Min Distance to Riemannian Mean (MDRM)

• MDRM is a measure that computes the average squared geodesic distance from manifold data points to their intrinsic Fréchet mean.
• It employs both stochastic and deterministic gradient descent methods with convergence guarantees under specific curvature constraints.
• Neural Eikonal solvers offer efficient, differentiable approximations of Riemannian distances for handling complex, high-dimensional manifolds.

The minimum distance to Riemannian mean (MDRM) quantifies the minimal average squared geodesic distance from empirical data points on a Riemannian manifold to their Fréchet mean. This notion is central to statistical inference, clustering, and geometric data analysis on manifolds. MDRM is rigorously defined via the empirical Fréchet energy and can be computed using both stochastic and deterministic gradient algorithms, with theoretical guarantees under curvature constraints (Arnaudon et al., 2011). Efficient computation of MDRM on complex manifolds is enabled by neural approximations of the metric-constrained Eikonal equation, providing differentiable surrogates of the Riemannian distance (Kelshaw et al., 2024).

1. Formal Definition and Variational Characterization

Let $(M,g)$ be a complete Riemannian manifold and $\{y_i\}_{i=1}^{n}\subset M$ a finite sample. The empirical measure is $\mu_n = \frac{1}{n}\sum_{i=1}^n \delta_{y_i}$. Define the empirical Fréchet energy:

$F_n(x) = \frac{1}{n} \sum_{i=1}^n d^2(x, y_i),$

where $d(\cdot,\cdot)$ denotes the geodesic distance. The sample Fréchet mean $\hat\mu_n$ is

$\hat\mu_n = \arg\min_{x\in M} F_n(x),$

and the minimum distance to mean (MDRM) statistic is

$$F_n(\hat\mu_n) = \frac{1}{n} \sum_{i=1}^n d^2(\hat\mu_n, y_i).$$

This functional measures point-cloud spread relative to their intrinsic mean under the manifold geometry (Arnaudon et al., 2011).

2. Optimality Conditions and Uniqueness

For sufficiently regular distributions and injectivity radius, the gradient of $F$ at $x$ is given by

$\{y_i\}_{i=1}^{n}\subset M$0

Hence, $\{y_i\}_{i=1}^{n}\subset M$1 is a Fréchet mean if and only if

$\{y_i\}_{i=1}^{n}\subset M$2

ensuring $\{y_i\}_{i=1}^{n}\subset M$3 zeroes the intrinsic mean of log-mapped vectors. On compact Riemannian manifolds and for empirical measures, the Fréchet mean is unique for generic data configurations (Arnaudon et al., 2011).

3. Curvature Bounds and Convexity Properties

Let $\{y_i\}_{i=1}^{n}\subset M$4 be a geodesic ball and sectional curvatures $\{y_i\}_{i=1}^{n}\subset M$5 on $\{y_i\}_{i=1}^{n}\subset M$6. When $\{y_i\}_{i=1}^{n}\subset M$7 (for $\{y_i\}_{i=1}^{n}\subset M$8),

$\{y_i\}_{i=1}^{n}\subset M$9

with $\mu_n = \frac{1}{n}\sum_{i=1}^n \delta_{y_i}$0 for $\mu_n = \frac{1}{n}\sum_{i=1}^n \delta_{y_i}$1, $\mu_n = \frac{1}{n}\sum_{i=1}^n \delta_{y_i}$2 for $\mu_n = \frac{1}{n}\sum_{i=1}^n \delta_{y_i}$3. Thus, $\mu_n = \frac{1}{n}\sum_{i=1}^n \delta_{y_i}$4 is strongly convex along geodesics, which guarantees uniqueness and stability of the minimizer, thereby affirming the robustness of the MDRM concept under curvature constraints (Arnaudon et al., 2011).

4. Algorithmic Computation of MDRM

For empirical measures, MDRM can be computed via gradient-based methods:

• Stochastic Gradient Update:

$\mu_n = \frac{1}{n}\sum_{i=1}^n \delta_{y_i}$5, with $\mu_n = \frac{1}{n}\sum_{i=1}^n \delta_{y_i}$6, $\mu_n = \frac{1}{n}\sum_{i=1}^n \delta_{y_i}$7. Step sizes $\mu_n = \frac{1}{n}\sum_{i=1}^n \delta_{y_i}$8 satisfy $\mu_n = \frac{1}{n}\sum_{i=1}^n \delta_{y_i}$9, $F_n(x) = \frac{1}{n} \sum_{i=1}^n d^2(x, y_i),$0 (Arnaudon et al., 2011).

