64 research outputs found
Approximate Kernel PCA Using Random Features: Computational vs. Statistical Trade-off
Kernel methods are powerful learning methodologies that provide a simple way
to construct nonlinear algorithms from linear ones. Despite their popularity,
they suffer from poor scalability in big data scenarios. Various approximation
methods, including random feature approximation have been proposed to alleviate
the problem. However, the statistical consistency of most of these approximate
kernel methods is not well understood except for kernel ridge regression
wherein it has been shown that the random feature approximation is not only
computationally efficient but also statistically consistent with a minimax
optimal rate of convergence. In this paper, we investigate the efficacy of
random feature approximation in the context of kernel principal component
analysis (KPCA) by studying the trade-off between computational and statistical
behaviors of approximate KPCA. We show that the approximate KPCA is both
computationally and statistically efficient compared to KPCA in terms of the
error associated with reconstructing a kernel function based on its projection
onto the corresponding eigenspaces. Depending on the eigenvalue decay behavior
of the covariance operator, we show that only features (polynomial
decay) or features (exponential decay) are needed to match the
statistical performance of KPCA. We also investigate their statistical
behaviors in terms of the convergence of corresponding eigenspaces wherein we
show that only features are required to match the performance of
KPCA and if fewer than features are used, then approximate KPCA has
a worse statistical behavior than that of KPCA.Comment: 46 page
Optimal Rates for Random Fourier Features
Kernel methods represent one of the most powerful tools in machine learning
to tackle problems expressed in terms of function values and derivatives due to
their capability to represent and model complex relations. While these methods
show good versatility, they are computationally intensive and have poor
scalability to large data as they require operations on Gram matrices. In order
to mitigate this serious computational limitation, recently randomized
constructions have been proposed in the literature, which allow the application
of fast linear algorithms. Random Fourier features (RFF) are among the most
popular and widely applied constructions: they provide an easily computable,
low-dimensional feature representation for shift-invariant kernels. Despite the
popularity of RFFs, very little is understood theoretically about their
approximation quality. In this paper, we provide a detailed finite-sample
theoretical analysis about the approximation quality of RFFs by (i)
establishing optimal (in terms of the RFF dimension, and growing set size)
performance guarantees in uniform norm, and (ii) presenting guarantees in
() norms. We also propose an RFF approximation to derivatives of
a kernel with a theoretical study on its approximation quality.Comment: To appear at NIPS-201
Minimax Estimation of Kernel Mean Embeddings
In this paper, we study the minimax estimation of the Bochner integral
also called as the kernel
mean embedding, based on random samples drawn i.i.d.~from , where
is a positive definite
kernel. Various estimators (including the empirical estimator),
of are studied in the literature wherein all of
them satisfy with
being the reproducing kernel Hilbert space induced by . The
main contribution of the paper is in showing that the above mentioned rate of
is minimax in and
-norms over the class of discrete measures and
the class of measures that has an infinitely differentiable density, with
being a continuous translation-invariant kernel on . The
interesting aspect of this result is that the minimax rate is independent of
the smoothness of the kernel and the density of (if it exists). This result
has practical consequences in statistical applications as the mean embedding
has been widely employed in non-parametric hypothesis testing, density
estimation, causal inference and feature selection, through its relation to
energy distance (and distance covariance)
A D.C. Programming Approach to the Sparse Generalized Eigenvalue Problem
In this paper, we consider the sparse eigenvalue problem wherein the goal is
to obtain a sparse solution to the generalized eigenvalue problem. We achieve
this by constraining the cardinality of the solution to the generalized
eigenvalue problem and obtain sparse principal component analysis (PCA), sparse
canonical correlation analysis (CCA) and sparse Fisher discriminant analysis
(FDA) as special cases. Unlike the -norm approximation to the
cardinality constraint, which previous methods have used in the context of
sparse PCA, we propose a tighter approximation that is related to the negative
log-likelihood of a Student's t-distribution. The problem is then framed as a
d.c. (difference of convex functions) program and is solved as a sequence of
convex programs by invoking the majorization-minimization method. The resulting
algorithm is proved to exhibit \emph{global convergence} behavior, i.e., for
any random initialization, the sequence (subsequence) of iterates generated by
the algorithm converges to a stationary point of the d.c. program. The
performance of the algorithm is empirically demonstrated on both sparse PCA
(finding few relevant genes that explain as much variance as possible in a
high-dimensional gene dataset) and sparse CCA (cross-language document
retrieval and vocabulary selection for music retrieval) applications.Comment: 40 page
Discussion of: Brownian distance covariance
Discussion on "Brownian distance covariance" by G\'{a}bor J. Sz\'{e}kely and
Maria L. Rizzo [arXiv:1010.0297]Comment: Published in at http://dx.doi.org/10.1214/09-AOAS312E the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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