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The Nystr\"om method is a convenient heuristic method to obtain low-rank approximations to kernel matrices in nearly linear complexity. Existing studies typically use the method to approximate positive semidefinite matrices with low or…

Numerical Analysis · Mathematics 2023-07-13 Jianlin Xia

Many kernel methods suffer from high time and space complexities and are thus prohibitive in big-data applications. To tackle the computational challenge, the Nystr\"om method has been extensively used to reduce time and space complexities…

Machine Learning · Computer Science 2014-04-02 Shusen Wang , Zhihua Zhang

The Nystr\"om method is a popular low-rank approximation technique for large matrices that arise in kernel methods and convex optimization. Yet, when the data exhibits heavy-tailed spectral decay, the effective dimension of the problem…

Data Structures and Algorithms · Computer Science 2025-07-22 Sachin Garg , Michał Dereziński

Low rank matrix approximations appear in a number of scientific computing applications. We consider the Nystr\"{o}m method for approximating a positive semidefinite matrix $A$. In the case that $A$ is very large or its entries can only be…

Numerical Analysis · Mathematics 2023-07-24 Erin Carson , Ieva Daužickaitė

Recently, Nystr\"{o}m method has proved its prominence empirically and theoretically in speeding up the training of kernel machines while retaining satisfactory performances and accuracy. So far, there are several different approaches…

Machine Learning · Computer Science 2021-09-21 Weida Li , Mingxia Liu , Daoqiang Zhang

The Nystr\"om method offers an effective way to obtain low-rank approximation of SPD matrices, and has been recently extended and analyzed to nonsymmetric matrices (leading to the generalized Nystr\"om method). It is a randomized,…

Numerical Analysis · Mathematics 2024-04-24 Alberto Bucci , Leonardo Robol

The Nystrom method is an efficient technique used to speed up large-scale learning applications by generating low-rank approximations. Crucial to the performance of this technique is the assumption that a matrix can be well approximated by…

Machine Learning · Computer Science 2014-08-12 Ameet Talwalkar , Afshin Rostamizadeh

The CUR matrix decomposition and the Nystr\"{o}m approximation are two important low-rank matrix approximation techniques. The Nystr\"{o}m method approximates a symmetric positive semidefinite matrix in terms of a small number of its…

Machine Learning · Computer Science 2013-10-02 Shusen Wang , Zhihua Zhang

The Nystrom method is a popular technique that uses a small number of landmark points to compute a fixed-rank approximation of large kernel matrices that arise in machine learning problems. In practice, to ensure high quality…

Machine Learning · Statistics 2019-12-04 Farhad Pourkamali-Anaraki , Stephen Becker

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,…

Machine Learning · Computer Science 2022-05-24 Roy Mitz , Yoel Shkolnisky

The Nystrom method is an efficient technique to speed up large-scale learning applications by generating low-rank approximations. Crucial to the performance of this technique is the assumption that a matrix can be well approximated by…

Artificial Intelligence · Computer Science 2010-04-13 Ameet Talwalkar , Afshin Rostamizadeh

The Nystr\"om methods have been popular techniques for scalable kernel based learning. They approximate explicit, low-dimensional feature mappings for kernel functions from the pairwise comparisons with the training data. However, Nystr\"om…

Machine Learning · Computer Science 2018-05-21 Mert Al , Thee Chanyaswad , Sun-Yuan Kung

The Nystrom method has been popular for generating the low-rank approximation of kernel matrices that arise in many machine learning problems. The approximation quality of the Nystrom method depends crucially on the number of selected…

Machine Learning · Statistics 2016-12-21 Farhad Pourkamali-Anaraki , Stephen Becker

This paper is concerned with the low-rank approximation for large-scale nonsymmetric matrices. Inspired by the classical Nystrom method, which is a popular method to find the low-rank approximation for symmetric positive semidefinite…

Numerical Analysis · Mathematics 2024-10-30 Yatian Wang , Hua Xiang , Chi Zhang , Songling Zhang

We give the first algorithm for kernel Nystr\"om approximation that runs in *linear time in the number of training points* and is provably accurate for all kernel matrices, without dependence on regularity or incoherence conditions. The…

Machine Learning · Computer Science 2017-11-06 Cameron Musco , Christopher Musco

Kernel $k$-means clustering can correctly identify and extract a far more varied collection of cluster structures than the linear $k$-means clustering algorithm. However, kernel $k$-means clustering is computationally expensive when the…

Machine Learning · Computer Science 2019-02-12 Shusen Wang , Alex Gittens , Michael W. Mahoney

The Nystr\"om method is a popular choice for finding a low-rank approximation to a symmetric positive semi-definite matrix. The method can fail when applied to symmetric indefinite matrices, for which the error can be unboundedly large. In…

Numerical Analysis · Mathematics 2023-10-10 Taejun Park , Yuji Nakatsukasa

We propose a novel class of kernels to alleviate the high computational cost of large-scale nonparametric learning with kernel methods. The proposed kernel is defined based on a hierarchical partitioning of the underlying data domain, where…

Machine Learning · Computer Science 2017-08-15 Jie Chen , Haim Avron , Vikas Sindhwani

Recent work has focused on combining kernel methods and deep learning to exploit the best of the two approaches. Here, we introduce a new architecture of neural networks in which we replace the top dense layers of standard convolutional…

Machine Learning · Computer Science 2019-12-02 Luc Giffon , Stéphane Ayache , Thierry Artières , Hachem Kadri

We propose a continuous optimization algorithm for the Column Subset Selection Problem (CSSP) and Nystr\"om approximation. The CSSP and Nystr\"om method construct low-rank approximations of matrices based on a predetermined subset of…

Methodology · Statistics 2023-04-20 Anant Mathur , Sarat Moka , Zdravko Botev
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