Nonlinear sufficient dimension reduction for functional data

A local–global mixed kernel with reproducing property

We propose a novel method of dimensionality reduction for supervised learning problems. Given a regression or classification problem in which we wish to predict a response variable Y from an explanatory variable X, we treat the problem of dimensionality reduction as that of finding a low-dimensional subspace for X which retains the statistical relationship between X and Y. We show that this problem can be formulated in terms of conditional independence. To turn this formulation into an optimization problem we establish a general nonparametric characterization of conditional independence using covariance operators on reproducing kernel Hilbert spaces. This characterization allows us to derive a contrast function for estimation of the effective subspace. Unlike many conventional methods for dimensionality reduction in supervised learning, the proposed method requires neither assumptions on the marginal distribution of X, nor a parametric model of the conditional distribution of Y. We present experiments that compare the performance of the method with conventional methods.

Active learning based on minimization of the expected path-length of random walks on the learned manifold structure

This article proposes a novel approach to linear dimension reduction for regression using nonparametric estimation with positive-definite kernels or reproducing kernel Hilbert spaces (RKHSs). The purpose of the dimension reduction is to find such directions in the explanatory variables that explain the response sufficiently: this is called sufficient dimension reduction. The proposed method is based on an estimator for the gradient of the regression function considered for the feature vectors mapped into RKHSs. It is proved that the method is able to estimate the directions that achieve sufficient dimension reduction. In comparison with other existing methods, the proposed one has wide applicability without strong assumptions on the distributions or the type of variables, and needs only eigendecomposition for estimating the projection matrix. The theoretical analysis shows that the estimator is consistent with certain rate under some conditions. The experimental results demonstrate that the proposed method successfully finds effective directions with efficient computation even for high-dimensional explanatory variables.

In this note, we prove that an operator between reproducing kernel Hilbert spaces is a multiplication operator if and only if it leaves invariant zero sets. To be more precise, it is shown that an operator T between reproducing kernel Hilbert spaces is a multiplication operator if and only if (Tf)(z)=0 holds for all f and z satisfying f(z)=0. As possible applications, we deduce a general reflexivity result for multiplier algebras, and furthermore prove fully vector-valued generalizations of mulitplier lifting results of Beatrous and Burbea.

In this paper, a contextual Support Vector Machine (SVM) technique based on the principle of Hilbert Space Embedding (HSE) of a local hyperspectral data distribution into an Reproducing Kernel Hilbert Space (RKHS) is proposed to optimally exploit the spectral and local spatial information of the hyperspectral image. The idea of embedding is to map hyperspectral pixels in a local neighborhood into a single point in the RKHS that can uniquely represent those pixels collectively. Previously, the authors have employed an HSE called empirical mean map to build the contextual SVM. In this work, a weighted empirical mean map is utilized to exploit the similarities and variation in the local spatial information. For every pixel, a small set of the neighboring pixels in a hyperspectral image are mapped into an RKHS induced by a certain kernel (Eg. Gaussian RBF kernel) and then, the embedded point of these group of pixels is obtained by calculating the weighted empirical mean of these mapped points. The weights are determined based on the distance between the pixel in consideration and its neighbors. An SVM separating hyperplane is built to maximize the margin between classes formed by weighted empirical means. The proposed technique showed significant improvement over the existing contextual and composite kernels on two hyperspectral image data sets.

The use of regional covariance descriptors to generate feature data represented by Symmetric Positive Definite (SPD) matrices from images or videos has become increasingly common in machine learning. However, SPD data itself does not constitute a vector space, and dictionary learning involves a large number of linear operations, so dictionary learning cannot be performed directly on SPD data. For this reason, a more common method is to map the SPD data to the Reproducing Kernel Hilbert Space (RKHS). The so-called kernel learning is to find the most suitable RKHS for specific tasks. RKHS can be uniquely generated by a kernel function. Therefore, RKHS learning can also be considered as kernel learning. In this article, there are two main contributions. The first contribution is to propose a framework which based on Kernel Learning and Riemannian Metric (KLRM). Usually the learnable kernel function framework is to learn some parameters in the kernel function. The second contribution is dictionary learning by applying KLRM to SPD data. The SPD data is transformed into the RKHS generated by KLRM, and RKHS after training provides the most suitable working space for dictionary learning. Under the proposed framework, we design a positive definite kernel function, which is defined by the Log-Euclidean metric. This function can be transformed into a corresponding Riemannian kernel. The experimental results provided in this paper is compared with other state-of-the-art algorithms for SPD data dictionary learning and show that the proposed algorithm achieves better results.

