Precise expressions for random projections: Low-rank approximation and randomized Newton
It is often desirable to reduce the dimensionality of a large dataset by projecting it onto a low-dimensional subspace. Matrix sketching has emerged as a powerful technique for performing such dimensionality reduction very efficiently. Even though there is an extensive literature on the worst-case p...
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Main Authors | , , , |
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Format | Journal Article |
Language | English |
Published |
18.06.2020
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Subjects | |
Online Access | Get full text |
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Summary: | It is often desirable to reduce the dimensionality of a large dataset by
projecting it onto a low-dimensional subspace. Matrix sketching has emerged as
a powerful technique for performing such dimensionality reduction very
efficiently. Even though there is an extensive literature on the worst-case
performance of sketching, existing guarantees are typically very different from
what is observed in practice. We exploit recent developments in the spectral
analysis of random matrices to develop novel techniques that provide provably
accurate expressions for the expected value of random projection matrices
obtained via sketching. These expressions can be used to characterize the
performance of dimensionality reduction in a variety of common machine learning
tasks, ranging from low-rank approximation to iterative stochastic
optimization. Our results apply to several popular sketching methods, including
Gaussian and Rademacher sketches, and they enable precise analysis of these
methods in terms of spectral properties of the data. Empirical results show
that the expressions we derive reflect the practical performance of these
sketching methods, down to lower-order effects and even constant factors. |
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DOI: | 10.48550/arxiv.2006.10653 |