### Abstract

Regression in its most common form where independent and dependent variables are in Rn is a ubiquitous tool in Sciences and Engineering. Recent advances in Medical Imaging has lead to a wide spread availability of manifoldvalued data leading to problems where the independent variables are manifoldvalued and dependent are real-valued or vice-versa. The most common method of regression on a manifold is the geodesic regression, which is the counterpart of linear regression in Euclidean space. Often, the relation between the variables is highly complex, and existing most commonly used geodesic regression can prove to be inaccurate. Thus, it is necessary to resort to a non-linear model for regression. In this work we present a novel Kernel based non-linear regression method when the mapping to be estimated is either from M → ℝn or ℝn → M, where M is a Riemannian manifold. A key advantage of this approach is that there is no requirement for the manifold-valued data to necessarily inherit an ordering from the data in ℝn.We present several synthetic and real data experiments along with comparisons to the state-of-the-art geodesic regression method in literature and thus validating the effectiveness of the proposed algorithm.

Original language | English (US) |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 719-727 |

Number of pages | 9 |

DOIs | |

State | Published - Oct 1 2015 |

Externally published | Yes |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 9349 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(pp. 719-727). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9349). Springer Verlag. https://doi.org/10.1007/978-3-319-24553-9_88

**Nonlinear regression on riemannian manifolds and its applications to neuro-image analysis.** / Banerjee, Monami; Chakraborty, Rudrasis; Ofori, Edward; Vaillancourt, David; Vemuri, Baba C.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9349, Springer Verlag, pp. 719-727. https://doi.org/10.1007/978-3-319-24553-9_88

}

TY - CHAP

T1 - Nonlinear regression on riemannian manifolds and its applications to neuro-image analysis

AU - Banerjee, Monami

AU - Chakraborty, Rudrasis

AU - Ofori, Edward

AU - Vaillancourt, David

AU - Vemuri, Baba C.

PY - 2015/10/1

Y1 - 2015/10/1

N2 - Regression in its most common form where independent and dependent variables are in Rn is a ubiquitous tool in Sciences and Engineering. Recent advances in Medical Imaging has lead to a wide spread availability of manifoldvalued data leading to problems where the independent variables are manifoldvalued and dependent are real-valued or vice-versa. The most common method of regression on a manifold is the geodesic regression, which is the counterpart of linear regression in Euclidean space. Often, the relation between the variables is highly complex, and existing most commonly used geodesic regression can prove to be inaccurate. Thus, it is necessary to resort to a non-linear model for regression. In this work we present a novel Kernel based non-linear regression method when the mapping to be estimated is either from M → ℝn or ℝn → M, where M is a Riemannian manifold. A key advantage of this approach is that there is no requirement for the manifold-valued data to necessarily inherit an ordering from the data in ℝn.We present several synthetic and real data experiments along with comparisons to the state-of-the-art geodesic regression method in literature and thus validating the effectiveness of the proposed algorithm.

AB - Regression in its most common form where independent and dependent variables are in Rn is a ubiquitous tool in Sciences and Engineering. Recent advances in Medical Imaging has lead to a wide spread availability of manifoldvalued data leading to problems where the independent variables are manifoldvalued and dependent are real-valued or vice-versa. The most common method of regression on a manifold is the geodesic regression, which is the counterpart of linear regression in Euclidean space. Often, the relation between the variables is highly complex, and existing most commonly used geodesic regression can prove to be inaccurate. Thus, it is necessary to resort to a non-linear model for regression. In this work we present a novel Kernel based non-linear regression method when the mapping to be estimated is either from M → ℝn or ℝn → M, where M is a Riemannian manifold. A key advantage of this approach is that there is no requirement for the manifold-valued data to necessarily inherit an ordering from the data in ℝn.We present several synthetic and real data experiments along with comparisons to the state-of-the-art geodesic regression method in literature and thus validating the effectiveness of the proposed algorithm.

UR - http://www.scopus.com/inward/record.url?scp=84947556741&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84947556741&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-24553-9_88

DO - 10.1007/978-3-319-24553-9_88

M3 - Chapter

AN - SCOPUS:84947556741

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 719

EP - 727

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

PB - Springer Verlag

ER -