Random Kernel Research Articles

Invariant moments based convolutional neural networks for image analysis

The paper proposes a method using convolutional neural network to effectively evaluate the discrimination between face and non face patterns, gender classification using facial images and facial expression recognition. The novelty of the method lies in the utilization of the initial trainable convolution kernels coefficients derived from the zernike moments by varying the moment order. The performance of the proposed method was compared with the convolutional neural network architecture that used random kernels as initial training parameters. The multilevel configuration of zernike moments was significant in extracting the shape information suitable for hierarchical feature learning to carry out image analysis and classification. Furthermore the results showed an outstanding performance of zernike moment based kernels in terms of the computation time and classification accuracy.

Random walk Green kernels in the neutral Moran model conditioned on survivors at a random time to origin

ABSTRACTIn the theory of finite discrete-time birth and death chains with absorbing endpoint boundaries, the evaluation of both additive and multiplicative path functionals is made possible by their Green and λ–potential kernels. These computations are addressed in the context of such Markov chains. The application to the neutral Moran model of population genetics yields first hitting and return times. A neutral Moran bridge model, forward and backward in time, for a given total number x of survivors of a single common ancestor at some random time T to the origin of times, yields the age of a mutant allele currently observed to have x copies of itself. This forward theory of age, made possible by Green kernels, is comparable to Watterson’s backward theory of age, which makes use of the reversibility of the Moran chain.

Conditional Distance Correlation

Statistical inference on conditional dependence is essential in many fields including genetic association studies and graphical models. The classic measures focus on linear conditional correlations and are incapable of characterizing nonlinear conditional relationship including nonmonotonic relationship. To overcome this limitation, we introduce a nonparametric measure of conditional dependence for multivariate random variables with arbitrary dimensions. Our measure possesses the necessary and intuitive properties as a correlation index. Briefly, it is zero almost surely if and only if two multivariate random variables are conditionally independent given a third random variable. More importantly, the sample version of this measure can be expressed elegantly as the root of a V or U-process with random kernels and has desirable theoretical properties. Based on the sample version, we propose a test for conditional independence, which is proven to be more powerful than some recently developed tests through our numerical simulations. The advantage of our test is even greater when the relationship between the multivariate random variables given the third random variable cannot be expressed in a linear or monotonic function of one random variable versus the other. We also show that the sample measure is consistent and weakly convergent, and the test statistic is asymptotically normal. By applying our test in a real data analysis, we are able to identify two conditionally associated gene expressions, which otherwise cannot be revealed. Thus, our measure of conditional dependence is not only an ideal concept, but also has important practical utility. Supplementary materials for this article are available online.

Fast Generation of Sparse Random Kernel Graphs.

The development of kernel-based inhomogeneous random graphs has provided models that are flexible enough to capture many observed characteristics of real networks, and that are also mathematically tractable. We specify a class of inhomogeneous random graph models, called random kernel graphs, that produces sparse graphs with tunable graph properties, and we develop an efficient generation algorithm to sample random instances from this model. As real-world networks are usually large, it is essential that the run-time of generation algorithms scales better than quadratically in the number of vertices n. We show that for many practical kernels our algorithm runs in time at most 𝒪(n(logn)2). As a practical example we show how to generate samples of power-law degree distribution graphs with tunable assortativity.

THE DENSITY OF THE PROCESS OF COLAESCING RANDOM WALKS

We study the density of the process of coalescing random walks starting from Z at time 0, where the random walk kernel associated to this model has finite second moment. It is shown that the density equals the survival probability of voter model with the initial condition being all 0’s except for a single 1 at the origin and it converges to 1 √ t . AMS Subject Classification: 60G05, 60G40, 60A10

