Trust has been extensively exploited to improve the predictive The dependency between the trust aspects is captured by accuracy of recommendations by ameliorating the issues a Guassian radial basis kernel function. Then, we incorporate such as data sparsity and cold start that recommender the trust information into the probabilistic matrix factorization systems inherently suffer from (Massa and Avesani 2007; model (Mnih and Salakhutdinov 2007) by modeling Ma et al. 2008). Basically, trust provides additional information trust as jointly conditioning on the trust value obtained from which user preference can be better modeled, alternative from the SVR model, as well as similarity between the corresponding or complementary to rating-based similarity.

Modeling videos and image-sets as linear subspaces has proven beneficial for many visual recognition tasks. However, it also incurs challenges arising from the fact that linear subspaces do not obey Euclidean geometry, but lie on a special type of Riemannian manifolds known as Grassmannian. To leverage the techniques developed for Euclidean spaces (e.g, support vector machines) with subspaces, several recent studies have proposed to embed the Grassmannian into a Hilbert space by making use of a positive definite kernel. Unfortunately, only two Grassmannian kernels are known, none of which -as we will show- is universal, which limits their ability to approximate a target function arbitrarily well. Here, we introduce several positive definite Grassmannian kernels, including universal ones, and demonstrate their superiority over previously-known kernels in various tasks, such as classification, clustering, sparse coding and hashing.

Despite the success of the popular kernelized support vector machines, they have two major limitations: they are restricted to Positive Semi-Definite (PSD) kernels, and their training complexity scales at least quadratically with the size of the data. Many natural measures of similarity between pairs of samples are not PSD e.g. invariant kernels, and those that are implicitly or explicitly defined by latent variable models. In this paper, we investigate scalable approaches for using indefinite similarity measures in large margin frameworks. In particular we show that a normalization of similarity to a subset of the data points constitutes a representation suitable for linear classifiers. The result is a classifier which is competitive to kernelized SVM in terms of accuracy, despite having better training and test time complexities. Experimental results demonstrate that on CIFAR-10 dataset, the model equipped with similarity measures invariant to rigid and non-rigid deformations, can be made more than 5 times sparser while being more accurate than kernelized SVM using RBF kernels.

Dimensionality reduction (DR) is often used as a preprocessing step in classification, but usually one first fixes the DR mapping, possibly using label information, and then learns a classifier (a filter approach). Best performance would be obtained by optimizing the classification error jointly over DR mapping and classifier (a wrapper approach), but this is a difficult nonconvex problem, particularly with nonlinear DR. Using the method of auxiliary coordinates, we give a simple, efficient algorithm to train a combination of nonlinear DR and a classifier, and apply it to a RBF mapping with a linear SVM. This alternates steps where we train the RBF mapping and a linear SVM as usual regression and classification, respectively, with a closed-form step that coordinates both. The resulting nonlinear low-dimensional classifier achieves classification errors competitive with the state-of-the-art but is fast at training and testing, and allows the user to trade off runtime for classification accuracy easily. We then study the role of nonlinear DR in linear classification, and the interplay between the DR mapping, the number of latent dimensions and the number of classes. When trained jointly, the DR mapping takes an extreme role in eliminating variation: it tends to collapse classes in latent space, erasing all manifold structure, and lay out class centroids so they are linearly separable with maximum margin.

This paper presents a study where semantic frames are used to mine financial news so as to quantify the impact of news on the stock market. We represent news documents in a novel semantic tree structure and use tree kernel support vector machines to predict the change of stock price. We achieve an efficient computation through linearization of tree kernels. In addition to two binary classification tasks, we rank news items according to their probability to affect change of price using two ranking methods that require vector space features. We evaluate our rankings based on receiver operating characteristic curves and analyze the predictive power of our semantic features. For both learning tasks, the proposed semantic features provide superior results.