• Deterministic Gradient Descent:

$F_n(x) = \frac{1}{n} \sum_{i=1}^n d^2(x, y_i),$1, where $F_n(x) = \frac{1}{n} \sum_{i=1}^n d^2(x, y_i),$2. Step sizes $F_n(x) = \frac{1}{n} \sum_{i=1}^n d^2(x, y_i),$3 as above ensure convergence to $F_n(x) = \frac{1}{n} \sum_{i=1}^n d^2(x, y_i),$4 (Arnaudon et al., 2011).

• Practical Recommendations:

Initialize at one of $F_n(x) = \frac{1}{n} \sum_{i=1}^n d^2(x, y_i),$5, iterate using chosen scheme until $F_n(x) = \frac{1}{n} \sum_{i=1}^n d^2(x, y_i),$6, then compute MDRM as $F_n(x) = \frac{1}{n} \sum_{i=1}^n d^2(x, y_i),$7. Error control is achieved by step-sizes of order $F_n(x) = \frac{1}{n} \sum_{i=1}^n d^2(x, y_i),$8, yielding $F_n(x) = \frac{1}{n} \sum_{i=1}^n d^2(x, y_i),$9 convergence in mean-square.

5. Neural Eikonal Solvers and Efficient MDRM Evaluation

Deep metric-constrained Eikonal solvers enable efficient approximation of the Riemannian distance function $d(\cdot,\cdot)$0 on manifold $d(\cdot,\cdot)$1 (Kelshaw et al., 2024). The MDRM for a query $d(\cdot,\cdot)$2 is evaluated as

$d(\cdot,\cdot)$3

with computation complexity $d(\cdot,\cdot)$4, where $d(\cdot,\cdot)$5 is the network FLOPs. For geodesic path recovery, backtracking using explicit ODE integration (Euler, RK4, symplectic) from $d(\cdot,\cdot)$6 to $d(\cdot,\cdot)$7 follows the negative Riemannian gradient field of $d(\cdot,\cdot)$8.

Accuracy guarantees are enforced by training loss $d(\cdot,\cdot)$9 to maintain $\hat\mu_n$0 uniformly, yielding relative $\hat\mu_n$1 errors of $\hat\mu_n$2 on benchmarks (Euclid, sphere, Gaussian mixture models). Handling of high-curvature regions involves Ricci-biased sampling; singularities and complex topologies require reparameterisation and reprojection strategies (Kelshaw et al., 2024).

6. Statistical Consistency and Interpretation

Given probability measure $\hat\mu_n$3 on $\hat\mu_n$4, the Fréchet variance is

$\hat\mu_n$5

Empirically, $\hat\mu_n$6 strongly consistently estimates $\hat\mu_n$7 as $\hat\mu_n$8. Under uniqueness of the Fréchet mean, almost sure convergence $\hat\mu_n$9 and $\hat\mu_n = \arg\min_{x\in M} F_n(x),$0 is guaranteed (Arnaudon et al., 2011). This affirms MDRM as a fundamental statistic for spread and centrality on Riemannian manifolds.

7. Applications and Computational Considerations

MDRM underlies statistical analysis on manifold-valued data in domains such as non-Euclidean machine learning, geometric statistics, and signal processing (e.g., radar target detection using Toeplitz covariance matrices (Arnaudon et al., 2011)). The neural Eikonal approach broadens MDRM computation to manifolds with complex metrics or high dimensions, previously intractable with classical solvers. Parameter choices (network depth/width, sampling method) must scale with manifold complexity to ensure accuracy (Kelshaw et al., 2024).

---

References

Kelshaw & Magri, “Computing distances and means on manifolds with a metric‐constrained Eikonal approach,” (Kelshaw et al., 2024) Arnaud et al., “Medians and means in Riemannian geometry: existence, uniqueness and computation,” (Arnaudon et al., 2011)