Kernel Principal Component Analysis (KPCA) is a popular generalization of linear PCA that allows non-linear feature extraction. In KPCA, data in the input space is mapped to higher (usually) dimensional feature space where the data can be linearly modeled. The feature space is typically induced implicitly by a kernel function, and linear PCA in the feature space is performed via the kernel trick. However, due to the implicitness of the feature space, some extensions of PCA such as robust PCA cannot be directly generalized to KPCA. This paper presents a technique to overcome this problem, and extends it to a unified framework for treating noise, missing data, and outliers in KPCA. Our method is based on a novel cost function to perform inference in KPCA. Extensive experiments, in both synthetic and real data, show that our algorithm outperforms existing methods.

We describe a new algorithmic framework for inference in probabilistic models, and apply it to inference for latent Dirichlet allocation. Our framework adopts the methodology of variational inference, but unlike existing variational methods such as mean field and expectation propagation it is not restricted to tractable classes of approximating distributions. Our approach can also be viewed as a sequential Monte Carlo (SMC) method, but unlike existing SMC methods there is no need to design the artificial sequence of distributions. Notably, our framework offers a principled means to exchange the variance of an importance sampling estimate for the bias incurred through variational approximation. Experiments on a challenging inference problem in population genetics demonstrate improvements in stability and accuracy over existing methods, and at a comparable cost.

One of the original goals of computer vision was to fully understand a natural scene. This requires solving several problems simultaneously, including object detection, labeling of meaningful regions, and 3d reconstruction. While great progress has been made in tackling each of these problems in isolation, only recently have researchers again been considering the difficult task of assembling various methods to the mutual benefit of all. We consider learning a set of such classification models in such a way that they both solve their own problem and help each other. We develop a framework known as Cascaded Classification Models (CCM), where repeated instantiations of these classifiers are coupled by their input/output variables in a cascade that improves performance at each level. Our method requires only a limited “black box” interface with the models, allowing us to use very sophisticated, state-of-the-art classifiers without having to look under the hood. We demonstrate the effectiveness of our method on a large set of natural images by combining the subtasks of scene categorization, object detection, multiclass image segmentation, and 3d scene reconstruction.

Covariance estimation for high dimensional vectors is a classically difficult problem in statistical analysis and machine learning due to limited sample size. In this paper, we propose a new approach to covariance estimation, which is based on constrained maximum likelihood (ML) estimation of the covariance. Specifically, the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform (SMT). The SMT is formed by a product of pairwise coordinate rotations known as Givens rotations. Using this framework, the covariance can be efficiently estimated using greedy minimization of the log likelihood function, and the number of Givens rotations can be efficiently computed using a cross-validation procedure. The estimator obtained using this method is always positive definite and well-conditioned even with limited sample size. Experiments on hyperspectral data show that SMT covariance estimation results in consistently better estimates of the covariance for a variety of different classes and sample sizes compared to traditional shrinkage estimators.

Extensive games are often used to model the interactions of multiple agents within an environment. Much recent work has focused on increasing the size of an extensive game that can be feasibly solved. Despite these improvements, many interesting games are still too large for such techniques. A common approach for computing strategies in these large games is to first employ an abstraction technique to reduce the original game to an abstract game that is of a manageable size. This abstract game is then solved and the resulting strategy is used in the original game. Most top programs in recent AAAI Computer Poker Competitions use this approach. The trend in this competition has been that strategies found in larger abstract games tend to beat strategies found in smaller abstract games. These larger abstract games have more expressive strategy spaces and therefore contain better strategies. In this paper we present a new method for computing strategies in large games. This method allows us to compute more expressive strategies without increasing the size of abstract games that we are required to solve. We demonstrate the power of the approach experimentally in both small and large games, while also providing a theoretical justification for the resulting improvement.

