A Kernel Log-Rank Test of Independence for Right-Censored Data

Tensor dimensionality reduction (TDR) is a hot research topic in machine learning, which learns data representations by preserving the original data structure while avoiding convert samples into vectors and solving the problem of the curse of dimensionality of tensor data. In the work, a novel TDR approach based on mode product and Hilbert–Schmidt Independence criterion (HSIC) is proposed. The contributions of authors' work is described as following: (1) HSIC measures the statistical correlation of two random variables. However, instead of measuring the statistical correlation of two random variables directly, HSIC first transforms the two random variables into two reproducing kernel Hilbert spaces (RKHSs), and then measures the statistical correlation of transformed random variables by using Hilbert–Schmidt operators between the two RKHSs. The exploitation of RKHS increases the flexibility and applicability of HSIC. Although HSIC is widely used in machine learning, the authors have not seen its application to dimensionality reduction (DR)(except for authors' previous work). (2) A novel HSIC‐based TDR approach is proposed, which first applies HSIC to capture statistical information of tensor data set for DR. The authors give the mathematical derivation of HSIC for tensor data and establish a framework of TDR based on HSIC, named HSIC‐TDR for short, which aims to improve the DR results of tensor by exploring and preserving the statistical information of original data set. (3) Furthermore, to solve the out‐of‐sample problem, the authors learn an explicit expression between the dimensionality‐reduced tensors and the higher‐dimensional tensors by introducing mode product to HSIC‐TDR. The experimental results between the proposed method and other state‐of‐the‐art algorithm on various datasets demonstrate the well performance of the proposed method.

Multivariate tests of independence based on a new class of measures of independence in Reproducing Kernel Hilbert Space

This paper presents a kernel-based adaptive filter that is applied for the digital domain self-interference cancellation (SIC) in a transceiver operating in full-duplex (FD) mode. In FD, the benefit of simultaneous transmission and receiving of signals comes at the price of strong self-interference (SI). In this work, we are primarily interested in suppressing the SI using an adaptive filter namely adaptive projected subgradient method (APSM) in a reproducing kernel Hilbert space (RKHS) of functions. Using the projection concept as a powerful tool, APSM is used to model and consequently remove the SI. A low-complexity and fast-tracking algorithm is provided taking advantage of parallel projections as well as the kernel trick in RKHS. The performance of the proposed method is evaluated on real measurement data. The method illustrates the good performance of the proposed adaptive filter, compared to the known popular benchmarks. They demonstrate that the kernel-based algorithm achieves a favorable level of digital SIC while enabling parallel computation-based implementation within a rich and nonlinear function space, thanks to the employed adaptive filtering method.

In this study, a kernel-based metric based on the Hilbert-Schmidt independence criterion (HSIC) is proposed in a computer-aided-prognosis system to monitor cancer therapy effects. In order to induce tumour cell death, sarcoma xenograft tumour-bearing mice were injected with microbubbles followed by ultrasound and X-ray radiation therapy successively as a new anti-vascular treatment. High frequency (central frequency 30 MHz) ultrasound imaging was performed before and at different times after treatment and using spectroscopy, quantitative ultrasound (QUS) parametric maps were derived from the radiofrequency (RF) signals. The intensity histogram of midband fit parametric maps was computed to represent the pre- and post-treatment images. Subsequently, the HSIC-based metric between preand post-treatment samples were computed for each animal as a measure of distance between the two distributions. The HSIC-based metrics computes the distance between two distributions in a reproducing kernel Hilbert space (RKHS), meaning that by using a kernel, the input vectors are non-linearly mapped into a different, possibly high dimensional feature space. Computing the population means in this new space, enhanced group separability (compared to, e.g., Euclidean distance in the original feature space) is ideally obtained. The pre- and post-treatment parametric maps for each animal were thus represented by a dissimilarity measure, in which a high value of this metric indicated more treatment effect on the animal. It was shown in this research that this metric has a high correlation with cell death and if it was used in supervised learning, a high accuracy classification was obtained using a k-nearest-neighbor (k-NN) classifier.

