epsilon$ in $O(\log(1/\epsilon))$, which is superior to the accelerated proximal gradient method [2,23] that depends on $\epsilon$ in $O(1/\sqrt{\epsilon})$. In addition, our GCG methods can be extended straightforwardly to solve box-constrained convex QP with finite convergence. Numerical results demonstrate that our methods are very favorable for solving ill-conditioned problems.

Multidimensional scaling (MDS) is a class of projective algorithms traditionally used in Euclidean space to produce two- or three-dimensional visualizations of datasets of multidimensional points or point distances. More recently however, several authors have pointed out that for certain datasets, hyperbolic target space may provide a better fit than Euclidean space. In this paper we develop PD-MDS, a metric MDS algorithm designed specifically for the Poincare disk (PD) model of the hyperbolic plane. Emphasizing the importance of proceeding from first principles in spite of the availability of various black box optimizers, our construction is based on an elementary hyperbolic line search and reveals numerous particulars that need to be carefully addressed when implementing this as well as more sophisticated iterative optimization methods in a hyperbolic space model.

We investigate the fundamental principles that drive the development of scalable algorithms for network optimization. Despite the significant amount of work on parallel and decentralized algorithms in the optimization community, the methods that have been proposed typically rely on strict separability assumptions for objective function and constraints. Beside sparsity, these methods typically do not exploit the strength of the interaction between variables in the system. We propose a notion of correlation in constrained optimization that is based on the sensitivity of the optimal solution upon perturbations of the constraints. We develop a general theory of sensitivity of optimizers the extends beyond the infinitesimal setting. We present instances in network optimization where the correlation decays exponentially fast with respect to the natural distance in the network, and we design algorithms that can exploit this decay to yield dimension-free optimization. Our results are the first of their kind, and open new possibilities in the theory of local algorithms.

The support vector machine (SVM) is an important class of learning machines for function approach, pattern recognition, and time-serious prediction, etc. It maps samples into the feature space by so-called support vectors of selected samples, and then feature vectors are separated by maximum margin hyperplane. The present paper presents the general vector machine (GVM) to replace the SVM. The support vectors are replaced by general project vectors selected from the usual vector space, and a Monte Carlo (MC) algorithm is developed to find the general vectors. The general project vectors improves the feature-extraction ability, and the MC algorithm can control the width of the separation margin of the hyperplane. By controlling the separation margin, we show that the maximum margin hyperplane can usually induce the overlearning, and the best learning machine is achieved with a proper separation margin. Applications in function approach, pattern recognition, and classification indicate that the developed method is very successful, particularly for small-set training problems. Additionally, our algorithm may induce some particular applications, such as for the transductive inference.

We study the problem of supervised learning for both binary and multiclass classification from a unified geometric perspective. In particular, we propose a geometric regularization technique to find the submanifold corresponding to a robust estimator of the class probability $P(y|\pmb{x})$. The regularization term measures the volume of this submanifold, based on the intuition that overfitting produces rapid local oscillations and hence large volume of the estimator. This technique can be applied to regularize any classification function that satisfies two requirements: firstly, an estimator of the class probability can be obtained; secondly, first and second derivatives of the class probability estimator can be calculated. In experiments, we apply our regularization technique to standard loss functions for classification, our RBF-based implementation compares favorably to widely used regularization methods for both binary and multiclass classification.

The public package registry npm is one of the biggest software registry. With its 216 911 software packages, it forms a big network of software dependencies. In this paper we evaluate various methods for finding similar packages in the npm network, using only the structure of the graph. Namely, we want to find a way of categorizing similar packages, which would be useful for recommendation systems. This size enables us to compute meaningful results, as it softened the particularities of the graph. Npm is also quite famous as it is the default package repository of Node.js. We believe that it will make our results interesting for more people than a less used package repository. This makes it a good subject of analysis of software networks.

Gaussian Processes (GPs) provide a general and analytically tractable way of modeling complex time-varying, nonparametric functions. The Automatic Bayesian Covariance Discovery (ABCD) system constructs natural-language description of time-series data by treating unknown time-series data nonparametrically using GP with a composite covariance kernel function. Unfortunately, learning a composite covariance kernel with a single time-series data set often results in less informative kernel that may not give qualitative, distinctive descriptions of data. We address this challenge by proposing two relational kernel learning methods which can model multiple time-series data sets by finding common, shared causes of changes. We show that the relational kernel learning methods find more accurate models for regression problems on several real-world data sets; US stock data, US house price index data and currency exchange rate data.

The emergence of crowdsourcing systems have provided a viable mechanism for incorporating humans into the computational loop at large scale and in real-time. This offers an unprecedent opportunity to study how artificial intelligence (AI) techniques and humans can collaborate to solve problems. An important challenge in crowdsourcing is how to make optimal use of human resources as people have different skills and their availability may be limited. In this paper, we provide the research community with a new dataset derived from an online game-based platform to address this challenge. Six crowdsourcing task allocation scenarios with different overall workload levels and worker population characteristics were presented to over 400 players to solve. With close to 3,000 game sessions and over 300,000 task allocation decisions from human and AI players, the dataset provides an efficient focal point for the research community to design solutions that can sustainably tap into the pool of human resources through crowdsourcing.

We present a Hidden Markov Model-based algorithm for constructing timescales for paleoclimate records by annual layer counting. This objective, statistics-based approach has a number of major advantages over the current manual approach, beginning with speed. Manual layer counting of a single core (up to 3km in length) can require multiple person-years of time; the StratiCounter algorithm can count up to 100 layers/min, corresponding to a full-length timescale constructed in a few days. Moreover, the algorithm gives rigorous uncertainty estimates for the resulting timescale, which are far smaller than those produced manually. We demonstrate the utility of StratiCounter by applying it to ice-core data from two cores from Greenland and Antarctica. Performance of the algorithm is comparable to a manual approach. When using all available data, false-discovery rates and miss rates are 1-1.2% and 1.2-1.6%, respectively, for the two cores. For one core, even better agreement is found when using only the chemistry series primarily employed by human experts in the manual approach.