The Conditional-Potts Clustering Model

This article presents a Bayesian kernel-based clustering method. The associated model arises as an embedding of the Potts density for class membership probabilities into an extended Bayesian model for joint data and class membership probabilities. The method may be seen as a principled extension of the super-paramagnetic clustering. The model depends on two parameters: the temperature and the kernel bandwidth. The clustering is obtained from the posterior marginal adjacency membership probabilities and does not depend on any particular value of the parameters. We elicit an informative prior based on random graph theory and kernel density estimation. A stochastic population Monte Carlo algorithm, based on parallel runs of the Wang–Landau algorithm, is developed to estimate the posterior adjacency membership probabilities and the parameter posterior. The convergence of the algorithm is also established. The method is applied to the whole human proteome to uncover human genes that share common evolutionary history. Our experiments and application show that good clustering results are obtained at many different values of the temperature and bandwidth parameters. Hence, instead of focusing on finding adequate values of the parameters, we advocate making clustering inference based on the study of the distribution of the posterior adjacency membership probabilities. This article has online supplementary material.

A new privacy-preserving proximal support vector machine for classification of vertically partitioned data

A new privacy-preserving proximal support vector machine (P3SVM) is formulated for classification of vertically partitioned data. Our classifier is based on the concept of global random reduced kernel which is composed of local reduced kernels. Each of them is computed using local reduced matrix with Gaussian perturbation, which is privately generated by only one of the parties, and never made public. This formulation leads to an extremely simple and fast privacy-preserving algorithm, for generating a linear or nonlinear classifier that merely requires the solution of a single system of linear equations. Comprehensive experiments are conducted on multiple publicly available benchmark datasets to evaluate the performance of the proposed algorithms and the results indicate that: (a) Our P3SVM achieves better performance than the recently proposed privacy-preserving SVM via random kernels in terms of both classification accuracy and computational time. (b) A significant improvement of accuracy is attained by our P3SVM when compared to classifiers generated only using each party’s own data. (c) The generated classifier has comparable accuracy to an ordinary PSVM classifier trained on the entire dataset, without releasing any private data.

Integrodifference models for persistence in temporally varying river environments

To fully understand population persistence in river ecosystems, it is necessary to consider the effect of the water flow, which varies tremendously with seasonal fluctuations of water runoff and snow melt. In this paper, we study integrodifference models for growth and dispersal in the presence of advective flow with both periodic (alternating) and random kernel parameters. For the alternating kernel model, we obtain the principal eigenvalue of the linearization operator to determine population persistence and derive a boundary value problem to calculate it. For the random model, we establish two persistence metrics: a generalized spectral radius and the asymptotic growth rate, which are mathematically equivalent but can be understood differently, to determine population persistence or extinction. The theoretical framework and methods for calculations are provided, and the framework is applied to calculating persistence in highly variable river environments.

Data-based decision rules about the convexity of the support of a distribution

Given $n$ independent, identically distributed random vectors in $\mathbb{R}^{d}$, drawn from a common density $f$, one wishes to find out whether the support of $f$ is convex or not. In this paper we describe a decision rule which decides correctly for sufficiently large $n$, with probability $1$, whenever $f$ is bounded away from zero in its compact support. We also show that the assumption of boundedness is necessary. The rule is based on a statistic that is a second-order $U$-statistic with a random kernel. Moreover, we suggest a way of approximating the distribution of the statistic under the hypothesis of convexity of the support. The performance of the proposed method is illustrated on simulated data sets. As an example of its potential statistical implications, the decision rule is used to automatically choose the tuning parameter of ISOMAP, a nonlinear dimensionality reduction method.

Stationary and invariant densities and disintegration kernels

Let \(G\) be a locally compact topological group, acting measurably on some Borel spaces \(S\) and \(T\), and consider some jointly stationary random measures \(\xi \) on \(S\times T\) and \(\eta \) on \(S\) such that \(\xi (\cdot \times T)\ll \eta \) a.s. Then there exists a stationary random kernel \(\zeta \) from \(S\) to \(T\) such that \(\xi =\eta \otimes \zeta \) a.s. This follows from the existence of an invariant kernel \(\varphi \) from \(S\times {\mathcal {M}}_{S\times T}\times {\mathcal {M}}_S\) to \(T\) such that \(\mu =\nu \otimes \varphi (\cdot ,\mu ,\nu )\) whenever \(\mu (\cdot \times T)\ll \nu \). Also included are some related results on stationary integration, absolute continuity, and ergodic decomposition.

