Consider a Hadamard manifold $\mathcal M$ with a lower sectional curvature bound $\kappa\leq 0$. Let $W=(w_{ij})\in\mathbb{R}^{n\times n}$ be a symmetric and doubly stochastic matrix, that is, $\sum_{i}w_{ij}=\sum_jw_{ij}=1$, $w_{ij}\geq 0$.

Now consider the following procedure. For any $n$ points $\{x_i\}_{i=1}^n\subseteq \mathcal M$, we can obtain $n$ points $\{y_i\}_{i=1}^n$ through 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. 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.

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.

Consider a Hadamard manifold $\cal M$ with a lower sectional curvature bound $\kappa\leq 0$. Let $W=(w_{ij})\in\mathbb{R}^{n\times n}$ be a symmetric and doubly stochastic matrix, that is, $\sum_{i}w_{ij}=\sum_jw_{ij}=1$, $w_{ij}\geq 0$.

Now consider the following procedure. For any $n$ points $\{x_i\}_{i=1}^n\subseteq {\cal M}$, we can obtain $n$ points $\{y_i\}_{i=1}^n$ through the weighted Frechet means: $$ y_i=\arg\min_{y\in{\cal M}}\sum_jw_{ij}d^2(y,x_j),\quad\forall i, $$ where $d(\cdot,\cdot)$ is the geodesic distance on ${\cal M}$ and $w_{ij}$ are the weights. Can we prove that the Frechet 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 Frechet means of $\{y_i\}$ and $\{x_i\}$, respectively.

Consider a Hadamard manifold $\mathcal M$ with a lower sectional curvature bound $\kappa\leq 0$. Let $W=(w_{ij})\in\mathbb{R}^{n\times n}$ be a symmetric and doubly stochastic matrix, that is, $\sum_{i}w_{ij}=\sum_jw_{ij}=1$, $w_{ij}\geq 0$.

Now consider the following procedure. For any $n$ points $\{x_i\}_{i=1}^n\subseteq \mathcal M$, we can obtain $n$ points $\{y_i\}_{i=1}^n$ through 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. 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.