44,633 research outputs found
Bayesian Approximate Kernel Regression with Variable Selection
Nonlinear kernel regression models are often used in statistics and machine
learning because they are more accurate than linear models. Variable selection
for kernel regression models is a challenge partly because, unlike the linear
regression setting, there is no clear concept of an effect size for regression
coefficients. In this paper, we propose a novel framework that provides an
effect size analog of each explanatory variable for Bayesian kernel regression
models when the kernel is shift-invariant --- for example, the Gaussian kernel.
We use function analytic properties of shift-invariant reproducing kernel
Hilbert spaces (RKHS) to define a linear vector space that: (i) captures
nonlinear structure, and (ii) can be projected onto the original explanatory
variables. The projection onto the original explanatory variables serves as an
analog of effect sizes. The specific function analytic property we use is that
shift-invariant kernel functions can be approximated via random Fourier bases.
Based on the random Fourier expansion we propose a computationally efficient
class of Bayesian approximate kernel regression (BAKR) models for both
nonlinear regression and binary classification for which one can compute an
analog of effect sizes. We illustrate the utility of BAKR by examining two
important problems in statistical genetics: genomic selection (i.e. phenotypic
prediction) and association mapping (i.e. inference of significant variants or
loci). State-of-the-art methods for genomic selection and association mapping
are based on kernel regression and linear models, respectively. BAKR is the
first method that is competitive in both settings.Comment: 22 pages, 3 figures, 3 tables; theory added; new simulations
presented; references adde
Moment-Matching Polynomials
We give a new framework for proving the existence of low-degree, polynomial
approximators for Boolean functions with respect to broad classes of
non-product distributions. Our proofs use techniques related to the classical
moment problem and deviate significantly from known Fourier-based methods,
which require the underlying distribution to have some product structure.
Our main application is the first polynomial-time algorithm for agnostically
learning any function of a constant number of halfspaces with respect to any
log-concave distribution (for any constant accuracy parameter). This result was
not known even for the case of learning the intersection of two halfspaces
without noise. Additionally, we show that in the "smoothed-analysis" setting,
the above results hold with respect to distributions that have sub-exponential
tails, a property satisfied by many natural and well-studied distributions in
machine learning.
Given that our algorithms can be implemented using Support Vector Machines
(SVMs) with a polynomial kernel, these results give a rigorous theoretical
explanation as to why many kernel methods work so well in practice
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