• Corpus ID: 16658740

# Subspace Embeddings for the Polynomial Kernel

@inproceedings{Avron2014SubspaceEF,
title={Subspace Embeddings for the Polynomial Kernel},
author={Haim Avron and Huy L. Nguyen and David P. Woodruff},
booktitle={NIPS},
year={2014}
}
• Published in NIPS 8 December 2014
• Computer Science
Sketching is a powerful dimensionality reduction tool for accelerating statistical learning algorithms. However, its applicability has been limited to a certain extent since the crucial ingredient, the so-called oblivious subspace embedding, can only be applied to data spaces with an explicit representation as the column span or row span of a matrix, while in many settings learning is done in a high-dimensional space implicitly defined by the data matrix via a kernel transformation. We propose…

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## References

SHOWING 1-10 OF 17 REFERENCES
Fast and scalable polynomial kernels via explicit feature maps
• Computer Science
KDD
• 2013
A novel randomized tensor product technique, called Tensor Sketching, is proposed for approximating any polynomial kernel in O(n(d+D \log{D})) time, and achieves higher accuracy and often runs orders of magnitude faster than the state-of-the-art approach for large-scale real-world datasets.
Low-Rank Approximation and Regression in Input Sparsity Time
• Computer Science
ArXiv
• 2012
We design a new distribution over m × n matrices S so that, for any fixed n × d matrix A of rank r, with probability at least 9/10, ∥SAx∥2 = (1 ± ε)∥Ax∥2 simultaneously for all x ∈ Rd. Here, m is
Compact Random Feature Maps
• Computer Science
ICML
• 2014
The error bounds of CRAFT maps are proved demonstrating their superior kernel reconstruction performance compared to the previous approximation schemes, and it is shown how structured random matrices can be used to efficiently generate CRAFTMaps.
Numerical linear algebra in the streaming model
• Computer Science
STOC '09
• 2009
Near-optimal space bounds are given in the streaming model for linear algebra problems that include estimation of matrix products, linear regression, low-rank approximation, and approximation of matrix rank; results for turnstile updates are proved.
Sketching Structured Matrices for Faster Nonlinear Regression
• Computer Science, Mathematics
NIPS
• 2013
This work considers a class of structured regression problems which involve Vandermonde matrices which arise naturally in various statistical modeling settings, and shows that this structure can be exploited to further accelerate the solution of the regression problem.
Random Features for Large-Scale Kernel Machines
• Computer Science
NIPS
• 2007
Two sets of random features are explored, provided convergence bounds on their ability to approximate various radial basis kernels, and it is shown that in large-scale classification and regression tasks linear machine learning algorithms applied to these features outperform state-of-the-art large- scale kernel machines.
OSNAP: Faster Numerical Linear Algebra Algorithms via Sparser Subspace Embeddings
• Computer Science
2013 IEEE 54th Annual Symposium on Foundations of Computer Science
• 2013
The main result is essentially a Bai-Yin type theorem in random matrix theory and is likely to be of independent interest: for any fixed U ∈ R<sup>n×d</sup> with orthonormal columns and random sparse Π, all singular values of ΠU lie in [1 - ε, 1 + ε] with good probability.
Fastfood: Approximate Kernel Expansions in Loglinear Time
• Computer Science
ICML 2013
• 2013
Improvements to Fastfood, an approximation that accelerates kernel methods significantly and achieves similar accuracy to full kernel expansions and Random Kitchen Sinks while being 100x faster and using 1000x less memory, make kernel methods more practical for applications that have large training sets and/or require real-time prediction.
Relative-Error CUR Matrix Decompositions
• Computer Science, Mathematics
SIAM J. Matrix Anal. Appl.
• 2008
These two algorithms are the first polynomial time algorithms for such low-rank matrix approximations that come with relative-error guarantees; previously, in some cases, it was not even known whether such matrix decompositions exist.
Randomized Algorithms for Matrices and Data
This monograph will provide a detailed overview of recent work on the theory of randomized matrix algorithms as well as the application of those ideas to the solution of practical problems in large-scale data analysis.