We propose in this work a new family of kernels for variable-length time series. Our work builds upon the vector autoregressive (VAR) model for multivariate stochastic processes: given a multivariate time series x, we consider the likelihood function p_{\theta}(x) of different parameters \theta in the VAR model as features to describe x. To compare two time series x and x', we form the product of their features p_{\theta}(x) p_{\theta}(x') which is integrated out w.r.t \theta using a matrix normal-inverse Wishart prior. Among other properties, this kernel can be easily computed when the dimension d of the time series is much larger than the lengths of the considered time series x and x'. It can also be generalized to time series taking values in arbitrary state spaces, as long as the state space itself is endowed with a kernel \kappa. In that case, the kernel between x and x' is a a function of the Gram matrices produced by \kappa on observations and subsequences of observations enumerated in x and x'. We describe a computationally efficient implementation of this generalization that uses low-rank matrix factorization techniques. These kernels are compared to other known kernels using a set of benchmark classification tasks carried out with support vector machines.

We prove rates of convergence in the statistical sense for kernel-based least squares regression using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is directly related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. The rates depend on two key quantities: first, on the regularity of the target regression function and second, on the effective dimensionality of the data mapped into the kernel space. Lower bounds on attainable rates depending on these two quantities were established in earlier literature, and we obtain upper bounds for the considered method that match these lower bounds (up to a log factor) if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates provided additional unlabeled data are available. The order of the learning rates in these two situations match state-of-the-art results that were recently obtained for the least squares support vector machine and for linear regularization operators.

Multi-task learning (MTL) improves the prediction performance on multiple, different but related, learning problems through shared parameters or representations. One of the most prominent multi-task learning algorithms is an extension to svms by Evgeniou et al. Although very elegant, multi-task svm is inherently restricted by the fact that support vector machines require each class to be addressed explicitly with its own weight vector which, in a multi-task setting, requires the different learning tasks to share the same set of classes. This paper proposes an alternative formulation for multi-task learning by extending the recently published large margin nearest neighbor (lmnn) algorithm to the MTL paradigm. Instead of relying on separating hyperplanes, its decision function is based on the nearest neighbor rule which inherently extends to many classes and becomes a natural fit for multitask learning. We evaluate the resulting multi-task lmnn on real-world insurance data and speech classification problems and show that it consistently outperforms single-task kNN under several metrics and state-of-the-art MTL classifiers.

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.

In a recent paper Joachims (2006) presented SVM-Perf, a cutting plane method (CPM) for training linear Support Vector Machines (SVMs) which converges to an $\epsilon$ accurate solution in $O(1/\epsilon^{2})$ iterations. By tightening the analysis, Teo et al. (2010) showed that $O(1/\epsilon)$ iterations suffice. Given the impressive convergence speed of CPM on a number of practical problems, it was conjectured that these rates could be further improved. In this paper we disprove this conjecture. We present counter examples which are not only applicable for training linear SVMs with hinge loss, but also hold for support vector methods which optimize a \emph{multivariate} performance score. However, surprisingly, these problems are not inherently hard. By exploiting the structure of the objective function we can devise an algorithm that converges in $O(1/\sqrt{\epsilon})$ iterations.

While clinicians can accurately identify different types of heartbeats in electrocardiograms (ECGs) from different patients, researchers have had limited success in applying supervised machine learning to the same task. The problem is made challenging by the variety of tasks, inter- and intra-patient differences, an often severe class imbalance, and the high cost of getting cardiologists to label data for individual patients. We address these difficulties using active learning to perform patient-adaptive and task-adaptive heartbeat classification. When tested on a benchmark database of cardiologist annotated ECG recordings, our method had considerably better performance than other recently proposed methods on the two primary classification tasks recommended by the Association for the Advancement of Medical Instrumentation. Additionally, our method required over 90% less patient-specific training data than the methods to which we compared it.

Multiple-Instance learning has been long known as a hard non-convex problem. In this work, we propose an approach that recasts it as a convex likelihood ratio estimation problem. Firstly, the constraint in multiple-instance learning is reformulated into a convex constraint on the likelihood ratio. Then we show that a joint estimation of a likelihood ratio function and the likelihood on training instances can be learned convexly. Theoretically, we prove a quantitative relationship between the risk estimated under the 0-1 classification loss, and under a loss function for likelihood ratio estimation. It is shown that our likelihood ratio estimation is generally a good surrogate for the 0-1 loss, and separates positive and negative instances well. However with the joint estimation it tends to underestimate the likelihood of an example to be positive. We propose to use these likelihood ratio estimates as features, and learn a linear combination on them to classify the bags. Experiments on synthetic and real datasets show the superiority of the approach.

We propose a new supervised learning framework for visual object counting tasks, such as estimating the number of cells in a microscopic image or the number of humans in surveillance video frames. We focus on the practically-attractive case when the training images are annotated with dots (one dot per object). Our goal is to accurately estimate the count. However, we evade the hard task of learning to detect and localize individual object instances. Instead, we cast the problem as that of estimating an image density whose integral over any image region gives the count of objects within that region. Learning to infer such density can be formulated as a minimization of a regularized risk quadratic cost function. We introduce a new loss function, which is well-suited for such learning, and at the same time can be computed efficiently via a maximum subarray algorithm. The learning can then be posed as a convex quadratic program solvable with cutting-plane optimization. The proposed framework is very flexible as it can accept any domain-specific visual features. Once trained, our system provides accurate object counts and requires a very small time overhead over the feature extraction step, making it a good candidate for applications involving real-time processing or dealing with huge amount of visual data.

We propose a discriminative model for recognizing group activities. Our model jointly captures the group activity, the individual person actions, and the interactions among them. Two new types of contextual information, group-person interaction and person-person interaction, are explored in a latent variable framework. Different from most of the previous latent structured models which assume a predefined structure for the hidden layer, e.g. a tree structure, we treat the structure of the hidden layer as a latent variable and implicitly infer it during learning and inference. Our experimental results demonstrate that by inferring this contextual information together with adaptive structures, the proposed model can significantly improve activity recognition performance.

In this paper we consider the problem of learning an n x n Kernel matrix from m similarity matrices under general convex loss. Past research have extensively studied the m =1 case and have derived several algorithms which require sophisticated techniques like ACCP, SOCP, etc. The existing algorithms do not apply if one uses arbitrary losses and often can not handle m > 1 case. We present several provably convergent iterative algorithms, where each iteration requires either an SVM or a Multiple Kernel Learning (MKL) solver for m > 1 case. One of the major contributions of the paper is to extend the well known Mirror Descent(MD) framework to handle Cartesian product of psd matrices. This novel extension leads to an algorithm, called EMKL, which solves the problem in O(m^2 log n) iterations; in each iteration one solves an MKL involving m kernels and m eigen-decomposition of n x n matrices. By suitably defining a restriction on the objective function, a faster version of EMKL is proposed, called REKL, which avoids the eigen-decomposition. An alternative to both EMKL and REKL is also suggested which requires only an SVM solver. Experimental results on real world protein data set involving several similarity matrices illustrate the efficacy of the proposed algorithms.