Lecture 1. Harmonic Analysis of Measures: Analysis on Fractals Lecture 2. Spectra of measures, tilings, and wandering vectors Lecture 3. The universal tiling conjecture in dimension one and operator fractals Lecture 4. Representations of Cuntz algebras associated to quasi-stationary Markov measures Lecture 5. The Cuntz relations and kernel decompositions Lecture 6. Harmonic analysis of wavelet ...

We introduce kernel nonparametric tests for Lancaster three-variable interaction and for total independence, using embeddings of signed measures into a reproducing kernel Hilbert space. The resulting test statistics are straightforward to compute, and are used in powerful interaction tests, which are consistent against all alternatives for a large family of reproducing kernels. We show the Lanc...

Recent results in geometry processing have shown that shape segmentation, comparison, and analysis can be successfully addressed through the heat diffusion kernel. In this paper, we focus our attention on the properties (e.g., scale-invariance, semi-group property, robustness to noise) of the wFEM heat kernel, recently proposed in [PF10], and its application to shape comparison and feature-driv...

We develop an extension of the sliced inverse regression (SIR) framework for dimension reduction using kernel models and Tikhonov regularization. The result is a numerically stable nonlinear dimension reduction method. We prove consistency of the method under weak conditions even when the reproducing kernel Hilbert space induced by the kernel is infinite dimensional. We illustrate the utility o...

A new method for performing a kernel principal component analysis is proposed. By kernelizing the generalized Hebbian algorithm, one can iteratively estimate the principal components in a reproducing kernel Hilbert space with only linear order memory complexity. The derivation of the method and preliminary applications in image hyperresolution are presented. In addition, we discuss the extensio...

In order to use the method of(least squares) collocation for the computation o f an approximation to the anomalous potential o f the Earth (T) it is necessary to specify a reproducing kernel Hilbert space the dual o.f which contain the (linear) functionals associated with the observation~ The specification includes the prescription o f an inner product or an equivalent norrrL It is demonstrated...

We characterize the reproducing kernel Hilbert spaces whose elements are p-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for p = 2 we show that the spectral decomposition of this integral operator gives a complete description of the reproducing kernel.

This paper considers minimax and adaptive prediction with functional predictors in the framework of functional linear model and reproducing kernel Hilbert space. Minimax rate of convergence for the excess prediction risk is established. It is shown that the optimal rate is determined jointly by the reproducing kernel and the covariance kernel. In particular, the alignment of these two kernels c...

Let F be a separable Banach space, and let (X, Y ) be a random pair taking values in F×R. Motivated by a broad range of potential applications, we investigate rates of convergence of the k-nearest neighbor estimate rn(x) of the regression function r(x) = E[Y |X = x], based on n independent copies of the pair (X, Y ). Using compact embedding theory, we present explicit and general finite sample ...