106,221 research outputs found
Forecasting and Granger Modelling with Non-linear Dynamical Dependencies
Traditional linear methods for forecasting multivariate time series are not
able to satisfactorily model the non-linear dependencies that may exist in
non-Gaussian series. We build on the theory of learning vector-valued functions
in the reproducing kernel Hilbert space and develop a method for learning
prediction functions that accommodate such non-linearities. The method not only
learns the predictive function but also the matrix-valued kernel underlying the
function search space directly from the data. Our approach is based on learning
multiple matrix-valued kernels, each of those composed of a set of input
kernels and a set of output kernels learned in the cone of positive
semi-definite matrices. In addition to superior predictive performance in the
presence of strong non-linearities, our method also recovers the hidden dynamic
relationships between the series and thus is a new alternative to existing
graphical Granger techniques.Comment: Accepted for ECML-PKDD 201
Supervised Learning with Similarity Functions
We address the problem of general supervised learning when data can only be
accessed through an (indefinite) similarity function between data points.
Existing work on learning with indefinite kernels has concentrated solely on
binary/multi-class classification problems. We propose a model that is generic
enough to handle any supervised learning task and also subsumes the model
previously proposed for classification. We give a "goodness" criterion for
similarity functions w.r.t. a given supervised learning task and then adapt a
well-known landmarking technique to provide efficient algorithms for supervised
learning using "good" similarity functions. We demonstrate the effectiveness of
our model on three important super-vised learning problems: a) real-valued
regression, b) ordinal regression and c) ranking where we show that our method
guarantees bounded generalization error. Furthermore, for the case of
real-valued regression, we give a natural goodness definition that, when used
in conjunction with a recent result in sparse vector recovery, guarantees a
sparse predictor with bounded generalization error. Finally, we report results
of our learning algorithms on regression and ordinal regression tasks using
non-PSD similarity functions and demonstrate the effectiveness of our
algorithms, especially that of the sparse landmark selection algorithm that
achieves significantly higher accuracies than the baseline methods while
offering reduced computational costs.Comment: To appear in the proceedings of NIPS 2012, 30 page
Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Equivariant Projected Kernels
Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas in the physical sciences. To do so, we present a general recipe for constructing gauge equivariant kernels, which induce Gaussian vector fields, i.e. vector-valued Gaussian processes coherent with geometry, from scalar-valued Riemannian kernels. We extend standard Gaussian process training methods, such as variational inference, to this setting. This enables vector-valued Gaussian processes on Riemannian manifolds to be trained using standard methods and makes them accessible to machine learning practitioners
Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels
Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas in the physical sciences. To do so, we present a general recipe for constructing gauge independent kernels, which induce Gaussian vector fields, i.e. vector-valued Gaussian processes coherent withgeometry, from scalar-valued Riemannian kernels. We extend standard Gaussian process training methods, such as variational inference, to this setting. This enables vector-valued Gaussian processes on Riemannian manifolds to be trained using standard methods and makes them accessible to machine learning practitioners
Multi-Output Learning via Spectral Filtering
In this paper we study a class of regularized kernel methods for vector-valued learning which are based on filtering the spectrum of the kernel matrix. The considered methods include Tikhonov regularization as a special case, as well as interesting alternatives such as vector-valued extensions of L2 boosting. Computational properties are discussed for various examples of kernels for vector-valued functions and the benefits of iterative techniques are illustrated. Generalizing previous results for the scalar case, we show finite sample bounds for the excess risk of the obtained estimator and, in turn, these results allow to prove consistency both for regression and multi-category classification. Finally, we present some promising results of the proposed algorithms on artificial and real data
Semi-supervised Vector-valued Learning: From Theory to Algorithm
Vector-valued learning, where the output space admits a vector-valued
structure, is an important problem that covers a broad family of important
domains, e.g. multi-label learning and multi-class classification. Using local
Rademacher complexity and unlabeled data, we derive novel data-dependent excess
risk bounds for learning vector-valued functions in both the kernel space and
linear space. The derived bounds are much sharper than existing ones, where
convergence rates are improved from to
and in special cases. Motivated
by our theoretical analysis, we propose a unified framework for learning
vector-valued functions, incorporating both local Rademacher complexity and
Laplacian regularization. Empirical results on a wide number of benchmark
datasets show that the proposed algorithm significantly outperforms baseline
methods, which coincides with our theoretical findings
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