For the general problem of minimizing a convex function over a compact convex domain, we will investigate a simple iterative approximation algorithm based on the method by Frank & Wolfe 1956, that does not need projection steps in order to stay inside the optimization domain. Instead of a projection step, the linearized problem defined by a current subgradient is solved, which gives a step direction that will naturally stay in the domain. Our framework generalizes the sparse greedy algorithm of Frank & Wolfe and its primal-dual analysis by Clarkson 2010 (and the low-rank SDP approach by Hazan 2008) to arbitrary convex domains. We give a convergence proof guaranteeing {\epsilon}-small duality gap after O(1/{\epsilon}) iterations. The method allows us to understand the sparsity of approximate solutions for any l1-regularized convex optimization problem (and for optimization over the simplex), expressed as a function of the approximation quality. We obtain matching upper and lower bounds of {\Theta}(1/{\epsilon}) for the sparsity for l1-problems. The same bounds apply to low-rank semidefinite optimization with bounded trace, showing that rank O(1/{\epsilon}) is best possible here as well. As another application, we obtain sparse matrices of O(1/{\epsilon}) non-zero entries as {\epsilon}-approximate solutions when optimizing any convex function over a class of diagonally dominant symmetric matrices. We show that our proposed first-order method also applies to nuclear norm and max-norm matrix optimization problems. For nuclear norm regularized optimization, such as matrix completion and low-rank recovery, we demonstrate the practical efficiency and scalability of our algorithm for large matrix problems, as e.g. the Netflix dataset. For general convex optimization over bounded matrix max-norm, our algorithm is the first with a convergence guarantee, to the best of our knowledge.

This paper considers the problem of completing a matrix with many missing entries under the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces. This generalizes the standard low-rank matrix completion problem to situations in which the matrix rank can be quite high or even full rank. Since the columns belong to a union of subspaces, this problem may also be viewed as a missing-data version of the subspace clustering problem. Let X be an n x N matrix whose (complete) columns lie in a union of at most k subspaces, each of rank <= r < n, and assume N >> kn. The main result of the paper shows that under mild assumptions each column of X can be perfectly recovered with high probability from an incomplete version so long as at least CrNlog^2(n) entries of X are observed uniformly at random, with C>1 a constant depending on the usual incoherence conditions, the geometrical arrangement of subspaces, and the distribution of columns over the subspaces. The result is illustrated with numerical experiments and an application to Internet distance matrix completion and topology identification.

Information theoretic active learning has been widely studied for probabilistic models. For simple regression an optimal myopic policy is easily tractable. However, for other tasks and with more complex models, such as classification with nonparametric models, the optimal solution is harder to compute. Current approaches make approximations to achieve tractability. We propose an approach that expresses information gain in terms of predictive entropies, and apply this method to the Gaussian Process Classifier (GPC). Our approach makes minimal approximations to the full information theoretic objective. Our experimental performance compares favourably to many popular active learning algorithms, and has equal or lower computational complexity. We compare well to decision theoretic approaches also, which are privy to more information and require much more computational time. Secondly, by developing further a reformulation of binary preference learning to a classification problem, we extend our algorithm to Gaussian Process preference learning.

Often, high dimensional data lie close to a low-dimensional submanifold and it is of interest to understand the geometry of these submanifolds. The homology groups of a manifold are important topological invariants that provide an algebraic summary of the manifold. These groups contain rich topological information, for instance, about the connected components, holes, tunnels and sometimes the dimension of the manifold. In this paper, we consider the statistical problem of estimating the homology of a manifold from noisy samples under several different noise models. We derive upper and lower bounds on the minimax risk for this problem. Our upper bounds are based on estimators which are constructed from a union of balls of appropriate radius around carefully selected points. In each case we establish complementary lower bounds using Le Cam's lemma.

