A Note on the Generalization Performance of Kernel Classifiers with Margin

We present distribution independent bounds on the generalization misclassification performance of a family of kernel classifiers with margin. Support Vector Machine classifiers (SVM) stem out of this class of machines. The bounds are derived through computations of the $V_gamma$ dimension of a family of loss functions where the SVM one belongs to. Bounds that use functions of margin distributions (i.e. functions of the slack variables of SVM) are derived.

On Convergence Properties of the EM Algorithm for Gaussian Mixtures

"Expectation-Maximization'' (EM) algorithm and gradient-based approaches for maximum likelihood learning of finite Gaussian mixtures. We show that the EM step in parameter space is obtained from the gradient via a projection matrix $P$, and we provide an explicit expression for the matrix. We then analyze the convergence of EM in terms of special properties of $P$ and provide new results analyzing the effect that $P$ has on the likelihood surface. Based on these mathematical results, we present a comparative discussion of the advantages and disadvantages of EM and other algorithms for the learning of Gaussian mixture models.

Sparse Correlation Kernel Analysis and Reconstruction

This paper presents a new paradigm for signal reconstruction and superresolution, Correlation Kernel Analysis (CKA), that is based on the selection of a sparse set of bases from a large dictionary of class- specific basis functions. The basis functions that we use are the correlation functions of the class of signals we are analyzing. To choose the appropriate features from this large dictionary, we use Support Vector Machine (SVM) regression and compare this to traditional Principal Component Analysis (PCA) for the tasks of signal reconstruction, superresolution, and compression. The testbed we use in this paper is a set of images of pedestrians. This paper also presents results of experiments in which we use a dictionary of multiscale basis functions and then use Basis Pursuit De-Noising to obtain a sparse, multiscale approximation of a signal. The results are analyzed and we conclude that 1) when used with a sparse representation technique, the correlation function is an effective kernel for image reconstruction and superresolution, 2) for image compression, PCA and SVM have different tradeoffs, depending on the particular metric that is used to evaluate the results, 3) in sparse representation techniques, L_1 is not a good proxy for the true measure of sparsity, L_0, and 4) the L_epsilon norm may be a better error metric for image reconstruction and compression than the L_2 norm, though the exact psychophysical metric should take into account high order structure in images.

On the V(subscript gamma) Dimension for Regression in Reproducing Kernel Hilbert Spaces

This paper presents a computation of the $V_gamma$ dimension for regression in bounded subspaces of Reproducing Kernel Hilbert Spaces (RKHS) for the Support Vector Machine (SVM) regression $epsilon$-insensitive loss function, and general $L_p$ loss functions. Finiteness of the RV_gamma$ dimension is shown, which also proves uniform convergence in probability for regression machines in RKHS subspaces that use the $L_epsilon$ or general $L_p$ loss functions. This paper presenta a novel proof of this result also for the case that a bias is added to the functions in the RKHS.

A comparison of the alr and ilr transformations for kernel density estimation of compositional data

In a seminal paper, Aitchison and Lauder (1985) introduced classical kernel density estimation techniques in the context of compositional data analysis. Indeed, they gave two options for the choice of the kernel to be used in the kernel estimator. One of these kernels is based on the use the alr transformation on the simplex SD jointly with the normal distribution on RD-1. However, these authors themselves recognized that this method has some deficiencies. A method for overcoming these dificulties based on recent developments for compositional data analysis and multivariate kernel estimation theory, combining the ilr transformation with the use of the normal density with a full bandwidth matrix, was recently proposed in Martín-Fernández, Chacón and Mateu- Figueras (2006). Here we present an extensive simulation study that compares both methods in practice, thus exploring the finite-sample behaviour of both estimators

The representation of conditional expectations of non-Gaussian noise

For Wiener spaces conditional expectations and $L^{2}$-martingales w.r.t. the natural filtration have a natural representation in terms of chaos expansion. In this note an extension to larger classes of processes is discussed. In particular, it is pointed out that orthogonality of the chaos expansion is not required.

