# Kernel Methods

### Inductive Regularized Learning of Kernel Functions

In this paper we consider the fundamental problem of semi-supervised kernel function learning. We propose a general regularized framework for learning a kernel matrix, and then demonstrate an equivalence between our proposed kernel matrix learning framework and a general linear transformation learning problem. Our result shows that the learned kernel matrices parameterize a linear transformation kernel function and can be applied inductively to new data points. Furthermore, our result gives a constructive method for kernelizing most existing Mahalanobis metric learning formulations. To make our results practical for large-scale data, we modify our framework to limit the number of parameters in the optimization process.

### Compressed Diffusion

Diffusion maps are a commonly used kernel-based method for manifold learning, which can reveal intrinsic structures in data and embed them in low dimensions. However, as with most kernel methods, its implementation requires a heavy computational load, reaching up to cubic complexity in the number of data points. This limits its usability in modern data analysis. Here, we present a new approach to computing the diffusion geometry, and related embeddings, from a compressed diffusion process between data regions rather than data points. Our construction is based on an adaptation of the previously proposed measure-based (MGC) kernel that robustly captures the local geometry around data points. We use this MGC kernel to efficiently compress diffusion relations from pointwise to data region resolution. Finally, a spectral embedding of the data regions provides coordinates that are used to interpolate and approximate the pointwise diffusion map embedding of data. We analyze theoretical connections between our construction and the original diffusion geometry of diffusion maps, and demonstrate the utility of our method in analyzing big datasets, where it outperforms competing approaches.

### Relating Leverage Scores and Density using Regularized Christoffel Functions

Statistical leverage scores emerged as a fundamental tool for matrix sketching and column sampling with applications to low rank approximation, regression, random feature learning and quadrature. Yet, the very nature of this quantity is barely understood. Borrowing ideas from the orthogonal polynomial literature, we introduce the regularized Christoffel function associated to a positive definite kernel. This uncovers a variational formulation for leverage scores for kernel methods and allows to elucidate their relationships with the chosen kernel as well as population density. Our main result quantitatively describes a decreasing relation between leverage score and population density for a broad class of kernels on Euclidean spaces. Numerical simulations support our findings.

### When is there a Representer Theorem? Reflexive Banach spaces

We consider a general regularised interpolation problem for learning a parameter vector from data. The well known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is the core of kernel methods in machine learning as it makes the problem computationally tractable. Most literature deals only with sufficient conditions for representer theorems in Hilbert spaces. We prove necessary and sufficient conditions for the existence of representer theorems in reflexive Banach spaces and illustrate why in a sense reflexivity is the minimal requirement on the function space. We further show that if the learning relies on the linear representer theorem the solution is independent of the regulariser and in fact determined by the function space alone. This in particular shows the value of generalising Hilbert space learning theory to Banach spaces.

### The Exact Equivalence of Distance and Kernel Methods for Hypothesis Testing

Distance-based methods, also called "energy statistics", are leading methods for two-sample and independence tests from the statistics community. Kernel methods, developed from "kernel mean embeddings", are leading methods for two-sample and independence tests from the machine learning community. Previous works demonstrated the equivalence of distance and kernel methods only at the population level, for each kind of test, requiring an embedding theory of kernels. We propose a simple, bijective transformation between semimetrics and nondegenerate kernels. We prove that for finite samples, two-sample tests are special cases of independence tests, and the distance-based statistic is equivalent to the kernel-based statistic, including the biased, unbiased, and normalized versions. In other words, upon setting the kernel or metric to be bijective of each other, running any of the four algorithms will yield the exact same answer up to numerical precision. This deepens and unifies our understanding of interpoint comparison based methods.