We consider the problem of classification using similarity/distance functions over data. Specifically, we propose a framework for defining the goodness of a (dis)similarity function with respect to a given learning task and propose algorithms that have guaranteed generalization properties when working with such good functions. Our framework unifies and generalizes the frameworks proposed by [Balcan-Blum ICML 2006] and [Wang et al ICML 2007]. An attractive feature of our framework is its adaptability to data - we do not promote a fixed notion of goodness but rather let data dictate it. We show, by giving theoretical guarantees that the goodness criterion best suited to a problem can itself be learned which makes our approach applicable to a variety of domains and problems. We propose a landmarking-based approach to obtaining a classifier from such learned goodness criteria. We then provide a novel diversity based heuristic to perform task-driven selection of landmark points instead of random selection. We demonstrate the effectiveness of our goodness criteria learning method as well as the landmark selection heuristic on a variety of similarity-based learning datasets and benchmark UCI datasets on which our method consistently outperforms existing approaches by a significant margin.

Machine learning is used to approximate density functionals. For the model problem of the kinetic energy of non-interacting fermions in 1d, mean absolute errors below 1 kcal/mol on test densities similar to the training set are reached with fewer than 100 training densities. A predictor identifies if a test density is within the interpolation region. Via principal component analysis, a projected functional derivative finds highly accurate self-consistent densities. Challenges for application of our method to real electronic structure problems are discussed.

Let us envision a new class of IT systems, the "Support Systems for Knowledge Works" or SSKW. An SSKW can be defined as a system built for providing comprehensive support to human knowledge-workers while performing instances of complex knowledge-works of a particular type within a particular domain of professional activities To get an idea what an SSKW-enabled work environment can be like, let us look into a hypothetical scenario that depicts the interaction between a physician and a patient-care SSKW during the activity of diagnosing a patient.

The success of kernel-based learning methods depend on the choice of kernel. Recently, kernel learning methods have been proposed that use data to select the most appropriate kernel, usually by combining a set of base kernels. We introduce a new algorithm for kernel learning that combines a {\em continuous set of base kernels}, without the common step of discretizing the space of base kernels. We demonstrate that our new method achieves state-of-the-art performance across a variety of real-world datasets. Furthermore, we explicitly demonstrate the importance of combining the right dictionary of kernels, which is problematic for methods based on a finite set of base kernels chosen a priori. Our method is not the first approach to work with continuously parameterized kernels. However, we show that our method requires substantially less computation than previous such approaches, and so is more amenable to multiple dimensional parameterizations of base kernels, which we demonstrate.

Visual reranking is effective to improve the performance of the text-based video search. However, existing reranking algorithms can only achieve limited improvement because of the well-known semantic gap between low level visual features and high level semantic concepts. In this paper, we adopt interactive video search reranking to bridge the semantic gap by introducing user's labeling effort. We propose a novel dimension reduction tool, termed sparse transfer learning (STL), to effectively and efficiently encode user's labeling information. STL is particularly designed for interactive video search reranking. Technically, it a) considers the pair-wise discriminative information to maximally separate labeled query relevant samples from labeled query irrelevant ones, b) achieves a sparse representation for the subspace to encodes user's intention by applying the elastic net penalty, and c) propagates user's labeling information from labeled samples to unlabeled samples by using the data distribution knowledge. We conducted extensive experiments on the TRECVID 2005, 2006 and 2007 benchmark datasets and compared STL with popular dimension reduction algorithms. We report superior performance by using the proposed STL based interactive video search reranking.

Nowadays, computer scientists have shown the interest in the study of social insect's behaviour in neural networks area for solving different combinatorial and statistical problems. Chief among these is the Artificial Bee Colony (ABC) algorithm. This paper investigates the use of ABC algorithm that simulates the intelligent foraging behaviour of a honey bee swarm. Multilayer Perceptron (MLP) trained with the standard back propagation algorithm normally utilises computationally intensive training algorithms. One of the crucial problems with the backpropagation (BP) algorithm is that it can sometimes yield the networks with suboptimal weights because of the presence of many local optima in the solution space. To overcome ABC algorithm used in this work to train MLP learning the complex behaviour of earthquake time series data trained by BP, the performance of MLP-ABC is benchmarked against MLP training with the standard BP. The experimental result shows that MLP-ABC performance is better than MLP-BP for time series data.