In this paper, we investigate the enhancement of speech by applying kernel adaptive filter. Noise removal is very important in many applications like telephone conversation, speech recognition, etc. Kernel methods have shown good results for other applications like handwriting recognition, inverse distance weightings, etc. To improve the speech quality and intelligibility, we can process the signals in new domain like Reproducing Kernel Hilbert Space (RKHS) unlike time and frequency domains. We have used the noisy speech corpus (NOIZEUS) for the experiments. The experimental results shown the noise removal in RKHS has good improvement in the Signal to Noise Ratio (SNR) values as compared the traditional methods.

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.

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.

Coherent states on quaternion slices and a measurable field of Hilbert spaces

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.

Dimension reduction has become an important tool for dealing with high dimensional data. Locally linear embedding (LLE) is a nonlinear dimension reduction method which can preserve local configurations of nearest neighbors. However, finding the nearest neighbors requires the definition of a distance measure, which is a critical step in LLE. In this paper, the Rank-order distance measure is used to substitute the traditional Euclidean distance measure in order to find better nearest neighbor candidates for preserving local configurations of the manifolds. The Rank-order distance between the data points is calculated using their neighbors’ ranking orders, and is shown to be able to improve the clustering of high dimensional data. The proposed method is called Rank-order based LLE (RLLE). The RLLE method is evaluated by comparing with the original LLE, ISO-LLE and IED-LLE on two handwritten datasets. It is shown that the effectiveness of a distance measure in the LLE method is closely related to whether it can be used to find good nearest neighbors. The experimental results show that the proposed RLLE method can improve the process of dimension reduction effectively, and C-index is another good candidate for evaluating the dimension reduction results.

The reproducing kernel Hilbert space (RKHS) methodology has shown to be a suitable tool for the resolution of a wide range of problems in statistical signal processing both in the real and complex domains. It relies on the idea of transforming the original functional data into an infinite series representation by projection onto an specific RKHS, which usually simplifies the statistical treatment without any loss of efficiency. Moreover, the advantages of quaternion algebra over real-valued three and four-dimensional vector algebra in the modelling of multidimensional data have been proven useful in much relatively recent research. This paper accordingly proposes a generic RKHS framework for the statistical analysis of augmented quaternion random vectors, which provide a complete description of their second order characteristics. It will allow us to exploit the full advantages of the RKHS theory in widely linear processing applications, such as signal detection. In particular, we address the detection of a quaternion signal disturbed by additive Gaussian noise and the discrimination between two quaternion Gaussian signals in continuous time.

Covariance matrices have attracted increasing attention for data representation in many computer vision tasks. The nonsingular covariance matrices are regarded as points on Riemannian manifolds rather than Euclidean space. A common technique for classification on Riemannian manifolds is to embed the covariance matrices into a reproducing kernel Hilbert space (RKHS), and then construct a map from RKHS to Euclidean space, while the explicit map from RKHS to Euclidean space in most kernel-based methods only depends on a linear hypothesis. In this paper, we propose a subspace learning framework to project Riemannian manifolds to Euclidean space, and give the theoretical derivation for it. Specifically, the Euclidean space is isomorphic to the subspace of RKHS. Under the framework, firstly we define an improved Log-Euclidean Gaussian radial basis function kernel for embedding. The first order statistical features of input images are incorporated into the kernel function to increase the discriminative power. After that we seek the optimal projection matrix of the subspace of the RKHS by conducting a graph embedding discriminant analysis. Texture recognition and object categorization experiments with region covariance descriptors demonstrate the considerable effectiveness of the improved Log-Euclidean Gaussian RBK kernel and the proposed method.