Manifold Geometry Meets Logistic Regression: The Rise of Hypergyroplanes

Abstract and 1. Introduction

2. Preliminaries

3. Proposed Approach

   3.1 Notation

   3.2 Nueral Networks on SPD Manifolds

   3.3 MLR in Structure Spaces

   3.4 Neural Networks on Grassmann Manifolds

4. Experiments

5. Conclusion and References

A. Notations

B. MLR in Structure Spaces

C. Formulation of MLR from the Perspective of Distances to Hyperplanes

D. Human Action Recognition

E. Node Classification

F. Limitations of our work

G. Some Related Definitions

H. Computation of Canonical Representation

I. Proof of Proposition 3.2

J. Proof of Proposition 3.4

K. Proof of Proposition 3.5

L. Proof of Proposition 3.6

M. Proof of Proposition 3.11

N. Proof of Proposition 3.12

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\ The key idea to generalize MLR to a Riemannian manifold is to change the margin to reflect the geometry of the considered manifold (a formulation of MLR from the perspective of distances to hyperplanes is given in Appendix C). This requires the notions of hyperplanes and margin in the considered manifold that are referred to as hypergyroplanes and pseudo-gyrodistances (Nguyen & Yang, 2023), respectively. In our case, the definition of hypergyroplanes in structure spaces, suggested by Proposition 3.2, can be given below.

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\ Proof See Appendix M.

\ The algorithm for computing the pseudo-gyrodistances is given in Appendix B.

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:::info Authors:

(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France ([email protected]);

(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France ([email protected]);

(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France ([email protected]).

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:::info This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license.

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