Stationary heat conduction processes in bodies of randomly inhomogeneous structure

This work is devoted to the mathematical modeling of stationary heat conduction processes in randomly inhomogeneous multiphase structures. An integro-differential equation with a random kernel, whose solution is constructed in the form of a Neumann series, is put in correspondence with the boundary-value problem of heat conduction. We establish the conditions of absolute and uniform convergence of this series, in particular, the condition of boundedness of the body volume. It is shown that, for unbounded bodies, the condition of boundedness of the domain occupied by inclusions is necessary for convergence of the Neumann series.

Perceptual distinctiveness in Native American maize (Zea mays L.) landraces has practical implications

The large variation in multifactorial and seemingly non-adaptive kernel colour traits displayed by Native American maize landraces is an evidence of recurring selection for perceptual distinctiveness. Native American farmers selected for colour traits that allowed them to distinguish between and maintain large diversity within maize landraces for traditional uses. Multivariate statistical procedures were employed to quantify the variation and interrelationships between physical traits, C:N ratio, protein content, micro- and macronutrient concentrations with kernel colour traits measured on random kernel samples of Northern flint maize landraces grown for 2 years in a common-garden experiment. The colour traits (L*, a* and b* indicating dark–light, red–green and yellow–blue colour continuum, respectively) were digitally quantified on 590 random kernels from each of 28 accessions in 11 landraces. Accessions within landraces exhibited the largest variation for all colour traits. The variation in the L*a*b*, L*a* and L*b* combinations explained significant variances in 37, 14 and 37% of 120 landrace–trait combinations, respectively; the remaining 12% were explained by L*, a* or b*. On average, 37.5% of the variation in protein content (range 19.2–64.5%) and 36.6% of the variation in the C:N ratio (range 15.7–65.0%) were explained by the combinations of colour traits in different landraces. Slightly larger average variations in K (43.7%), S (43.0%), Fe (42.2%) and P (40.8%) were accounted for by colour traits. A hierarchical and joint clustering procedure of landraces and traits was developed to facilitate the identification of large variation and the selection of single or multiple traits based on kernel colour traits with reasonable certainty.

Privacy-preserving linear and nonlinear approximation via linear programming

We propose a novel privacy-preserving random kernel approximation based on a data matrix A∈R m×n whose rows are divided into privately owned blocks. Each block of rows belongs to a different entity that is unwilling to share its rows or make them public. We wish to obtain an accurate function approximation for a given y∈R m corresponding to each of the m rows of A. Our approximation of y is a real function on R n evaluated at each row of A and is based on the concept of a reduced kernel K(A, B′), where B′ is the transpose of a completely random matrix B. The proposed linear-programming-based approximation, which is public but does not reveal the privately held data matrix A, has accuracy comparable to that of an ordinary kernel approximation based on a publicly disclosed data matrix A.

Multiple integrals and expansion of solutions of differential equations driven by rough paths and by fractional Brownian motions

Multiple integrals with respect to several Hölder continuous functions of exponent are studied by using fractional calculus. They are applied to obtain the Volterra expansion (with remainder) for the solution of a differential system driven by rough paths. The results are applied to stochastic differential equations driven by fractional Brownian motions of Hurst parameter . For the solution of a stochastic differential equation driven by fractional Brownian motion, we obtain its chaos expansion, as well as the finite chaos expansion with remainder. To this end, we study the multiple Itô integral with random kernels. The Hu–Meyer formulas between multiple Itô and multiple pathwise integrals with random kernels are also obtained.

