We present a method for approximate inference for a broad class of non-conjugate probabilistic models. In particular, for the family of generalized linear model target densities we describe a rich class of variational approximating densities which can be best fit to the target by minimizing the Kullback-Leibler divergence. Our approach is based on using the Fourier representation which we show results in efficient and scalable inference.

Multiple-output regression models require estimating multiple functions, one for each output. To improve parameter estimation in such models, methods based on structural regularization of the model parameters are usually needed. In this paper, we present a multiple-output regression model that leverages the covariance structure of the functions (i.e., how the multiple functions are related with each other) as well as the conditional covariance structure of the outputs. This is in contrast with existing methods that usually take into account only one of these structures. More importantly, unlike most of the other existing methods, none of these structures need be known a priori in our model, and are learned from the data. Several previously proposed structural regularization based multiple-output regression models turn out to be special cases of our model. Moreover, in addition to being a rich model for multiple-output regression, our model can also be used in estimating the graphical model structure of a set of variables (multivariate outputs) conditioned on another set of variables (inputs). Experimental results on both synthetic and real datasets demonstrate the effectiveness of our method.

A challenging problem in hierarchical classification is to leverage the hierarchical relations among classes for improving classification performance. An even greater challenge is to do so in a manner that is computationally feasible for the large scale problems usually encountered in practice. This paper proposes a set of Bayesian methods to model hierarchical dependencies among class labels using multivari- ate logistic regression. Specifically, the parent-child relationships are modeled by placing a hierarchical prior over the children nodes centered around the parame- ters of their parents; thereby encouraging classes nearby in the hierarchy to share similar model parameters. We present new, efficient variational algorithms for tractable posterior inference in these models, and provide a parallel implementa- tion that can comfortably handle large-scale problems with hundreds of thousands of dimensions and tens of thousands of classes. We run a comparative evaluation on multiple large-scale benchmark datasets that highlights the scalability of our approach, and shows a significant performance advantage over the other state-of- the-art hierarchical methods.

In many applications, one has information, e.g., labels that are provided in a semi-supervised manner, about a specific target region of a large data set, and one wants to perform machine learning and data analysis tasks nearby that pre-specified target region. Locally-biased problems of this sort are particularly challenging for popular eigenvector-based machine learning and data analysis tools. At root, the reason is that eigenvectors are inherently global quantities. In this paper, we address this issue by providing a methodology to construct semi-supervised eigenvectors of a graph Laplacian, and we illustrate how these locally-biased eigenvectors can be used to perform locally-biased machine learning. These semi-supervised eigenvectors capture successively-orthogonalized directions of maximum variance, conditioned on being well-correlated with an input seed set of nodes that is assumed to be provided in a semi-supervised manner. We also provide several empirical examples demonstrating how these semi-supervised eigenvectors can be used to perform locally-biased learning.

We consider the estimation of an i.i.d.\ vector $\xbf \in \R^n$ from measurements $\ybf \in \R^m$ obtained by a general cascade model consisting of a known linear transform followed by a probabilistic componentwise (possibly nonlinear) measurement channel. We present a method, called adaptive generalized approximate message passing (Adaptive GAMP), that enables joint learning of the statistics of the prior and measurement channel along with estimation of the unknown vector $\xbf$. The proposed algorithm is a generalization of a recently-developed method by Vila and Schniter that uses expectation-maximization (EM) iterations where the posteriors in the E-steps are computed via approximate message passing. The techniques can be applied to a large class of learning problems including the learning of sparse priors in compressed sensing or identification of linear-nonlinear cascade models in dynamical systems and neural spiking processes. We prove that for large i.i.d.\ Gaussian transform matrices the asymptotic componentwise behavior of the adaptive GAMP algorithm is predicted by a simple set of scalar state evolution equations. This analysis shows that the adaptive GAMP method can yield asymptotically consistent parameter estimates, which implies that the algorithm achieves a reconstruction quality equivalent to the oracle algorithm that knows the correct parameter values. The adaptive GAMP methodology thus provides a systematic, general and computationally efficient method applicable to a large range of complex linear-nonlinear models with provable guarantees.

A key problem in statistics and machine learning is the determination of network structure from data. We consider the case where the structure of the graph to be reconstructed is known to be scale-free. We show that in such cases it is natural to formulate structured sparsity inducing priors using submodular functions, and we use their Lovász extension to obtain a convex relaxation. For tractable classes such as Gaussian graphical models, this leads to a convex optimization problem that can be efficiently solved. We show that our method results in an improvement in the accuracy of reconstructed networks for synthetic data. We also show how our prior encourages scale-free reconstructions on a bioinfomatics dataset.

We offer a regularized, kernel extension of the multi-set, orthogonal Procrustes problem, or hyperalignment. Our new method, called Kernel Hyperalignment, expands the scope of hyperalignment to include nonlinear measures of similarity and enables the alignment of multiple datasets with a large number of base features. With direct application to fMRI data analysis, kernel hyperalignment is well-suited for multi-subject alignment of large ROIs, including the entire cortex. We conducted experiments using real-world, multi-subject fMRI data.

We present a novel method in the family of particle MCMC methods that we refer to as particle Gibbs with ancestor sampling (PG-AS). Similarly to the existing PG with backward simulation (PG-BS) procedure, we use backward sampling to (considerably) improve the mixing of the PG kernel. Instead of using separate forward and backward sweeps as in PG-BS, however, we achieve the same effect in a single forward sweep. We apply the PG-AS framework to the challenging class of non-Markovian state-space models. We develop a truncation strategy of these models that is applicable in principle to any backward-simulation-based method, but which is particularly well suited to the PG-AS framework. In particular, as we show in a simulation study, PG-AS can yield an order-of-magnitude improved accuracy relative to PG-BS due to its robustness to the truncation error. Several application examples are discussed, including Rao-Blackwellized particle smoothing and inference in degenerate state-space models.

Topic modeling is a generalization of clustering that posits that observations (words in a document) are generated by \emph{multiple} latent factors (topics), as opposed to just one. This increased representational power comes at the cost of a more challenging unsupervised learning problem of estimating the topic-word distributions when only words are observed, and the topics are hidden. This work provides a simple and efficient learning procedure that is guaranteed to recover the parameters for a wide class of topic models, including Latent Dirichlet Allocation (LDA). For LDA, the procedure correctly recovers both the topic-word distributions and the parameters of the Dirichlet prior over the topic mixtures, using only trigram statistics (\emph{i.e.}, third order moments, which may be estimated with documents containing just three words). The method, called Excess Correlation Analysis, is based on a spectral decomposition of low-order moments via two singular value decompositions (SVDs). Moreover, the algorithm is scalable, since the SVDs are carried out only on $k \times k$ matrices, where $k$ is the number of latent factors (topics) and is typically much smaller than the dimension of the observation (word) space.

Finding maximum aposteriori (MAP) assignments in graphical models is an important task in many applications. Since the problem is generally hard, linear programming (LP) relaxations are often used. Solving these relaxations efficiently is thus an important practical problem. In recent years, several authors have proposed message passing updates corresponding to coordinate descent in the dual LP. However,these are generally not guaranteed to converge to a global optimum. One approach to remedy this is to smooth the LP, and perform coordinate descent on the smoothed dual. However, little is known about the convergence rate of this procedure. Here we perform a thorough rate analysis of such schemes and derive primal and dual convergence rates. We also provide a simple dual to primal mapping that yields feasible primal solutions with a guaranteed rate of convergence. Empirical evaluation supports our theoretical claims and shows that the method is highly competitive with state of the art approaches that yield global optima.