# Mean Estimation on the Diagonal of Product Manifolds

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### Contribution

## 2. Background

#### 2.1. Riemannian Geometry

#### 2.2. Weighted Fréchet Mean

#### 2.3. Weighted Diffusion Mean

#### 2.4. Diffusion Bridges

#### Euclidean Diffusion Bridges

#### 2.5. Manifold Diffusion Processes

#### 2.6. Manifold Bridges

## 3. Diffusion Mean Estimation

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

#### 3.1. Fermi Bridges to the Diagonal

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

- (i)
- the sectional curvature of planes containing the radial direction is non-negative or the Ricci curvature in the radial direction is non-negative;
- (ii)
- $Cut\left(N\right)$ is polar for the Fermi bridge ${Y}^{F}$ and either the sectional curvature of planes containing the radial direction is non-positive or the Ricci curvature in the radial direction is non-positive;

**Proof.**

#### 3.2. Simulation in Coordinates

**Lemma**

**1.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 3.3. Accounting for $\phi $

Algorithm 1:weighted Diffusion Mean |

## 4. Experiments

#### 4.1. Mean Estimation on ${\mathbb{S}}^{2}$

#### 4.2. LDDMM Landmarks

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Fréchet, M. Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. L’Institut Henri Poincaré
**1948**, 10, 215–310. [Google Scholar] - Arnaudon, M.; Li, X.M. Barycenters of measures transported by stochastic flows. Ann. Probab.
**2005**, 33, 1509–1543. [Google Scholar] [CrossRef] [Green Version] - Pennec, X. Barycentric Subspace Analysis on Manifolds. Ann. Stat.
**2018**, 46, 2711–2746. [Google Scholar] [CrossRef] [Green Version] - Hansen, P.; Eltzner, B.; Sommer, S. Diffusion Means and Heat Kernel on Manifolds. In Geometric Science of Information; Lecture Notes in Computer Science; Springer Nature: Cham, Switzerland, 2021; pp. 111–118. [Google Scholar] [CrossRef]
- Hansen, P.; Eltzner, B.; Huckemann, S.F.; Sommer, S. Diffusion Means in Geometric Spaces. arXiv
**2021**, arXiv:2105.12061. [Google Scholar] - Chakraborty, R.; Bouza, J.; Manton, J.H.; Vemuri, B.C. ManifoldNet: A Deep Neural Network for Manifold-Valued Data With Applications. IEEE Trans. Pattern Anal. Mach. Intell.
**2022**, 44, 799–810. [Google Scholar] [CrossRef] - Sommer, S.; Bronstein, A. Horizontal Flows and Manifold Stochastics in Geometric Deep Learning. IEEE Trans. Pattern Anal. Mach. Intell.
**2022**, 44, 811–822. [Google Scholar] [CrossRef] - Thompson, J. Submanifold Bridge Processes. Ph.D. Thesis, University of Warwick, Coventry, UK, 2015. [Google Scholar]
- Pennec, X.; Sommer, S.; Fletcher, T. Riemannian Geometric Statistics in Medical Image Analysis; Elsevier: Amsterdam, The Netherlands, 2020. [Google Scholar]
- Pennec, X.; Fillard, P.; Ayache, N. A Riemannian Framework for Tensor Computing. Int. J. Comput. Vis.
**2006**, 66, 41–66. [Google Scholar] [CrossRef] [Green Version] - Hsu, E.P. Stochastic Analysis on Manifolds; American Mathematical Society: Providence, RI, USA, 2002; Volume 38. [Google Scholar]
- Grong, E.; Sommer, S. Most Probable Paths for Anisotropic Brownian Motions on Manifolds. arXiv
**2021**, arXiv:2110.15634. [Google Scholar] - Sommer, S.; Arnaudon, A.; Kuhnel, L.; Joshi, S. Bridge Simulation and Metric Estimation on Landmark Manifolds. In Graphs in Biomedical Image Analysis, Computational Anatomy and Imaging Genetics; Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2017; pp. 79–91. [Google Scholar]
- Papaspiliopoulos, O.; Roberts, G. Importance sampling techniques for estimation of diffusion models. Stat. Methods Stoch. Differ. Equ.
**2012**, 124, 311–340. [Google Scholar] - Jensen, M.H.; Sommer, S. Simulation of Conditioned Semimartingales on Riemannian Manifolds. arXiv
**2021**, arXiv:2105.13190. [Google Scholar] - Lyons, T.J.; Zheng, W.A. On conditional diffusion processes. Proc. R. Soc. Edinb. Sect. Math.
**1990**, 115, 243–255. [Google Scholar] [CrossRef] - Delyon, B.; Hu, Y. Simulation of Conditioned Diffusion and Application to Parameter Estimation. Stoch. Process. Appl.
**2006**, 116, 1660–1675. [Google Scholar] [CrossRef] - Schauer, M.; Van Der Meulen, F.; Van Zanten, H. Guided proposals for simulating multi-dimensional diffusion bridges. Bernoulli
**2017**, 23, 2917–2950. [Google Scholar] [CrossRef] [Green Version] - Marchand, J.L. Conditioning diffusions with respect to partial observations. arXiv
**2011**, arXiv:1105.1608. [Google Scholar] - van der Meulen, F.; Schauer, M. Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals. Electron. J. Stat.
**2017**, 11, 2358–2396. [Google Scholar] [CrossRef] [Green Version] - Elworthy, D. Geometric aspects of diffusions on manifolds. In École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87; Springer: Berlin/Heidelberg, Germany, 1988; pp. 277–425. [Google Scholar]
- Emery, M. Stochastic Calculus in Manifolds; Springer: Berlin/Heidelberg, Germany, 1989. [Google Scholar]
- Elworthy, K.; Truman, A. The diffusion equation and classical mechanics: An elementary formula. In Stochastic Processes in Quantum Theory and Statistical Physics; Springer: Berlin/Heidelberg, Germany, 1982; pp. 136–146. [Google Scholar]
- Li, X.M. On the semi-classical Brownian bridge measure. Electron. Commun. Probab.
**2017**, 22, 1–15. [Google Scholar] [CrossRef] - Ndumu, M.N. Brownian Motion and the Heat Kernel on Riemannian Manifolds. Ph.D. Thesis, University of Warwick, Coventry, UK, 1991. [Google Scholar]
- Malliavin, P.; Stroock, D.W. Short time behavior of the heat kernel and its logarithmic derivatives. J. Differ. Geom.
**1996**, 44, 550–570. [Google Scholar] [CrossRef] - Stroock, D.W.; Turetsky, J. Short time behavior of logarithmic derivatives of the heat kernel. Asian J. Math.
**1997**, 1, 17–33. [Google Scholar] [CrossRef] [Green Version] - Thompson, J. Brownian bridges to submanifolds. Potential Anal.
**2018**, 49, 555–581. [Google Scholar] [CrossRef] [Green Version] - Joshi, S.; Miller, M. Landmark Matching via Large Deformation Diffeomorphisms. IEEE Trans. Image Process.
**2000**, 9, 1357–1370. [Google Scholar] [CrossRef] [Green Version] - Younes, L. Shapes and Diffeomorphisms; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Kühnel, L.; Sommer, S.; Arnaudon, A. Differential Geometry and Stochastic Dynamics with Deep Learning Numerics. Appl. Math. Comput.
**2019**, 356, 411–437. [Google Scholar] [CrossRef] [Green Version] - Stegmann, M.B.; Fisker, R.; Ersbøll, B.K. Extending and Applying Active Appearance Models for Automated, High Precision Segmentation in Different Image Modalities. Available online: http://www2.imm.dtu.dk/pubdb/edoc/imm118.pdf (accessed on 4 February 2022).

