679 research outputs found
Hot new directions for quasi-Monte Carlo research in step with applications
This article provides an overview of some interfaces between the theory of
quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC
theoretical settings: first order QMC methods in the unit cube and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
under appropriate conditions on the function spaces. Another important feature
is that good parameters for these QMC methods can be obtained by fast efficient
algorithms even when is large. We outline three different applications and
explain how they can tap into the different QMC theory. We also discuss three
cost saving strategies that can be combined with QMC in these applications.
Many of these recent QMC theory and methods are developed not in isolation, but
in close connection with applications
Recycling Randomness with Structure for Sublinear time Kernel Expansions
We propose a scheme for recycling Gaussian random vectors into structured
matrices to approximate various kernel functions in sublinear time via random
embeddings. Our framework includes the Fastfood construction as a special case,
but also extends to Circulant, Toeplitz and Hankel matrices, and the broader
family of structured matrices that are characterized by the concept of
low-displacement rank. We introduce notions of coherence and graph-theoretic
structural constants that control the approximation quality, and prove
unbiasedness and low-variance properties of random feature maps that arise
within our framework. For the case of low-displacement matrices, we show how
the degree of structure and randomness can be controlled to reduce statistical
variance at the cost of increased computation and storage requirements.
Empirical results strongly support our theory and justify the use of a broader
family of structured matrices for scaling up kernel methods using random
features
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