Based on benchmarking various methods and analyzing multiple real scRNA-seq datasets, a computational platform/workflow and a set of tips for best practices are developed for analyzing causal interactions or relationships in single cells.

This paper describes a rigorous framework for reconstructing MR images of the heart, acquired continuously over the cardiac and respiratory cycle. The framework generalizes existing techniques, commonly referred to as retrospective gating, and is based on the properties of reproducing kernel Hilbert spaces. The reconstruction problem is formulated as a moment problem in a multidimensional reproducing kernel Hilbert spaces (a two-dimensional space for cardiac and respiratory resolved imaging). Several reproducing kernel Hilbert spaces were tested and compared, including those corresponding to commonly used interpolation techniques (sinc-based and splines kernels) and a more specific kernel allowed by the framework (based on a first-order Sobolev RKHS). The Sobolev reproducing kernel Hilbert spaces was shown to allow improved reconstructions in both simulated and real data from healthy volunteers, acquired in free breathing.

Underdetermined blind separation of nonnegative dependent sources consists in decomposing a set of observed mixed signals into greater number of original nonnegative and dependent component (source) signals. That is an important problem for which very few algorithms exist. It is also practically relevant for contemporary metabolic profiling of biological samples, such as biomarker identification studies, where sources (a.k.a. pure components or analytes) are aimed to be extracted from mass spectra of complex multicomponent mixtures. This paper presents a method for underdetermined blind separation of nonnegative dependent sources. The method performs nonlinear mixture‐wise mapping of observed data in high‐dimensional reproducible kernel Hilbert space (RKHS) of functions and sparseness‐constrained nonnegative matrix factorization (NMF) therein. Thus, the original problem is converted into new one with increased number of mixtures, increased number of dependent sources, and higher‐order (error) terms generated by nonlinear mapping. Provided that amplitudes of original components are sparsely distributed, which is the case for mass spectra of analytes, sparseness‐constrained NMF in RKHS yields, with significant probability, improved accuracy relative to the case when the same NMF algorithm is performed on the original problem. The method is exemplified on numerical and experimental examples related respectively to extraction of 10 dependent components from five mixtures and to extraction of 10 dependent analytes from mass spectra of two to five mixtures. Thereby, analytes mimic complexity of components expected to be found in biological samples. Copyright © 2013 John Wiley & Sons, Ltd.

We characterize all weighted composition operators v C ψ vC_{\psi } that commute with a J μ J_{\mu } -symmetric weighted composition operator u C φ uC_{\varphi } on the reproducing kernel Hilbert space H γ H_{\gamma } of analytic functions on the unit disk D {\mathbb D} . It turns out these commuting operators v C ψ vC_{\psi } are necessarily J μ J_{\mu } -symmetric. Furthermore, we obtain characterization(s) for the commuting operator v C ψ vC_{\psi } to be self-adjoint, normal or unitary.

An online spectrum cartography algorithm is proposed to reconstruct power spectral density (PSD) maps in space and frequency based on compressed and quantized sensor measurements. The emerging regression task is addressed by decomposing the PSD at every location into a linear combination of the power spectra (due to individual transmitters and background noise) scaled by attenuation functions capturing propagation effects. The attenuation functions are, in turn, postulated to be a sum of two terms: the first is a linear combination of a collection of basis functions whereas the second is an element of a reproducing kernel Hilbert space (RKHS) of vector-valued functions. A novel stochastic gradient descent algorithm is proposed to compute both components in an online fashion. Numerical tests verify the map estimation performance of the proposed technique.

We present a general nonlinear Bayesian filter for high-dimensional state estimation using the theory of reproducing kernel Hilbert space (RKHS). By applying the kernel method and the representer theorem to perform linear quadratic estimation in a functional space, we derive a Bayesian recursive state estimator for a general nonlinear dynamical system in the original input space. Unlike existing nonlinear extensions of the Kalman filter where the system dynamics are assumed known, the state-space representation for the Functional Bayesian Filter (FBF) is completely learned online from measurement data in the form of an infinite impulse response (IIR) filter or recurrent network in the RKHS, with universal approximation property. Using a positive definite kernel function satisfying Mercer’s conditions to compute and evolve information quantities, the FBF exploits both the statistical and time-domain information about the signal, extracts higher-order moments, and preserves the properties of covariances without the ill effects due to conventional arithmetic operations. We apply this novel kernel adaptive filtering (KAF) to recurrent network training, chaotic time-series estimation and cooperative filtering using Gaussian and non-Gaussian noises, and inverse kinematics modeling. Simulation results show FBF outperforms existing Kalman-based algorithms.