Sensitivity analysis and model selection for a generalized convolution model for spatial processes

We examine a ∞exible class of nonstationary stochastic models for multivariate spatial data proposed by Majumdar et al. (2010). This covariance model is based on convolutions of spatially varying covariance kernels with cen- ters corresponding to the centers of \local stationarity. A Bayesian method for estimation of the parameters in the model based on a Gibbs sampler is applied to simulated data to check for model sensitivity. The efiect of a perturbation of the model in terms of kernel centers is also examined. Finally, the method is applied to a bivariate soil chemistry data from the the Central Arizona Phoenix Long Term Ecological Project (CAP LTER). Prediction bias, prediction standard deviation and predictive coverage are examined for difierent candidate models. In addition, a comparison with the bivariate stationary coregionalization model introduced by Wackernagel (2003) is carried out. A variant of the model proposed in Majumdar et al. (2010), with random kernel centers, is also examined. The latter model is seen to work much better than the stationary coregionalization model, and to per- form comparably with the model with random kernel centers. Simulations indicate that the model is sensitive to under- or over-speciflcation of kernel centers. On the other hand, application to real data seems to indicate that centroids of the regions that are homogeneous can be used as means of the random kernel centers. Cross validation can be used as a way of flnding the best model with an appropriate number of kernels.

Aspects of large random Markov kernels

We briefly review certain asymptotic properties of random Markov kernels on a finite state space. These models can be thought of as finite Markov chains in random environment. Here, the asymptotics are taken with respect to the cardinality of the state space. We study, for instance, the behaviour of the normalized invariant vector, the global behaviour of the spectrum and the extremal eigenvalues. The analysis of these models involves Random Matrix Theory, convex compact polytopes and finite graphs with random weights. We also give some open problems related to these models.

Novel kernels for error-tolerant graph classification

One of the major difficulties in graph classification is the lack of mathematical structure in the space of graphs. The use of kernel machines allows us to overcome this fundamental limitation in an elegant manner by addressing the pattern recognition problem in an implicitly existing feature vector space instead of the original space of graphs. In this paper we propose three novel error-tolerant graph kernels -- a diffusion kernel, a convolution kernel, and a random walk kernel. The kernels are closely related to one of the most flexible graph matching methods, graph edit distance. Consequently, our kernels are applicable to virtually any kind of graph. They also show a high degree of robustness against various types of distortion. In an experimental evaluation involving the classification of line drawings, images, diatoms, fingerprints, and molecules, we demonstrate the superior performance of the proposed kernels in conjunction with support vector machines over a standard nearest-neighbor reference method and several other graph kernels including a standard random walk kernel.

Privacy-preserving classification of vertically partitioned data via random kernels

We propose a novel privacy-preserving support vector machine (SVM) classifier for a data matrix A whose input feature columns are divided into groups belonging to different entities. Each entity is unwilling to share its group of columns or make it public. Our classifier is based on the concept of a reduced kernel K ( A , B ′), where B ′ is the transpose of a random matrix B . The column blocks of B corresponding to the different entities are privately generated by each entity and never made public. The proposed linear or nonlinear SVM classifier, which is public but does not reveal any of the privately held data, has accuracy comparable to that of an ordinary SVM classifier that uses the entire set of input features directly.

Performance Analysis of a Real-Time Control Network Test Bed in a Linux-Based System With Sporadic Message Arrivals

This paper describes the results from a real-time network test bed for a Linux-based control system with event-driven message traffic. Interest has grown in the use of Linux as a cost-effective alternative to expensive proprietary real-time operating systems. However, kernel tasks in Linux can cause random delays in message transmissions. Event-driven traffic can result in unpredictable message inter-arrival rates. Nondeterministic delays in message transmissions and sporadic message arrival times complicate the design of the network as they require the use of an estimation technique to analyze feasibility. Random temporal behavior can lead to network overloads, where it becomes unavoidable that some messages will be late. We discuss the design and feasibility analysis for the network and evaluate the network's performance in the presence of random kernel delay and sporadic message arrivals. Three message scheduling approaches are evaluated using different Linux configurations at different levels of network loading

The existence theorems of the random solutions for random Hammerstein equation with random kernel (II)

In this paper, we use the random fixed theorems of cone expansion and compression of random operator to obtain the existence theorems of random solutions for random Hammerstein integral equation of polynomial type with random kernel.