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A Unifying Framework in Vector-valued Reproducing Kernel Hilbert Spaces for Manifold Regularization and Co-Regularized Multi-view Learning
This paper presents a general vector-valued reproducing kernel Hilbert spaces
(RKHS) framework for the problem of learning an unknown functional dependency
between a structured input space and a structured output space. Our formulation
encompasses both Vector-valued Manifold Regularization and Co-regularized
Multi-view Learning, providing in particular a unifying framework linking these
two important learning approaches. In the case of the least square loss
function, we provide a closed form solution, which is obtained by solving a
system of linear equations. In the case of Support Vector Machine (SVM)
classification, our formulation generalizes in particular both the binary
Laplacian SVM to the multi-class, multi-view settings and the multi-class
Simplex Cone SVM to the semi-supervised, multi-view settings. The solution is
obtained by solving a single quadratic optimization problem, as in standard
SVM, via the Sequential Minimal Optimization (SMO) approach. Empirical results
obtained on the task of object recognition, using several challenging datasets,
demonstrate the competitiveness of our algorithms compared with other
state-of-the-art methods.Comment: 72 page
Out-of-sample generalizations for supervised manifold learning for classification
Supervised manifold learning methods for data classification map data samples
residing in a high-dimensional ambient space to a lower-dimensional domain in a
structure-preserving way, while enhancing the separation between different
classes in the learned embedding. Most nonlinear supervised manifold learning
methods compute the embedding of the manifolds only at the initially available
training points, while the generalization of the embedding to novel points,
known as the out-of-sample extension problem in manifold learning, becomes
especially important in classification applications. In this work, we propose a
semi-supervised method for building an interpolation function that provides an
out-of-sample extension for general supervised manifold learning algorithms
studied in the context of classification. The proposed algorithm computes a
radial basis function (RBF) interpolator that minimizes an objective function
consisting of the total embedding error of unlabeled test samples, defined as
their distance to the embeddings of the manifolds of their own class, as well
as a regularization term that controls the smoothness of the interpolation
function in a direction-dependent way. The class labels of test data and the
interpolation function parameters are estimated jointly with a progressive
procedure. Experimental results on face and object images demonstrate the
potential of the proposed out-of-sample extension algorithm for the
classification of manifold-modeled data sets
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