When learning models that are represented in matrix forms, enforcing a low-rank constraint can dramatically improve the memory and run time complexity, while providing a natural regularization of the model. However, naive approaches for minimizing functions over the set of low-rank matrices are either prohibitively time consuming (repeated singular value decomposition of the matrix) or numerically unstable (optimizing a factored representation of the low rank matrix). We build on recent advances in optimization over manifolds, and describe an iterative online learning procedure, consisting of a gradient step, followed by a second-order retraction back to the manifold. While the ideal retraction is hard to compute, and so is the projection operator that approximates it, we describe another second-order retraction that can be computed efficiently, with run time and memory complexity of O((n+m)k) for a rank-k matrix of dimension m x n, given rank one gradients. We use this algorithm, LORETA, to learn a matrix-form similarity measure over pairs of documents represented as high dimensional vectors. LORETA improves the mean average precision over a passive- aggressive approach in a factorized model, and also improves over a full model trained over pre-selected features using the same memory requirements. LORETA also showed consistent improvement over standard methods in a large (1600 classes) multi-label image classification task.

Continuous Markov random fields are a general formalism to model joint probability distributions over events with continuous outcomes. We prove that marginal computation for constrained continuous MRFs is #P-hard in general and present a polynomial-time approximation scheme under mild assumptions on the structure of the random field. Moreover, we introduce a sampling algorithm to compute marginal distributions and develop novel techniques to increase its efficiency. Continuous MRFs are a general purpose probabilistic modeling tool and we demonstrate how they can be applied to statistical relational learning. On the problem of collective classification, we evaluate our algorithm and show that the standard deviation of marginals serves as a useful measure of confidence.

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.

We define a data dependent permutation complexity for a hypothesis set H, which is similar to a Rademacher complexity or maximum discrepancy. The permutation complexity is based (like the maximum discrepancy) on dependent sampling. We prove a uniform bound on the generalization error, as well as a concentration result which means that the permutation estimate can be efficiently estimated.

Gaussian graphical models are of great interest in statistical learning. Because the conditional independencies between different nodes correspond to zero entries in the inverse covariance matrix of the Gaussian distribution, one can learn the structure of the graph by estimating a sparse inverse covariance matrix from sample data, by solving a convex maximum likelihood problem with an $\ell_1$-regularization term. In this paper, we propose a first-order method based on an alternating linearization technique that exploits the problem's special structure; in particular, the subproblems solved in each iteration have closed-form solutions. Moreover, our algorithm obtains an $\epsilon$-optimal solution in $O(1/\epsilon)$ iterations. Numerical experiments on both synthetic and real data from gene association networks show that a practical version of this algorithm outperforms other competitive algorithms.

Multi-class classification becomes challenging at test time when the number of classes is very large and testing against every possible class can become computationally infeasible. This problem can be alleviated by imposing (or learning) a structure over the set of classes. We propose an algorithm for learning a tree-structure of classifiers which, by optimizing the overall tree loss, provides superior accuracy to existing tree labeling methods. We also propose a method that learns to embed labels in a low dimensional space that is faster than non-embedding approaches and has superior accuracy to existing embedding approaches. Finally we combine the two ideas resulting in the label embedding tree that outperforms alternative methods including One-vs-Rest while being orders of magnitude faster.

We propose a discriminative latent model for annotating images with unaligned object-level textual annotations. Instead of using the bag-of-words image representation currently popular in the computer vision community, our model explicitly captures more intricate relationships underlying visual and textual information. In particular, we model the mapping that translates image regions to annotations. This mapping allows us to relate image regions to their corresponding annotation terms. We also model the overall scene label as latent information. This allows us to cluster test images. Our training data consist of images and their associated annotations. But we do not have access to the ground-truth region-to-annotation mapping or the overall scene label. We develop a novel variant of the latent SVM framework to model them as latent variables. Our experimental results demonstrate the effectiveness of the proposed model compared with other baseline methods.

We present a novel algorithm, Random Conic Pursuit, that solves semidefinite programs (SDPs) via repeated optimization over randomly selected two-dimensional subcones of the PSD cone. This scheme is simple, easily implemented, applicable to very general SDPs, scalable, and theoretically interesting. Its advantages are realized at the expense of an ability to readily compute highly exact solutions, though useful approximate solutions are easily obtained. This property renders Random Conic Pursuit of particular interest for machine learning applications, in which the relevant SDPs are generally based upon random data and so exact minima are often not a priority. Indeed, we present empirical results to this effect for various SDPs encountered in machine learning; these experiments demonstrate the potential practical usefulness of Random Conic Pursuit. We also provide a preliminary analysis that yields insight into the theoretical properties and convergence of the algorithm.

A standard approach to learning object category detectors is to provide strong supervision in the form of a region of interest (ROI) specifying each instance of the object in the training images. In this work are goal is to learn from heterogeneous labels, in which some images are only weakly supervised, specifying only the presence or absence of the object or a weak indication of object location, whilst others are fully annotated. To this end we develop a discriminative learning approach and make two contributions: (i) we propose a structured output formulation for weakly annotated images where full annotations are treated as latent variables; and (ii) we propose to optimize a ranking objective function, allowing our method to more effectively use negatively labeled images to improve detection average precision performance. The method is demonstrated on the benchmark INRIA pedestrian detection dataset of Dalal and Triggs and the PASCAL VOC dataset, and it is shown that for a significant proportion of weakly supervised images the performance achieved is very similar to the fully supervised (state of the art) results.

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.