## Abstract

Consider unknown smooth function which maps high-dimensional inputs to multidimensional outputs and whose domain of definition is unknown low-dimensional input manifold embedded in an ambient high-dimensional input space. Given training dataset consisting of ‘input-output’ pairs, regression on input manifold problem is to estimate the unknown function and its Jacobian matrix, as well to estimate the input manifold. By transforming high-dimensional inputs in their low-dimensional features, initial regression problem is reduced to certain regression on feature space problem. The paper presents a new geometrically motivated method for solving both interrelated regression problems.

Original language | English |
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Pages (from-to) | 209-240 |

Number of pages | 32 |

Journal | Annals of Mathematics and Artificial Intelligence |

Volume | 81 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 1 Oct 2017 |

## Keywords

- Dimensionality reduction
- Manifold estimation
- Manifold learning
- Regression on feature space
- Regression on manifolds