Statistical Learning on Manifold-Valued Data

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9729)

Abstract

Regression on manifolds problem is to estimate an unknown smooth function f that maps p-dimensional manifold-valued inputs, whose values lie on unknown Input manifold M of lower dimensionality q < p embedded in an ambient high-dimensional input space Rp, to m-dimensional outputs from training sample consisting of given ‘input-output’ pairs. We consider this problem in which Jacobian Jf(X) of function f and Input manifold M should be also estimated. The paper presents a new geometrically motivated method for estimating a triple (f(X), Jf(X), M) from given sample. The proposed solution is based on solving a Tangent bundle manifold learning problem for specific unknown Regression manifold embedded in input-output space Rp+m and consisting of input-output pairs (X, f(X)), X ∈ M.

Keywords

Regression on manifolds Regression on features Input manifold reconstruction Jacobian estimation Tangent bundle manifold learning 

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Skolkovo Institute of Science and TechnologyMoscowRussia
  2. 2.Institute for Systems Analysis, FRC CSC RASMoscowRussia
  3. 3.Kharkevich Institute for Information Transmission Problems RASMoscowRussia

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