Concept

# Manifold alignment

Résumé
Manifold alignment is a class of machine learning algorithms that produce projections between sets of data, given that the original data sets lie on a common manifold. The concept was first introduced as such by Ham, Lee, and Saul in 2003, adding a manifold constraint to the general problem of correlating sets of high-dimensional vectors. Manifold alignment assumes that disparate data sets produced by similar generating processes will share a similar underlying manifold representation. By learning projections from each original space to the shared manifold, correspondences are recovered and knowledge from one domain can be transferred to another. Most manifold alignment techniques consider only two data sets, but the concept extends to arbitrarily many initial data sets. Consider the case of aligning two data sets, and , with and . Manifold alignment algorithms attempt to project both and into a new d-dimensional space such that the projections both minimize distance between corresponding points and preserve the local manifold structure of the original data. The projection functions are denoted: Let represent the binary correspondence matrix between points in and : Let and represent pointwise similarities within data sets. This is usually encoded as the heat kernel of the adjacency matrix of a k-nearest neighbor graph. Finally, introduce a coefficient , which can be tuned to adjust the weight of the 'preserve manifold structure' goal, versus the 'minimize corresponding point distances' goal. With these definitions in place, the loss function for manifold alignment can be written: Solving this optimization problem is equivalent to solving a generalized eigenvalue problem using the graph laplacian of the joint matrix, G: The algorithm described above requires full pairwise correspondence information between input data sets; a supervised learning paradigm. However, this information is usually difficult or impossible to obtain in real world applications. Recent work has extended the core manifold alignment algorithm to semi-supervised unsupervised and multiple-instance settings.
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