arXiv subjects

Roy Mitz

Publications and source records attributed to Roy Mitz.

3 recordsLinked to original sources

arXiv2020

A Perturbation-Based Kernel Approximation Framework

Kernel methods are powerful tools in various data analysis tasks. Yet, in many cases, their time and space complexity render them impractical for large datasets. Various kernel approximation methods were proposed to overcome this issue, with the most prominent method being the Nystr{\"o}m method. In this paper, we derive a perturbation-based kernel approximation framework building upon results from classical perturbation theory. We provide an error analysis for this framework, and prove that in fact, it generalizes the Nystr{\"o}m method and several of its variants. Furthermore, we show that our framework gives rise to new kernel approximation schemes, that can be tuned to take advantage of the structure of the approximated kernel matrix. We support our theoretical results numerically and demonstrate the advantages of our approximation framework on both synthetic and real-world data.

cs.LG↗

arXiv2019

ROIPCA: An online memory-restricted PCA algorithm based on rank-one updates

Principal components analysis (PCA) is a fundamental algorithm in data analysis. Its memory-restricted online versions are useful in many modern applications, where the data are too large to fit in memory, or when data arrive as a stream of items. In this paper, we propose ROIPCA and fROIPCA, two online PCA algorithms that are based on rank-one updates. While ROIPCA is typically more accurate, fROIPCA is faster and has comparable accuracy. We show the relation between fROIPCA and an existing popular gradient algorithm for online PCA, and in particular, prove that fROIPCA is in fact a gradient algorithm with an optimal learning rate. We demonstrate numerically the advantages of our algorithms over existing state-of-the-art algorithms in terms of accuracy and runtime.

cs.LG↗

arXiv2017

Symmetric rank-one updates from partial spectrum with an application to out-of-sample extension

Rank-one update of the spectrum of a matrix is a fundamental problem in classical perturbation theory. In this paper, we consider its variant where only part of the spectrum is known. We address this variant using an efficient scheme for updating the known eigenpairs with guaranteed error bounds. Then, we apply our scheme to the extension of the top eigenvectors of the graph Laplacian to a new data sample. In particular, we model this extension as a perturbation problem and show how to solve it using our rank-one updating scheme. We provide a theoretical analysis of this extension method, and back it up with numerical results that illustrate its advantages.

math.NA↗