Given $n$ noisy samples with $p$ dimensions, where $n \ll p$, we show that the multi-stage thresholding procedures can accurately estimate a sparse vector $\beta \in \R^p$ in a linear model, under the restricted eigenvalue conditions (Bickel-Ritov-Tsybakov 09). Thus our conditions for model selection consistency are considerably weaker than what has been achieved in previous works. More importantly, this method allows very significant values of $s$, which is the number of non-zero elements in the true parameter $\beta$. For example, it works for cases where the ordinary Lasso would have failed. Finally, we show that if $X$ obeys a uniform uncertainty principle and if the true parameter is sufficiently sparse, the Gauss-Dantzig selector (Cand\{e}s-Tao 07) achieves the $\ell_2$ loss within a logarithmic factor of the ideal mean square error one would achieve with an oracle which would supply perfect information about which coordinates are non-zero and which are above the noise level, while selecting a sufficiently sparse model.

Multitask learning addressed the problem of learning related tasks whose information can be shared each other. Traditional problem usually deal with homogeneous tasks such as regression, classification individually. In this paper we consider the problem learning multiple related tasks where tasks consist of both continuous and discrete outputs from a common set of input variables that lie in a high-dimensional space. All of the tasks are related in the sense that they share the same set of relevant input variables, but the amount of influence of each input on different outputs may vary. We formulate this problem as a combination of linear regression and logistic regression and model the joint sparsity as L1/Linf and L1/L2-norm of the model parameters. Among several possible applications, our approach addresses an important open problem in genetic association mapping, where we are interested in discovering genetic markers that influence multiple correlated traits jointly. In our experiments, we demonstrate our method in the scenario of association mapping, using simulated and asthma data, and show that the algorithm can effectively recover the relevant inputs with respect to all of the tasks.

Implementations of topic models typically use symmetric Dirichlet priors with fixed concentration parameters, with the implicit assumption that such smoothing parameters" have little practical effect. In this paper, we explore several classes of structured priors for topic models. We find that an asymmetric Dirichlet prior over the document-topic distributions has substantial advantages over a symmetric prior, while an asymmetric prior over the topic-word distributions provides no real benefit. Approximation of this prior structure through simple, efficient hyperparameter optimization steps is sufficient to achieve these performance gains. The prior structure we advocate substantially increases the robustness of topic models to variations in the number of topics and to the highly skewed word frequency distributions common in natural language. Since this prior structure can be implemented using efficient algorithms that add negligible cost beyond standard inference techniques, we recommend it as a new standard for topic modeling."

We propose a new approach to the problem of robust estimation in multiview geometry. Inspired by recent advances in the sparse recovery problem of statistics, our estimator is defined as a Bayesian maximum a posteriori with multivariate Laplace prior on the vector describing the outliers. This leads to an estimator in which the fidelity to the data is measured by the $L_\infty$-norm while the regularization is done by the $L_1$-norm. The proposed procedure is fairly fast since the outlier removal is done by solving one linear program (LP). An important difference compared to existing algorithms is that for our estimator it is not necessary to specify neither the number nor the proportion of the outliers. The theoretical results, as well as the numerical example reported in this work, confirm the efficiency of the proposed approach.

Distributed learning is a problem of fundamental interest in machine learning and cognitive science. In this paper, we present asynchronous distributed learning algorithms for two well-known unsupervised learning frameworks: Latent Dirichlet Allocation (LDA) and Hierarchical Dirichlet Processes (HDP). In the proposed approach, the data are distributed across P processors, and processors independently perform Gibbs sampling on their local data and communicate their information in a local asynchronous manner with other processors. We demonstrate that our asynchronous algorithms are able to learn global topic models that are statistically as accurate as those learned by the standard LDA and HDP samplers, but with significant improvements in computation time and memory. We show speedup results on a 730-million-word text corpus using 32 processors, and we provide perplexity results for up to 1500 virtual processors. As a stepping stone in the development of asynchronous HDP, a parallel HDP sampler is also introduced.