This paper is about the asymptotic behavior of the Kalman filter [11]. The Kalman-Bucy filter merges predictions from a trusted model of the dynamics of the system with incoming measurements in order to get an accurate, real-time estimate of the unknown internal state of the system. The estimation relies on the computation of a positive semidefinite matrix P, the covariance of the estimation error. The difference equation verified by P is a discrete-time algebraic Riccati equation. Kalman showed that, for a linear time-invariant system, under detectability conditions, the Riccati equation converges to a fixed point, which is unique under certain stabilizability conditions ([10], see also [9]). The classical convergence analysis requires several steps, showing that the error covariance is upper bounded, that, with zero initial value, it is monotone increasing, so that it admits a limit, and then proving that the corresponding filter is stable and that the limit is the same for all initial covariances. In [4] Bougerol proposed a more geometric convergence analysis by showing that the discrete-time Riccati iteration is a contraction for the Riemannian metric associated to the cone of positive definite matrices. Other authors elaborated along these lines (see e.g.

In this paper we show that for the purposes of dimensionality reduction certain class of structured random matrices behave similarly to random Gaussian matrices. This class includes several matrices for which matrix-vector multiply can be computed in log-linear time, providing efficient dimensionality reduction of general sets. In particular, we show that using such matrices any set from high dimensions can be embedded into lower dimensions with near optimal distortion. We obtain our results by connecting dimensionality reduction of any set to dimensionality reduction of sparse vectors via a chaining argument.

The nowadays massive amounts of generated and communicated data present major challenges in their processing. While capable of successfully classifying nonlinearly separable objects in various settings, subspace clustering (SC) methods incur prohibitively high computational complexity when processing large-scale data. Inspired by the random sampling and consensus (RANSAC) approach to robust regression, the present paper introduces a randomized scheme for SC, termed sketching and validation (SkeVa-)SC, tailored for large-scale data. At the heart of SkeVa-SC lies a randomized scheme for approximating the underlying probability density function of the observed data by kernel smoothing arguments. Sparsity in data representations is also exploited to reduce the computational burden of SC, while achieving high clustering accuracy. Performance analysis as well as extensive numerical tests on synthetic and real data corroborate the potential of SkeVa-SC and its competitive performance relative to state-of-the-art scalable SC approaches. Keywords: Subspace clustering, big data, kernel smoothing, randomization, sketching, validation, sparsity.

We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows us to optimize over subsets of valid difference of convex decompositions (dcds) and find ones that speed up the convex-concave procedure (CCP). We prove, however, that optimizing over the entire set of dcds is NP-hard.

A common strategy for sparse linear regression is to introduce regularization, which eliminates irrelevant features by letting the corresponding weights be zeros. However, regularization often shrinks the estimator for relevant features, which leads to incorrect feature selection. Motivated by the above-mentioned issue, we propose Bayesian masking (BM), a sparse estimation method which imposes no regularization on the weights. The key concept of BM is to introduce binary latent variables that randomly mask features. Estimating the masking rates determines the relevance of the features automatically. We derive a variational Bayesian inference algorithm that maximizes the lower bound of the factorized information criterion (FIC), which is a recently developed asymptotic criterion for evaluating the marginal log-likelihood. In addition, we propose reparametrization to accelerate the convergence of the derived algorithm. Finally, we show that BM outperforms Lasso and automatic relevance determination (ARD) in terms of the sparsity-shrinkage trade-off.

Similarity between objects is multi-faceted and it can be easier for human annotators to measure it when the focus is on a specific aspect. We consider the problem of mapping objects into view-specific embeddings where the distance between them is consistent with the similarity comparisons of the form "from the t-th view, object A is more similar to B than to C". Our framework jointly learns view-specific embeddings exploiting correlations between views. Experiments on a number of datasets, including one of multi-view crowdsourced comparison on bird images, show the proposed method achieves lower triplet generalization error when compared to both learning embeddings independently for each view and all views pooled into one view. Our method can also be used to learn multiple measures of similarity over input features taking class labels into account and compares favorably to existing approaches for multi-task metric learning on the ISOLET dataset.

