相关论文: Inner-product Kernels are Asymptotically Equivalen…

This paper is concerned with the asymptotic distribution of the largest eigenvalues for some nonlinear random matrix ensemble stemming from the study of neural networks. More precisely we consider $M= \frac{1}{m} YY^\top$ with $Y=f(WX)$…

We investigate random matrices whose entries are obtained by applying a nonlinear kernel function to pairwise inner products between $n$ independent data vectors, drawn uniformly from the unit sphere in $\mathbb{R}^d$. This study is…

Kernel matrices are of central importance to many applied fields. In this manuscript, we focus on spectral properties of kernel matrices in the so-called ``flat limit'', which occurs when points are close together relative to the scale of…

We find the asymptotic distribution of the multi-dimensional multi-scale and kernel estimators for high-frequency financial data with microstructure. Sampling times are allowed to be asynchronous and endogenous. In the process, we show that…

Standard kernels such as Mat\'ern or RBF kernels only encode simple monotonic dependencies within the input space. Spectral mixture kernels have been proposed as general-purpose, flexible kernels for learning and discovering more…

We place ourselves in the setting of high-dimensional statistical inference where the number of variables $p$ in a dataset of interest is of the same order of magnitude as the number of observations $n$. We consider the spectrum of certain…

We consider n-by-n matrices whose (i, j)-th entry is f(X_i^T X_j), where X_1, ...,X_n are i.i.d. standard Gaussian random vectors in R^p, and f is a real-valued function. The eigenvalue distribution of these random kernel matrices is…

Recent work in random matrix theory (RMT) has developed the notion of deterministic equivalents: typically linear surrogate models that approximate the spectral behavior of large nonlinear random matrices, such as nonlinear feature maps in…

One of the main computational bottlenecks when working with kernel based learning is dealing with the large and typically dense kernel matrix. Techniques dealing with fast approximations of the matrix vector product for these kernel…

In this article, we investigate the spectral behavior of random features kernel matrices of the type ${\bf K} = \mathbb{E}_{{\bf w}} \left[\sigma\left({\bf w}^{\sf T}{\bf x}_i\right)\sigma\left({\bf w}^{\sf T}{\bf…

In data science, individual observations are often assumed to come independently from an underlying probability space. Kernel matrices formed from large sets of such observations arise frequently, for example during classification tasks. It…

We study the spectrum of inner-product kernel matrices, i.e., $n \times n$ matrices with entries $h (\langle \textbf{x}_i ,\textbf{x}_j \rangle/d)$ where the $( \textbf{x}_i)_{i \leq n}$ are i.i.d.~random covariates in $\mathbb{R}^d$. In…

This study presents an efficient approach for incomplete data classification, where the entries of samples are missing or masked due to privacy preservation. To deal with these incomplete data, a new kernel function with asymmetric…

This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a…

Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to…

In this paper we study the concentration properties for the eigenvalues of kernel matrices, which are central objects in a wide range of kernel methods and, more recently, in network analysis. We present a set of concentration inequalities…

We study an "inner-product kernel" random matrix model, whose empirical spectral distribution was shown by Xiuyuan Cheng and Amit Singer to converge to a deterministic measure in the large $n$ and $p$ limit. We provide an interpretation of…

Kernel method is a very powerful tool in machine learning. The trick of kernel has been effectively and extensively applied in many areas of machine learning, such as support vector machine (SVM) and kernel principal component analysis…

Positive definite kernels and their associated Reproducing Kernel Hilbert Spaces provide a mathematically compelling and practically competitive framework for learning from data. In this paper we take the approximation theory point of view…

Approximating non-linear kernels using feature maps has gained a lot of interest in recent years due to applications in reducing training and testing times of SVM classifiers and other kernel based learning algorithms. We extend this line…