6,169 research outputs found
Rates of Convergence for Sparse Variational Gaussian Process Regression
Excellent variational approximations to Gaussian process posteriors have been developed which avoid the O(N³) scaling with dataset size N. They reduce the computational cost to O(NM²), with M≪N being the number of inducing variables, which summarise the process. While the computational cost seems to be linear in N, the true complexity of the algorithm depends on how M must increase to ensure a certain quality of approximation. We address this by characterising the behavior of an upper bound on the KL divergence to the posterior. We show that with high probability the KL divergence can be made arbitrarily small by growing M more slowly than N. A particular case of interest is that for regression with normally distributed inputs in D-dimensions with the popular Squared Exponential kernel, M = O(log^DN) is sufficient. Our results show that as datasets grow, Gaussian process posteriors can truly be approximated cheaply, and provide a concrete rule for how to increase M in continual learning scenarios
Variational Bayesian multinomial probit regression with Gaussian process priors
It is well known in the statistics literature that augmenting binary and polychotomous response models with Gaussian latent variables enables exact Bayesian analysis via Gibbs sampling from the parameter posterior. By adopting such a data augmentation strategy, dispensing with priors over regression coefficients in favour of Gaussian Process (GP) priors over functions, and employing variational approximations to the full posterior we obtain efficient computational methods for Gaussian Process classification in the multi-class setting. The model augmentation with additional latent variables ensures full a posteriori class coupling whilst retaining the simple a priori independent GP covariance structure from which sparse approximations, such as multi-class Informative Vector Machines (IVM), emerge in a very natural and straightforward manner. This is the first time that a fully Variational Bayesian treatment for multi-class GP classification has been developed without having to resort to additional explicit approximations to the non-Gaussian likelihood term. Empirical comparisons with exact analysis via MCMC and Laplace approximations illustrate the utility of the variational approximation as a computationally economic alternative to full MCMC and it is shown to be more accurate than the Laplace approximation
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
Bayesian Compressed Regression
As an alternative to variable selection or shrinkage in high dimensional
regression, we propose to randomly compress the predictors prior to analysis.
This dramatically reduces storage and computational bottlenecks, performing
well when the predictors can be projected to a low dimensional linear subspace
with minimal loss of information about the response. As opposed to existing
Bayesian dimensionality reduction approaches, the exact posterior distribution
conditional on the compressed data is available analytically, speeding up
computation by many orders of magnitude while also bypassing robustness issues
due to convergence and mixing problems with MCMC. Model averaging is used to
reduce sensitivity to the random projection matrix, while accommodating
uncertainty in the subspace dimension. Strong theoretical support is provided
for the approach by showing near parametric convergence rates for the
predictive density in the large p small n asymptotic paradigm. Practical
performance relative to competitors is illustrated in simulations and real data
applications.Comment: 29 pages, 4 figure
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