The Group-Lasso is a well-known tool for joint regularization in machine learning methods. While the l_{1,2} and the l_{1,\infty} version have been studied in detail and efficient algorithms exist, there are still open questions regarding other l_{1,p} variants. We characterize conditions for solutions of the l_{1,p} Group-Lasso for all p-norms with 1 <= p <= \infty, and we present a unified active set algorithm. For all p-norms, a highly efficient projected gradient algorithm is presented. This new algorithm enables us to compare the prediction performance of many variants of the Group-Lasso in a multi-task learning setting, where the aim is to solve many learning problems in parallel which are coupled via the Group-Lasso constraint. We conduct large-scale experiments on synthetic data and on two real-world data sets. In accordance with theoretical characterizations of the different norms we observe that the weak-coupling norms with p between 1.5 and 2 consistently outperform the strong-coupling norms with p >> 2.

We consider principal component analysis for contaminated data-set in the high dimensional regime, where the dimensionality of each observation is comparable or even more than the number of observations. We propose a deterministic high-dimensional robust PCA algorithm which inherits all theoretical properties of its randomized counterpart, i.e., it is tractable, robust to contaminated points, easily kernelizable, asymptotic consistent and achieves maximal robustness -- a breakdown point of 50%. More importantly, the proposed method exhibits significantly better computational efficiency, which makes it suitable for large-scale real applications.

We present a robust multiple manifolds structure learning (RMMSL) scheme to robustly estimate data structures under the multiple low intrinsic dimensional manifolds assumption. In the local learning stage, RMMSL efficiently estimates local tangent space by weighted low-rank matrix factorization. In the global learning stage, we propose a robust manifold clustering method based on local structure learning results. The proposed clustering method is designed to get the flattest manifolds clusters by introducing a novel curved-level similarity function. Our approach is evaluated and compared to state-of-the-art methods on synthetic data, handwritten digit images, human motion capture data and motorbike videos. We demonstrate the effectiveness of the proposed approach, which yields higher clustering accuracy, and produces promising results for challenging tasks of human motion segmentation and motion flow learning from videos.

Driven by the multi-level structure of human intracranial electroencephalogram (iEEG) recordings of epileptic seizures, we introduce a new variant of a hierarchical Dirichlet Process---the multi-level clustering hierarchical Dirichlet Process (MLC-HDP)---that simultaneously clusters datasets on multiple levels. Our seizure dataset contains brain activity recorded in typically more than a hundred individual channels for each seizure of each patient. The MLC-HDP model clusters over channels-types, seizure-types, and patient-types simultaneously. We describe this model and its implementation in detail. We also present the results of a simulation study comparing the MLC-HDP to a similar model, the Nested Dirichlet Process and finally demonstrate the MLC-HDP's use in modeling seizures across multiple patients. We find the MLC-HDP's clustering to be comparable to independent human physician clusterings. To our knowledge, the MLC-HDP model is the first in the epilepsy literature capable of clustering seizures within and between patients.

We propose a general information-theoretic approach called Seraph (SEmi-supervised metRic leArning Paradigm with Hyper-sparsity) for metric learning that does not rely upon the manifold assumption. Given the probability parameterized by a Mahalanobis distance, we maximize the entropy of that probability on labeled data and minimize it on unlabeled data following entropy regularization, which allows the supervised and unsupervised parts to be integrated in a natural and meaningful way. Furthermore, Seraph is regularized by encouraging a low-rank projection induced from the metric. The optimization of Seraph is solved efficiently and stably by an EM-like scheme with the analytical E-Step and convex M-Step. Experiments demonstrate that Seraph compares favorably with many well-known global and local metric learning methods.

This paper considers the multi-task learning problem and in the setting where some relevant features could be shared across few related tasks. Most of the existing methods assume the extent to which the given tasks are related or share a common feature space to be known apriori. In real-world applications however, it is desirable to automatically discover the groups of related tasks that share a feature space. In this paper we aim at searching the exponentially large space of all possible groups of tasks that may share a feature space. The main contribution is a convex formulation that employs a graph-based regularizer and simultaneously discovers few groups of related tasks, having close-by task parameters, as well as the feature space shared within each group. The regularizer encodes an important structure among the groups of tasks leading to an efficient algorithm for solving it: if there is no feature space under which a group of tasks has close-by task parameters, then there does not exist such a feature space for any of its supersets. An efficient active set algorithm that exploits this simplification and performs a clever search in the exponentially large space is presented. The algorithm is guaranteed to solve the proposed formulation (within some precision) in a time polynomial in the number of groups of related tasks discovered. Empirical results on benchmark datasets show that the proposed formulation achieves good generalization and outperforms state-of-the-art multi-task learning algorithms in some cases.

We present a hybrid algorithm for optimizing a convex, smooth function over the cone of positive semidefinite matrices. Our algorithm converges to the global optimal solution and can be used to solve general large-scale semidefinite programs and hence can be readily applied to a variety of machine learning problems. We show experimental results on three machine learning problems (matrix completion, metric learning, and sparse PCA) . Our approach outperforms state-of-the-art algorithms.

We present a framework for online inference in the presence of a nonexhaustively defined set of classes that incorporates supervised classification with class discovery and modeling. A Dirichlet process prior (DPP) model defined over class distributions ensures that both known and unknown class distributions originate according to a common base distribution. In an attempt to automatically discover potentially interesting class formations, the prior model is coupled with a suitably chosen data model, and sequential Monte Carlo sampling is used to perform online inference. Our research is driven by a biodetection application, where a new class of pathogen may suddenly appear, and the rapid increase in the number of samples originating from this class indicates the onset of an outbreak.

A wide variety of machine learning algorithms such as support vector machine (SVM), minimax probability machine (MPM), and Fisher discriminant analysis (FDA), exist for binary classification. The purpose of this paper is to provide a unified classification model that includes the above models through a robust optimization approach. This unified model has several benefits. One is that the extensions and improvements intended for SVM become applicable to MPM and FDA, and vice versa. Another benefit is to provide theoretical results to above learning methods at once by dealing with the unified model. We give a statistical interpretation of the unified classification model and propose a non-convex optimization algorithm that can be applied to non-convex variants of existing learning methods.

Probabilistic principal component analysis (PPCA) seeks a low dimensional representation of a data set in the presence of independent spherical Gaussian noise. The maximum likelihood solution for the model is an eigenvalue problem on the sample covariance matrix. In this paper we consider the situation where the data variance is already partially explained by other factors, for example sparse conditional dependencies between the covariates, or temporal correlations leaving some residual variance. We decompose the residual variance into its components through a generalised eigenvalue problem, which we call residual component analysis (RCA). We explore a range of new algorithms that arise from the framework, including one that factorises the covariance of a Gaussian density into a low-rank and a sparse-inverse component. We illustrate the ideas on the recovery of a protein-signaling network, a gene expression time-series data set and the recovery of the human skeleton from motion capture 3-D cloud data.