Sharp estimates for the covering numbers of the Weierstrass fractal kernel

The nonparametric multivariate analysis of variance (NPMANOVA) testing procedure has been proven to be a valuable tool for comparing groups. In the present paper, we propose a kernel extension of this technique in order to effectively confront high-dimensionality, a recurrent problem in many fields of science. The new method is called kernel multivariate analysis of variance (KMANOVA). The basic idea is to take advantage of the kernel framework: we propose to project the data from the original data space to a Hilbert space generated by a given kernel function and then perform the NPMANOVA method in the reproducing kernel Hilbert space (RKHS). Dispersion of the embedded points can be measured by the distance induced by the inner product in the RKHS but also by many other distances best suited in high-dimensional settings. For this purpose, we study two promising distances: a Manhattan-type distance and a distance based on an orthogonal projection of the embedded points in the direction of the group centroids. We show that the NPMANOVA method and the KMANOVA method with the induced distance are essentially equivalent. We also show that the KMANOVA method with the other two distances performs considerably better than the NPMANOVA method. We illustrate the advantages of our approach in the context of genetic association studies and demonstrate its usefulness on Alzheimer’s disease data. We also provide a software implementation of the method that is available on GitHub https://github.com/8699vicente/Kmanova.

Cubature formulas and geometrical designs are described in terms of reproducing kernels for Hilbert spaces of functions on the one hand, and Markov operators associated to orthogonal group representations on the other hand. In this way, several known results for spheres in Euclidean spaces, involving cubature formulas for polynomial functions and spherical designs, are shown to generalize to large classes of finite measure spaces (Ω, σ) and appropriate spaces of functions inside L2(Ω, σ). The last section points out how spherical designs are related to a class of reflection groups which are (in general dense) subgroups of orthogonal groups.

We propose a general theory and the estimation procedures for nonlinear sufficient dimension reduction where both the predictor and the response may be random functions. The relation between the response and predictor can be arbitrary and the sets of observed time points can vary from subject to subject. The functional and nonlinear nature of the problem leads to construction of two functional spaces: the first representing the functional data, assumed to be a Hilbert space, and the second characterizing nonlinearity, assumed to be a reproducing kernel Hilbert space. A particularly attractive feature of our construction is that the two spaces are nested, in the sense that the kernel for the second space is determined by the inner product of the first. We propose two estimators for this general dimension reduction problem, and establish the consistency and convergence rate for one of them. These asymptotic results are flexible enough to accommodate both fully and partially observed functional data. We investigate the performances of our estimators by simulations, and applied them to data sets about speech recognition and handwritten symbols.

Measures of statistical dependence between random variables have been successfully applied in many machine learning tasks, such as independent component analysis, feature selection, clustering and dimensionality reduction. The success is based on the fact that many existing learning tasks can be cast into problems of dependence maximization (or minimization). Motivated by this, we present a unifying view of kernel learning via statistical dependence estimation. The key idea is that good kernels should maximize the statistical dependence between the kernels and the class labels. The dependence is measured by the Hilbert–Schmidt independence criterion (HSIC), which is based on computing the Hilbert–Schmidt norm of the cross-covariance operator of mapped samples in the corresponding Hilbert spaces and is traditionally used to measure the statistical dependence between random variables. As a special case of kernel learning, we propose a Gaussian kernel optimization method for classification by maximizing the HSIC, where two forms of Gaussian kernels (spherical kernel and ellipsoidal kernel) are considered. Extensive experiments on real-world data sets from UCI benchmark repository validate the superiority of the proposed approach in terms of both prediction accuracy and computational efficiency.