Contraction and consensus on Hadamard manifolds

Question

Let $\mathcal M$ be a Hadamard manifold and $\{x_i\}_{i=1}^n\subseteq{\cal M}$ be $n$ points. Define $\{y_i\}_{i=1}^n$ as the weighted Fréchet means: $$ y_i=\arg\min_{y\in\mathcal M}\sum_jw_{ij}d^2(y,x_j),\quad\forall i, $$ where $d(\cdot,\cdot)$ is the geodesic distance on ${\cal M}$ and $w_{ij}$ are the weights. Assume the weight matrix $W=(w_{ij})\in\mathbb{R}^{n\times n}$ is a symmetric and doubly stochastic matrix, that is, $\sum_{i}w_{ij}=\sum_jw_{ij}=1$, $w_{ij}\geq 0$. Can we prove that the Fréchet variance significantly reduces after this procedure in the sense that $$ \sum_id^2(y_i,\bar y)\leq \sigma_2^2(W)\cdot \sum_id^2(x_i,\bar x), $$ where $\sigma_2(W)$ is the second largest singular vlaue of $W$ and $\bar y$, $\bar x$ are the Fréchet means of $\{y_i\}$ and $\{x_i\}$, respectively.

Answer 1

I have solved this question. One can refer to Lemma 3 in the paper Decentralized Online Riemannian Optimization with Dynamic Environments. arXiv:2410.05128v1