We present a novel approach to inference in conditionally Gaussian continuous time stochastic processes, where the latent process is a Markovian jump process. We first consider the case of jump-diffusion processes, where the drift of a linear stochastic differential equation can jump at arbitrary time points. We derive partial differential equations for exact inference and present a very efficient mean field approximation. By introducing a novel lower bound on the free energy, we then generalise our approach to Gaussian processes with arbitrary covariance, such as the non-Markovian RBF covariance. We present results on both simulated and real data, showing that the approach is very accurate in capturing latent dynamics and can be useful in a number of real data modelling tasks.

Regularization technique has become a principled tool for statistics and machine learning research and practice. However, in most situations, these regularization terms are not well interpreted, especially on how they are related to the loss function anddata. In this paper, we propose a robust minimax framework to interpret the relationship between data and regularization terms for a large class of loss functions. We show that various regularization terms are essentially corresponding todifferent distortions to the original data matrix. This minimax framework includes ridge regression, lasso, elastic net, fused lasso, group lasso, local coordinate coding,multiple kernel learning, etc., as special cases. Within this minimax framework, we further give mathematically exact definition for a novel representation calledsparse grouping representation (SGR), and prove a set of sufficient conditions for generating such group level sparsity. Under these sufficient conditions, alarge set of consistent regularization terms can be designed. This SGR is essentially different from group lasso in the way of using class or group information, andit outperforms group lasso when there appears group label noise. We also provide some generalization bounds in a classification setting.

As increasing amounts of sensitive personal information finds its way into data repositories, it is important to develop analysis mechanisms that can derive aggregate information from these repositories without revealing information about individual data instances. Though the differential privacy model provides a framework to analyze such mechanisms for databases belonging to a single party, this framework has not yet been considered in a multi-party setting. In this paper, we propose a privacy-preserving protocol for composing a differentially private aggregate classifier using classifiers trained locally by separate mutually untrusting parties. The protocol allows these parties to interact with an untrusted curator to construct additive shares of a perturbed aggregate classifier. We also present a detailed theoretical analysis containing a proof of differential privacy of the perturbed aggregate classifier and a bound on the excess risk introduced by the perturbation. We verify the bound with an experimental evaluation on a real dataset.

We present original empirical Bernstein inequalities for U-statistics with bounded symmetric kernels q. They are expressed with respect to empirical estimates of either the variance of q or the conditional variance that appears in the Bernstein-type inequality for U-statistics derived by Arcones [2]. Our result subsumes other existing empirical Bernstein inequalities, as it reduces to them when U-statistics of order 1 are considered. In addition, it is based on a rather direct argument using two applications of the same (non-empirical) Bernstein inequality for U-statistics. We discuss potential applications of our new inequalities, especially in the realm of learning ranking/scoring functions. In the process, we exhibit an efficient procedure to compute the variance estimates for the special case of bipartite ranking that rests on a sorting argument. We also argue that our results may provide test set bounds and particularly interesting empirical racing algorithms for the problem of online learning of scoring functions.

We describe a model based on a Boltzmann machine with third-order connections that can learn how to accumulate information about a shape over several fixations. The model uses a retina that only has enough high resolution pixels to cover a small area of the image, so it must decide on a sequence of fixations and it must combine the glimpse" at each fixation with the location of the fixation before integrating the information with information from other glimpses of the same object. We evaluate this model on a synthetic dataset and two image classification datasets, showing that it can perform at least as well as a model trained on whole images."

Gaussian graphical models with sparsity in the inverse covariance matrix are of significant interest in many modern applications. For the problem of recovering the graphical structure, information criteria provide useful optimization objectives for algorithms searching through sets of graphs or for selection of tuning parameters of other methods such as the graphical lasso, which is a likelihood penalization technique. In this paper we establish the asymptotic consistency of an extended Bayesian information criterion for Gaussian graphical models in a scenario where both the number of variables p and the sample size n grow. Compared to earlier work on the regression case, our treatment allows for growth in the number of non-zero parameters in the true model, which is necessary in order to cover connected graphs. We demonstrate the performance of this criterion on simulated data when used in conjuction with the graphical lasso, and verify that the criterion indeed performs better than either cross-validation or the ordinary Bayesian information criterion when p and the number of non-zero parameters q both scale with n.

Divisive normalization (DN) has been advocated as an effective nonlinear {\em efficient coding} transform for natural sensory signals with applications in biology and engineering. In this work, we aim to establish a connection between the DN transform and the statistical properties of natural sensory signals. Our analysis is based on the use of multivariate {\em t} model to capture some important statistical properties of natural sensory signals. The multivariate {\em t} model justifies DN as an approximation to the transform that completely eliminates its statistical dependency. Furthermore, using the multivariate {\em t} model and measuring statistical dependency with multi-information, we can precisely quantify the statistical dependency that is reduced by the DN transform. We compare this with the actual performance of the DN transform in reducing statistical dependencies of natural sensory signals. Our theoretical analysis and quantitative evaluations confirm DN as an effective efficient coding transform for natural sensory signals. On the other hand, we also observe a previously unreported phenomenon that DN may increase statistical dependencies when the size of pooling is small.

We study learning curves for Gaussian process regression which characterise performance interms of the Bayes error averaged over datasets of a given size. Whilst learning curves are in general very difficult to calculate we show that for discrete input domains, where similarity between input points is characterised in terms of a graph, accurate predictions can be obtained. These should in fact become exact for large graphs drawn from a broad range of random graph ensembles with arbitrary degree distributions where each input (node) is connected only to a finite number of others. Our approach is based on translating the appropriate belief propagation equations to the graph ensemble. We demonstrate the accuracy of the predictions for Poisson (Erdos-Renyi) and regular random graphs, and discuss when and why previous approximations of the learning curve fail.

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.