Time-multiplexed Laguerre-Gaussian holographic optical tweezers for biological applications

A ferroelectric liquid crystal spatial light modulator is used to generate up to 24 independently controllable traps in a holographic optical tweezers system using time-multiplexed Fresnel zone plates. For use in biological applications, helical zone plates are used to generate Laguerre-Gaussian laser modes. The high speed switching of the ferroelectric device together with recent advances in computer technology enable fast, smooth movement of traps that can be independently controlled in real time. This is demonstrated by the trapping and manipulation of yeast cells and fungal spores. (c) 2006 Optical Society of America.

Bayesian analysis of an inverse Gaussian correlated frailty model

In survival analysis frailty is often used to model heterogeneity between individuals or correlation within clusters. Typically frailty is taken to be a continuous random effect, yielding a continuous mixture distribution for survival times. A Bayesian analysis of a correlated frailty model is discussed in the context of inverse Gaussian frailty. An MCMC approach is adopted and the deviance information criterion is used to compare models. As an illustration of the approach a bivariate data set of corneal graft survival times is analysed. (C) 2006 Elsevier B.V. All rights reserved.

Antioxidant and tyrosinase inhibitory activity of mango seed kernel by product

The antioxidant and tyrosinase inhibitory properties of extracts of mango seed kernel (Mangifera indica L.), which is normally discarded when the fruit is processed, were studied. Extracts contained phenolic components by a high antioxidant activity, which was assessed in homogeneous solution by the 2,2-diphenyt-1-picrylhydrazyl radical and 2,2'-azinobis (3-ethylbenzothialozinesulfonic acid) radical cation-scavenging assays and in an emulsion with the ferric thiocyanate test. The extracts also possessed tyrosinase inhibitory activity. Drying conditions and extraction solvent were varied, and optimum conditions for preparation of mango seed kernel extract were found to be sun-drying with ethanol extraction at room temperature. Refluxing in acidified ethanol gave an increase in yield and the obtained extract had the highest content of total phenolics, and also was the most effective antioxidant with the highest radical-scavenging, metal-chelating and tyrosinase inhibitory activity. The extracts did not cause acute irritation of rabbit skins. Our study for the first time reveals the high total phenol content, radical-scavenging, metal-chelating and tyrosinase inhibitory activities of the extract from mango seed kernel. This extract may be suitable for use in food, cosmetic, nutraceutical and pharmaceutical applications. (C) 2009 Elsevier Ltd. All rights reserved.

A sparse kernel density estimation algorithm using forward constrained regression

Using the classical Parzen window (PW) estimate as the target function, the sparse kernel density estimator is constructed in a forward constrained regression manner. The leave-one-out (LOO) test score is used for kernel selection. The jackknife parameter estimator subject to positivity constraint check is used for the parameter estimation of a single parameter at each forward step. As such the proposed approach is simple to implement and the associated computational cost is very low. An illustrative example is employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with comparable accuracy to that of the classical Parzen window estimate.

Sparse kernel modelling: a unified approach

A unified approach is proposed for sparse kernel data modelling that includes regression and classification as well as probability density function estimation. The orthogonal-least-squares forward selection method based on the leave-one-out test criteria is presented within this unified data-modelling framework to construct sparse kernel models that generalise well. Examples from regression, classification and density estimation applications are used to illustrate the effectiveness of this generic sparse kernel data modelling approach.

Sparse kernel density estimator using orthogonal regression based on D-optimality experimental design

A novel sparse kernel density estimator is derived based on a regression approach, which selects a very small subset of significant kernels by means of the D-optimality experimental design criterion using an orthogonal forward selection procedure. The weights of the resulting sparse kernel model are calculated using the multiplicative nonnegative quadratic programming algorithm. The proposed method is computationally attractive, in comparison with many existing kernel density estimation algorithms. Our numerical results also show that the proposed method compares favourably with other existing methods, in terms of both test accuracy and model sparsity, for constructing kernel density estimates.