We consider a regularized least squares problem, with regularization by structured sparsity-inducing norms, which extend the usual $\ell_1$ and the group lasso penalty, by allowing the subsets to overlap. Such regularizations lead to nonsmooth problems that are difficult to optimize, and we propose in this paper a suitable version of an accelerated proximal method to solve them. We prove convergence of a nested procedure, obtained composing an accelerated proximal method with an inner algorithm for computing the proximity operator. By exploiting the geometrical properties of the penalty, we devise a new active set strategy, thanks to which the inner iteration is relatively fast, thus guaranteeing good computational performances of the overall algorithm. Our approach allows to deal with high dimensional problems without pre-processing for dimensionality reduction, leading to better computational and prediction performances with respect to the state-of-the art methods, as shown empirically both on toy and real data.

The technological applications of hidden Markov models have been extremely diverse and successful, including natural language processing, gesture recognition, gene sequencing, and Kalman filtering of physical measurements. HMMs are highly non-linear statistical models, and just as linear models are amenable to linear algebraic techniques, non-linear models are amenable to commutative algebra and algebraic geometry. This paper closely examines HMMs in which all the hidden random variables are binary. Its main contributions are (1) a birational parametrization for every such HMM, with an explicit inverse for recovering the hidden parameters in terms of observables, (2) a semialgebraic model membership test for every such HMM, and (3) minimal defining equations for the 4-node fully binary model, comprising 21 quadrics and 29 cubics, which were computed using Grobner bases in the cumulant coordinates of Sturmfels and Zwiernik. The new model parameters in (1) are rationally identifiable in the sense of Sullivant, Garcia-Puente, and Spielvogel, and each model's Zariski closure is therefore a rational projective variety of dimension 5. Grobner basis computations for the model and its graph are found to be considerably faster using these parameters. In the case of two hidden states, item (2) supersedes a previous algorithm of Schonhuth which is only generically defined, and the defining equations (3) yield new invariants for HMMs of all lengths $\geq 4$. Such invariants have been used successfully in model selection problems in phylogenetics, and one can hope for similar applications in the case of HMMs.

Supply chain management is a very dynamic operation research problem where one has to quickly adapt according to the changes perceived in environment in order to maximize the benefit or minimize the loss. Therefore we require a system which changes as per the changing requirements. Multi agent system technology in recent times has emerged as a possible way of efficient solution implementation for many such complex problems. Our research here focuses on building a Multi Agent System (MAS), which implements a modified version of Gravitational Search swarm intelligence Algorithm (GSA) to find out an optimal strategy in managing the demand supply chain. We target the grains distribution system among various centers of Food Corporation of India (FCI) as application domain. We assume centers with larger stocks as objects of greater mass and vice versa. Applying Newtonian law of gravity as suggested in GSA, larger objects attract objects of smaller mass towards itself, creating a virtual grain supply source. As heavier object sheds its mass by supplying some to the one in demand, it loses its gravitational pull and thus keeps the whole system of supply chain per-fectly in balance. The multi agent system helps in continuous updation of the whole system with the help of autonomous agents which react to the change in environment and act accordingly. This model also reduces the communication bottleneck to greater extents.

We develop an improved bound for the approximation error of the Nystr\"{o}m method under the assumption that there is a large eigengap in the spectrum of kernel matrix. This is based on the empirical observation that the eigengap has a significant impact on the approximation error of the Nystr\"{o}m method. Our approach is based on the concentration inequality of integral operator and the theory of matrix perturbation. Our analysis shows that when there is a large eigengap, we can improve the approximation error of the Nystr\"{o}m method from $O(N/m^{1/4})$ to $O(N/m^{1/2})$ when measured in Frobenius norm, where $N$ is the size of the kernel matrix, and $m$ is the number of sampled columns.

A statistical model or a learning machine is called regular if the map taking a parameter to a probability distribution is one-to-one and if its Fisher information matrix is always positive definite. If otherwise, it is called singular. In regular statistical models, the Bayes free energy, which is defined by the minus logarithm of Bayes marginal likelihood, can be asymptotically approximated by the Schwarz Bayes information criterion (BIC), whereas in singular models such approximation does not hold. Recently, it was proved that the Bayes free energy of a singular model is asymptotically given by a generalized formula using a birational invariant, the real log canonical threshold (RLCT), instead of half the number of parameters in BIC. Theoretical values of RLCTs in several statistical models are now being discovered based on algebraic geometrical methodology. However, it has been difficult to estimate the Bayes free energy using only training samples, because an RLCT depends on an unknown true distribution. In the present paper, we define a widely applicable Bayesian information criterion (WBIC) by the average log likelihood function over the posterior distribution with the inverse temperature $1/\log n$, where $n$ is the number of training samples. We mathematically prove that WBIC has the same asymptotic expansion as the Bayes free energy, even if a statistical model is singular for and unrealizable by a statistical model. Since WBIC can be numerically calculated without any information about a true distribution, it is a generalized version of BIC onto singular statistical models.

