Recently several generalizations to higher dimension of the classical Fourier transform (FT) using Clifford geometric algebra have been introduced, including the two-dimensional (2D) Clifford Fourier transform (CFT). Based on the 2D CFT, we establish the two-dimensional Clifford windowed Fourier transform (CWFT). Using the spectral representation of the CFT, we derive several important properti...

Multiple Kernel Learning (MKL) synthesizes a single kernel from a set of multiple kernels for use in a support vector machine. We propose that MKL be preceded by feature extraction. Given a set of kernels and a vector y of class labels, Multiple Kernel Basis Extraction (MKBE) constructs orthogonal vectors {v1, . . . , vm} whose corresponding kernels, {v1v 1 , . . . , vmv m}, are maximally align...

We show that the average characteristic polynomial P n (z) = E[det(zI−M)] of the random Hermitian matrix ensemble Z −1 n exp(−Tr(V (M) − AM))dM is characterized by multiple orthogonality conditions that depend on the eigenvalues of the external source A. For each eigenvalue a j of A, there is a weight and P n has n j orthogonality conditions with respect to this weight, if n j is the multiplici...

Using the direct relation between Gegenbauer polynomials C λn (x) and Ferrers function of first kind P μν (x), we compute interrelations certain Jacobi polynomials, Meixner functions second kind. We then Rodrigues-type, standard integral orthogonality Sobolev relations for kinds. In remainder paper using derive connection linearization relations, some definite series expansions, Christoffel-Dar...

Efforts to construct synthetic networks in living cells have been hindered by the limited number of regulatory components that provide wide dynamic range and low crosstalk. Here, we report a class of de-novo-designed prokaryotic riboregulators called toehold switches that activate gene expression in response to cognate RNAs with arbitrary sequences. Toehold switches provide a high level of orth...

We show that a bounded linear operator on Banach space with closed range has range-kernel complementarity if and only its generalized amplitude is less than $\pi$. An application to the strong convergence of iterations also given.

Abstract: A key question in modern statistics is how to make fast and reliable inferences for complex, high-dimensional data. While there has been much interest in sparse techniques, current methods do not generalize well to data with nonlinear structure. In this work, we present an orthogonal series estimator for predictors that are complex aggregate objects, such as natural images, galaxy spe...

Kernel-based methods exhibit well-documented performance in various nonlinear learning tasks. Most of them rely on a preselected kernel, whose prudent choice presumes task-specific prior information. Especially when the latter is not available, multi-kernel learning has gained popularity thanks to its flexibility in choosing kernels from a prescribed kernel dictionary. Leveraging the random fea...