### Relating Leverage Scores and Density using Regularized Christoffel Functions

Statistical leverage scores emerged as a fundamental tool for matrix sketching and column sampling with applications to low rank approximation, regression, random feature learning and quadrature. Yet, the very nature of this quantity is barely understood. Borrowing ideas from the orthogonal polynomial literature, we introduce the regularized Christoffel function associated to a positive definite kernel. This uncovers a variational formulation for leverage scores for kernel methods and allows to elucidate their relationships with the chosen kernel as well as population density. Our main result quantitatively describes a decreasing relation between leverage score and population density for a broad class of kernels on Euclidean spaces. Numerical simulations support our findings.

### When is there a Representer Theorem? Nondifferentiable Regularisers and Banach spaces

We consider a general regularised interpolation problem for learning a parameter vector from data. The well known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is the core of kernel methods in machine learning as it makes the problem computationally tractable. Necessary and sufficient conditions for differentiable regularisers on Hilbert spaces to admit a representer theorem have been proved. We extend those results to nondifferentiable regularisers on uniformly convex and uniformly smooth Banach spaces. This gives a (more) complete answer to the question when there is a representer theorem. We then note that for regularised interpolation in fact the solution is determined by the function space alone and independent of the regulariser, making the extension to Banach spaces even more valuable.

### Adaptivity for Regularized Kernel Methods by Lepskii's Principle

We address the problem of {\it adaptivity} in the framework of reproducing kernel Hilbert space (RKHS) regression. More precisely, we analyze estimators arising from a linear regularization scheme $g_\lam$. In practical applications, an important task is to choose the regularization parameter $\lam$ appropriately, i.e. based only on the given data and independently on unknown structural assumptions on the regression function. An attractive approach avoiding data-splitting is the {\it Lepskii Principle} (LP), also known as the {\it Balancing Principle} is this setting. We show that a modified parameter choice based on (LP) is minimax optimal adaptive, up to $\log\log(n)$. A convenient result is the fact that balancing in $L^2(\nu)-$ norm, which is easiest, automatically gives optimal balancing in all stronger norms, interpolating between $L^2(\nu)$ and the RKHS. An analogous result is open for other classical approaches to data dependent choices of the regularization parameter, e.g. for Hold-Out.

### Randomized Kernel Selection With Spectra of Multilevel Circulant Matrices

Kernel selection aims at choosing an appropriate kernel function for kernel-based learning algorithms to avoid either underfitting or overfitting of the resulting hypothesis. One of the main problems faced by kernel selection is the evaluation of the goodness of a kernel, which is typically difficult and computationally expensive. In this paper, we propose a randomized kernel selection approach to evaluate and select the kernel with the spectra of the specifically designed multilevel circulant matrices (MCMs), which is statistically sound and computationally efficient. Instead of constructing the kernel matrix, we construct the randomized MCM to encode the kernel function and all data points together with labels. We build a one-to-one correspondence between all candidate kernel functions and the spectra of the randomized MCMs by Fourier transform. We prove the statistical properties of the randomized MCMs and the randomized kernel selection criteria, which theoretically qualify the utility of the randomized criteria in kernel selection. With the spectra of the randomized MCMs, we derive a series of randomized criteria to conduct kernel selection, which can be computed in log-linear time and linear space complexity by fast Fourier transform (FFT). Experimental results demonstrate that our randomized kernel selection criteria are significantly more efficient than the existing classic and widely-used criteria while preserving similar predictive performance.

### Alternating Circulant Random Features for Semigroup Kernels

The random features method is an efficient method to approximate the kernel function. In this paper, we propose novel random features called "alternating circulant random features,'' which consist of a random mixture of independent random structured matrices. Existing fast random features exploit random sign flipping to reduce the correlation between features. Sign flipping works well on random Fourier features for real-valued shift-invariant kernels because the corresponding weight distribution is symmetric. However, this method cannot be applied to random Laplace features directly because the distribution is not symmetric. The method proposed herein yields alternating circulant random features, with the correlation between features being reduced through the random sampling of weights from multiple independent random structured matrices instead of via random sign flipping. The proposed method facilitates rapid calculation by employing structured matrices. In addition, the weight distribution is preserved because sign flipping is not implemented. The performance of the proposed alternating circulant random features method is theoretically and empirically evaluated.