The aim of a Content-Based Image Retrieval (CBIR) system, also known as Query by Image Content (QBIC), is to help users to retrieve relevant images based on their contents. CBIR technologies provide a method to find images in large databases by using unique descriptors from a trained image. The image descriptors include texture, color, intensity and shape of the object inside an image. Several feature-extraction techniques viz., Average RGB, Color Moments, Co-occurrence, Local Color Histogram, Global Color Histogram and Geometric Moment have been critically compared in this paper. However, individually these techniques result in poor performance. So, combinations of these techniques have also been evaluated and results for the most efficient combination of techniques have been presented and optimized for each class of image query. We also propose an improvement in image retrieval performance by introducing the idea of Query modification through image cropping. It enables the user to identify a region of interest and modify the initial query to refine and personalize the image retrieval results.

Gaia will obtain astrometry and spectrophotometry for essentially all sources in the sky down to a broad band magnitude limit of G=20, an expected yield of 10^9 stars. Its main scientific objective is to reveal the formation and evolution of our Galaxy through chemo-dynamical analysis. In addition to inferring positions, parallaxes and proper motions from the astrometry, we must also infer the astrophysical parameters of the stars from the spectrophotometry, the BP/RP spectrum. Here we investigate the performance of three different algorithms (SVM, ILIUM, Aeneas) for estimating the effective temperature, line-of-sight interstellar extinction, metallicity and surface gravity of A-M stars over a wide range of these parameters and over the full magnitude range Gaia will observe (G=6-20mag). One of the algorithms, Aeneas, infers the posterior probability density function over all parameters, and can optionally take into account the parallax and the Hertzsprung-Russell diagram to improve the estimates. For all algorithms the accuracy of estimation depends on G and on the value of the parameters themselves, so a broad summary of performance is only approximate. For stars at G=15 with less than two magnitudes extinction, we expect to be able to estimate Teff to within 1%, logg to 0.1-0.2dex, and [Fe/H] (for FGKM stars) to 0.1-0.2dex, just using the BP/RP spectrum (mean absolute error statistics are quoted). Performance degrades at larger extinctions, but not always by a large amount. Extinction can be estimated to an accuracy of 0.05-0.2mag for stars across the full parameter range with a priori unknown extinction between 0 and 10mag. Performance degrades at fainter magnitudes, but even at G=19 we can estimate logg to better than 0.2dex for all spectral types, and [Fe/H] to within 0.35dex for FGKM stars, for extinctions below 1mag.

Machine learning algorithms are rarely parameter-free; whether via the properties of a regularizer, the hyperprior of a generative model, or the step size of a gradient-based optimization, learning procedures almost always require a set of high-level choices that significantly impact generalization performance. As a practitioner, one is usually able to specify the general framework of an inductive bias much more easily than the particular weighting that it should have relative to training data. As a result, these high-level parameters are often considered a nuisance, making it desirable to develop algorithms with as few of these "knobs" as possible. Another, more flexible take on this issue is to view the optimization of high-level parameters as a procedure to be automated. Specifically, we could view such tuning as the optimization of an unknown black-box function that reflects generalization performance and invoke algorithms developed for such problems. These optimization problems have a somewhat different flavor than the low-level objectives one often encounters as part of a training procedure: here function evaluations are very expensive, as they involve running the primary machine learning algorithm to completion. In this setting where function evaluations are expensive, it is desirable to spend computational time making better choices about where to seek the best parameters. Bayesian optimization (Mockus et al., 1978) provides an elegant approach and has been shown to outperform other state of the art global optimization algorithms on a number of challenging optimization benchmark functions (Jones, 2001).

Bayesian learning is often hampered by large computational expense. As a powerful generalization of popular belief propagation, expectation propagation (EP) efficiently approximates the exact Bayesian computation. Nevertheless, EP can be sensitive to outliers and suffer from divergence for difficult cases. To address this issue, we propose a new approximate inference approach, relaxed expectation propagation (REP). It relaxes the moment matching requirement of expectation propagation by adding a relaxation factor into the KL minimization. We penalize this relaxation with a $l_1$ penalty. As a result, when two distributions in the relaxed KL divergence are similar, the relaxation factor will be penalized to zero and, therefore, we obtain the original moment matching; In the presence of outliers, these two distributions are significantly different and the relaxation factor will be used to reduce the contribution of the outlier. Based on this penalized KL minimization, REP is robust to outliers and can greatly improve the posterior approximation quality over EP. To examine the effectiveness of REP, we apply it to Gaussian process classification, a task known to be suitable to EP. Our classification results on synthetic and UCI benchmark datasets demonstrate significant improvement of REP over EP and Power EP--in terms of algorithmic stability, estimation accuracy and predictive performance.

Divergence from a random baseline is a technique for the evaluation of document clustering. It ensures cluster quality measures are performing work that prevents ineffective clusterings from giving high scores to clusterings that provide no useful result. These concepts are defined and analysed using intrinsic and extrinsic approaches to the evaluation of document cluster quality. This includes the classical clusters to categories approach and a novel approach that uses ad hoc information retrieval. The divergence from a random baseline approach is able to differentiate ineffective clusterings encountered in the INEX XML Mining track. It also appears to perform a normalisation similar to the Normalised Mutual Information (NMI) measure but it can be applied to any measure of cluster quality. When it is applied to the intrinsic measure of distortion as measured by RMSE, subtraction from a random baseline provides a clear optimum that is not apparent otherwise. This approach can be applied to any clustering evaluation. This paper describes its use in the context of document clustering evaluation.