**Figure 1.**(

**left**) The mean estimator viewed as a projection onto the diagonal of a product manifold. Given a set ${x}_{1},\cdots ,{x}_{n}\in M$, the tuple $({x}_{1},\cdots ,{x}_{n})$ (blue dot) belongs to the product manifold $M\times \cdots \times M$. The mean estimator $\widehat{\mu}$ can be identified with the projection of $({x}_{1},\cdots ,{x}_{n})$ onto the diagonal N (red dot). (

**right**) Diffusion mean estimator in ${\mathbb{R}}^{2}$ using Brownian bridges conditioned on the diagonal. Here a Brownian bridge ${X}_{t}=({X}_{1,t},\cdots ,{X}_{4,t})$ in ${\mathbb{R}}^{8}$ is conditioned on hitting the diagonal $N\subseteq {\mathbb{R}}^{8}$ at time $T>0$. The components ${X}_{j}$ each being two-dimensional processes are shown in the plot.

**Figure 2.**The mean estimator viewed as a projection onto the diagonal of a product manifold. Conditioning on the closest point in the diagonal yields a density on the diagonal depending on the time to arrival $T>0$. As T tends to zero the density convergence to the Dirac-delta distribution (grey), whereas as T increases the variance of the distribution increases (rouge).

**Figure 3.**3 points on ${\mathbb{S}}^{2}$ together with a sample mean (red) and the diagonal process in ${\left({\mathbb{S}}^{2}\right)}^{n}$, $n=3$ with $T=0.2$ conditioned on the diagonal.

**Figure 4.**(

**left**) 256 sampled data points on ${\mathbb{S}}^{2}$ (north pole being population mean). (

**right**) 32 samples of the diffusion mean conditioned on the diagonal of ${\left({\mathbb{S}}^{2}\right)}^{n}$, $n=256$, $T=0.2$. As can be seen, the variation in the mean samples is limited.

**Figure 5.**(

**left**) One configuration of 17 landmarks overlayed the MR image from which the configuration was annotated. (

**right**) All 14 landmark configurations plotted together (one color for each configuration of 17 landmarks).

**Figure 6.**Samples from the diagonal process with $T=0.2$ (

**left**) and $T=1$ (

**right**). The effect of varying the Brownian motion end time T is clearly visible.

**Figure 7.**One sampled diffusion mean with the sampling scheme (blue configuration) together with estimated Fréchet mean (green configuration). The forward sampling scheme is significantly faster than the iterative optimization needed for the Fréchet mean on the landmark manifold where closed-form solutions of the geodesic equations are not available.

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**MDPI and ACS Style**

Jensen, M.H.; Sommer, S.
Mean Estimation on the Diagonal of Product Manifolds. *Algorithms* **2022**, *15*, 92.
https://doi.org/10.3390/a15030092

**AMA Style**

Jensen MH, Sommer S.
Mean Estimation on the Diagonal of Product Manifolds. *Algorithms*. 2022; 15(3):92.
https://doi.org/10.3390/a15030092

**Chicago/Turabian Style**

Jensen, Mathias Højgaard, and Stefan Sommer.
2022. "Mean Estimation on the Diagonal of Product Manifolds" *Algorithms* 15, no. 3: 92.
https://doi.org/10.3390/a15030092