Electronic discovery is an interesting subproblem of information retrieval in which one identifies documents that are potentially relevant to issues and facts of a legal case from an electronically stored document collection (a corpus). In this paper, we consider representing documents in a topic space using the well-known topic models such as latent Dirichlet allocation and latent semantic indexing, and solving the information retrieval problem via finding document similarities in the topic space rather doing it in the corpus vocabulary space. We also develop an iterative SMART ranking and categorization framework including human-in-the-loop to label a set of seed (training) documents and using them to build a semi-supervised binary document classification model based on Support Vector Machines. To improve this model, we propose a method for choosing seed documents from the whole population via an active learning strategy. We report the results of our experiments on a real dataset in the electronic discovery domain.

We consider the question of learning in general topological vector spaces. By exploiting known (or parametrized) covariance structures, our Main Theorem demonstrates that any continuous linear map corresponds to a certain isomorphism of embedded Hilbert spaces. By inverting this isomorphism and extending continuously, we construct a version of the Ordinary Least Squares estimator in absolute generality. Our Gauss-Markov theorem demonstrates that OLS is a "best linear unbiased estimator", extending the classical result. We construct a stochastic version of the OLS estimator, which is a continuous disintegration exactly for the class of "uncorrelated implies independent" (UII) measures. As a consequence, Gaussian measures always exhibit continuous disintegrations through continuous linear maps, extending a theorem of the first author. Applying this framework to some problems in machine learning, we prove a useful representation theorem for covariance tensors, and show that OLS defines a good kriging predictor for vector-valued arrays on general index spaces. We also construct a support-vector machine classifier in this setting. We hope that our article shines light on some deeper connections between probability theory, statistics and machine learning, and may serve as a point of intersection for these three communities.

Random projection (RP) is a classical technique for reducing storage and computational costs. We analyze RP-based approximations of convex programs, in which the original optimization problem is approximated by the solution of a lower-dimensional problem. Such dimensionality reduction is essential in computation-limited settings, since the complexity of general convex programming can be quite high (e.g., cubic for quadratic programs, and substantially higher for semidefinite programs). In addition to computational savings, random projection is also useful for reducing memory usage, and has useful properties for privacy-sensitive optimization. We prove that the approximation ratio of this procedure can be bounded in terms of the geometry of constraint set. For a broad class of random projections, including those based on various sub-Gaussian distributions as well as randomized Hadamard and Fourier transforms, the data matrix defining the cost function can be projected down to the statistical dimension of the tangent cone of the constraints at the original solution, which is often substantially smaller than the original dimension. We illustrate consequences of our theory for various cases, including unconstrained and $\ell_1$-constrained least squares, support vector machines, low-rank matrix estimation, and discuss implications on privacy-sensitive optimization and some connections with de-noising and compressed sensing.

We investigate the relation of two fundamental tools in machine learning and signal processing, that is the support vector machine (SVM) for classification, and the Lasso technique used in regression. We show that the resulting optimization problems are equivalent, in the following sense. Given any instance of an $\ell_2$-loss soft-margin (or hard-margin) SVM, we construct a Lasso instance having the same optimal solutions, and vice versa. As a consequence, many existing optimization algorithms for both SVMs and Lasso can also be applied to the respective other problem instances. Also, the equivalence allows for many known theoretical insights for SVM and Lasso to be translated between the two settings. One such implication gives a simple kernelized version of the Lasso, analogous to the kernels used in the SVM setting. Another consequence is that the sparsity of a Lasso solution is equal to the number of support vectors for the corresponding SVM instance, and that one can use screening rules to prune the set of support vectors. Furthermore, we can relate sublinear time algorithms for the two problems, and give a new such algorithm variant for the Lasso. We also study the regularization paths for both methods.

The term "CoRE kernel" stands for correlation-resemblance kernel. In many applications (e.g., vision), the data are often high-dimensional, sparse, and non-binary. We propose two types of (nonlinear) CoRE kernels for non-binary sparse data and demonstrate the effectiveness of the new kernels through a classification experiment. CoRE kernels are simple with no tuning parameters. However, training nonlinear kernel SVM can be (very) costly in time and memory and may not be suitable for truly large-scale industrial applications (e.g. search). In order to make the proposed CoRE kernels more practical, we develop basic probabilistic hashing algorithms which transform nonlinear kernels into linear kernels.