Answer set programming is a declarative programming paradigm oriented towards difficult combinatorial search problems. A fundamental task in answer set programming is to compute stable models, i.e., solutions of logic programs. Answer set solvers are the programs that perform this task. The problem of deciding whether a disjunctive program has a stable model is $\Sigma^P_2$-complete. The high complexity of reasoning within disjunctive logic programming is responsible for few solvers capable of dealing with such programs, namely DLV, GnT, Cmodels, CLASP and WASP. In this paper we show that transition systems introduced by Nieuwenhuis, Oliveras, and Tinelli to model and analyze satisfiability solvers can be adapted for disjunctive answer set solvers. Transition systems give a unifying perspective and bring clarity in the description and comparison of solvers. They can be effectively used for analyzing, comparing and proving correctness of search algorithms as well as inspiring new ideas in the design of disjunctive answer set solvers. In this light, we introduce a general template, which accounts for major techniques implemented in disjunctive solvers. We then illustrate how this general template captures solvers DLV, GnT and Cmodels. We also show how this framework provides a convenient tool for designing new solving algorithms by means of combinations of techniques employed in different solvers.

In this paper, a Bayesian inference technique based on Taylor series approximation of the logarithm of the likelihood function is presented. The proposed approximation is devised for the case, where the prior distribution belongs to the exponential family of distributions. The logarithm of the likelihood function is linearized with respect to the sufficient statistic of the prior distribution in exponential family such that the posterior obtains the same exponential family form as the prior. Similarities between the proposed method and the extended Kalman filter for nonlinear filtering are illustrated. Furthermore, an extended target measurement update for target models where the target extent is represented by a random matrix having an inverse Wishart distribution is derived. The approximate update covers the important case where the spread of measurement is due to the target extent as well as the measurement noise in the sensor.

The task of determining labels of all network nodes based on the knowledge about network structure and labels of some training subset of nodes is called the within-network classification. It may happen that none of the labels of the nodes is known and additionally there is no information about number of classes to which nodes can be assigned. In such a case a subset of nodes has to be selected for initial label acquisition. The question that arises is: "labels of which nodes should be collected and used for learning in order to provide the best classification accuracy for the whole network?". Active learning and inference is a practical framework to study this problem. A set of methods for active learning and inference for within network classification is proposed and validated. The utility score calculation for each node based on network structure is the first step in the process. The scores enable to rank the nodes. Based on the ranking, a set of nodes, for which the labels are acquired, is selected (e.g. by taking top or bottom N from the ranking). The new measure-neighbour methods proposed in the paper suggest not obtaining labels of nodes from the ranking but rather acquiring labels of their neighbours. The paper examines 29 distinct formulations of utility score and selection methods reporting their impact on the results of two collective classification algorithms: Iterative Classification Algorithm and Loopy Belief Propagation. We advocate that the accuracy of presented methods depends on the structural properties of the examined network. We claim that measure-neighbour methods will work better than the regular methods for networks with higher clustering coefficient and worse than regular methods for networks with low clustering coefficient. According to our hypothesis, based on clustering coefficient we are able to recommend appropriate active learning and inference method.

Work in the classification literature has shown that in computing a classification function, one need not know the class membership of all observations in the training set; the unlabeled observations still provide information on the marginal distribution of the feature set, and can thus contribute to increased classification accuracy for future observations. The present paper will show that this scheme can also be used for the estimation of class prior probabilities, which would be very useful in applications in which it is difficult or expensive to determine class membership. Both parametric and nonparametric estimators are developed. Asymptotic distributions of the estimators are derived, and it is proven that the use of the unlabeled observations does reduce asymptotic variance. This methodology is also extended to the estimation of subclass probabilities.