Support vector machines (SVM) are increasingly used in brain image analyses since they allow capturing complex multivariate relationships in the data. Moreover, when the kernel is linear, SVMs can be used to localize spatial patterns of discrimination between two groups of subjects. However, the features' spatial distribution is not taken into account. As a consequence, the optimal margin hyperplane is often scattered and lacks spatial coherence, making its anatomical interpretation difficult. This paper introduces a framework to spatially regularize SVM for brain image analysis. We show that Laplacian regularization provides a flexible framework to integrate various types of constraints and can be applied to both cortical surfaces and 3D brain images. The proposed framework is applied to the classification of MR images based on gray matter concentration maps and cortical thickness measures from 30 patients with Alzheimer's disease and 30 elderly controls. The results demonstrate that the proposed method enables natural spatial and anatomical regularization of the classifier.

We present simple and computationally efficient nonparametric estimators of R\'enyi entropy and mutual information based on an i.i.d. sample drawn from an unknown, absolutely continuous distribution over $\R^d$. The estimators are calculated as the sum of $p$-th powers of the Euclidean lengths of the edges of the `generalized nearest-neighbor' graph of the sample and the empirical copula of the sample respectively. For the first time, we prove the almost sure consistency of these estimators and upper bounds on their rates of convergence, the latter of which under the assumption that the density underlying the sample is Lipschitz continuous. Experiments demonstrate their usefulness in independent subspace analysis.

The problem of controlling the margin of a classifier is studied. A detailed analytical study is presented on how properties of the classification risk, such as its optimal link and minimum risk functions, are related to the shape of the loss, and its margin enforcing properties. It is shown that for a class of risks, denoted canonical risks, asymptotic Bayes consistency is compatible with simple analytical relationships between these functions. These enable a precise characterization of the loss for a popular class of link functions. It is shown that, when the risk is in canonical form and the link is inverse sigmoidal, the margin properties of the loss are determined by a single parameter. Novel families of Bayes consistent loss functions, of variable margin, are derived. These families are then used to design boosting style algorithms with explicit control of the classification margin. The new algorithms generalize well established approaches, such as LogitBoost. Experimental results show that the proposed variable margin losses outperform the fixed margin counterparts used by existing algorithms. Finally, it is shown that best performance can be achieved by cross-validating the margin parameter.

We propose a new variational EM algorithm for fitting factor analysis models with mixed continuous and categorical observations. The algorithm is based on a simple quadratic bound to the log-sum-exp function. In the special case of fully observed binary data, the bound we propose is significantly faster than previous variational methods. We show that EM is significantly more robust in the presence of missing data compared to treating the latent factors as parameters, which is the approach used by exponential family PCA and other related matrix-factorization methods. A further benefit of the variational approach is that it can easily be extended to the case of mixtures of factor analyzers, as we show. We present results on synthetic and real data sets demonstrating several desirable properties of our proposed method.

The commute distance between two vertices in a graph is the expected time it takes a random walk to travel from the first to the second vertex and back. We study the behavior of the commute distance as the size of the underlying graph increases. We prove that the commute distance converges to an expression that does not take into account the structure of the graph at all and that is completely meaningless as a distance function on the graph. Consequently, the use of the raw commute distance for machine learning purposes is strongly discouraged for large graphs and in high dimensions. As an alternative we introduce the amplified commute distance that corrects for the undesired large sample effects.

We deal with the problem of variable selection when variables must be selected group-wise, with possibly overlapping groups defined a priori. In particular we propose a new optimization procedure for solving the regularized algorithm presented in Jacob et al. 09, where the group lasso penalty is generalized to overlapping groups of variables. While in Jacob et al. 09 the proposed implementation requires explicit replication of the variables belonging to more than one group, our iterative procedure is based on a combination of proximal methods in the primal space and constrained Newton method in a reduced dual space, corresponding to the active groups. This procedure provides a scalable alternative with no need for data duplication, and allows to deal with high dimensional problems without pre-processing to reduce the dimensionality of the data. The computational advantages of our scheme with respect to state-of-the-art algorithms using data duplication are shown empirically with numerical simulations.

With the increase in available data parallel machine learning has become an increasingly pressing problem. In this paper we present the first parallel stochastic gradient descent algorithm including a detailed analysis and experimental evidence. Unlike prior work on parallel optimization algorithms our variant comes with parallel acceleration guarantees and it poses no overly tight latency constraints, which might only be available in the multicore setting. Our analysis introduces a novel proof technique --- contractive mappings to quantify the speed of convergence of parameter distributions to their asymptotic limits. As a side effect this answers the question of how quickly stochastic gradient descent algorithms reach the asymptotically normal regime.

Upstream supervised topic models have been widely used for complicated scene understanding. However, existing maximum likelihood estimation (MLE) schemes can make the prediction model learning independent of latent topic discovery and result in an imbalanced prediction rule for scene classification. This paper presents a joint max-margin and max-likelihood learning method for upstream scene understanding models, in which latent topic discovery and prediction model estimation are closely coupled and well-balanced. The optimization problem is efficiently solved with a variational EM procedure, which iteratively solves an online loss-augmented SVM. We demonstrate the advantages of the large-margin approach on both an 8-category sports dataset and the 67-class MIT indoor scene dataset for scene categorization.

Dimensionality reduction is often needed in many applications due to the high dimensionality of the data involved. In this paper, we first analyze the scatter measures used in the conventional linear discriminant analysis~(LDA) model and note that the formulation is based on the average-case view. Based on this analysis, we then propose a new dimensionality reduction method called worst-case linear discriminant analysis~(WLDA) by defining new between-class and within-class scatter measures. This new model adopts the worst-case view which arguably is more suitable for applications such as classification. When the number of training data points or the number of features is not very large, we relax the optimization problem involved and formulate it as a metric learning problem. Otherwise, we take a greedy approach by finding one direction of the transformation at a time. Moreover, we also analyze a special case of WLDA to show its relationship with conventional LDA. Experiments conducted on several benchmark datasets demonstrate the effectiveness of WLDA when compared with some related dimensionality reduction methods.

Recently, some variants of the $l_1$ norm, particularly matrix norms such as the $l_{1,2}$ and $l_{1,\infty}$ norms, have been widely used in multi-task learning, compressed sensing and other related areas to enforce sparsity via joint regularization. In this paper, we unify the $l_{1,2}$ and $l_{1,\infty}$ norms by considering a family of $l_{1,q}$ norms for $1 < q\le\infty$ and study the problem of determining the most appropriate sparsity enforcing norm to use in the context of multi-task feature selection. Using the generalized normal distribution, we provide a probabilistic interpretation of the general multi-task feature selection problem using the $l_{1,q}$ norm. Based on this probabilistic interpretation, we develop a probabilistic model using the noninformative Jeffreys prior. We also extend the model to learn and exploit more general types of pairwise relationships between tasks. For both versions of the model, we devise expectation-maximization~(EM) algorithms to learn all model parameters, including $q$, automatically. Experiments have been conducted on two cancer classification applications using microarray gene expression data.