Sorted eigenvalue comparison $d_{\mathsf{Eig}}$: A simple alternative to $d_{\mathsf{FID}}$

For $i = 1, 2$, let $\mathbf{S}_i$ be the sample covariance of $\mathbf{Z}_i$ with $n_i$ $p$-dimensional vectors. First, we theoretically justify an improved Fréchet Inception Distance ($d_{\mathsf{FID}}$) algorithm that replaces np.trace(sqrtm($\mathbf{S}_1 \mathbf{S}_2$)) with np.sqrt(eigvals($\mathbf{S}_1 \mathbf{S}_2$)).sum(). With the appearance of unsorted eigenvalues in the improved $d_{\mathsf{FID}}$, we are then motivated to propose sorted eigenvalue comparison ($d_{\mathsf{Eig}}$) as a simple alternative: $d_{\mathsf{Eig}}(\mathbf{S}_1, \mathbf{S}_2)^2=\sum_{j=1}^p (\sqrt{\lambda_j^1} - \sqrt{\lambda_j^2})^2$, and $\lambda_j^i$ is the $j$-th largest eigenvalue of $\mathbf{S}_i$. Second, we present two main takeaways for the improved $d_{\mathsf{FID}}$ and proposed $d_{\mathsf{Eig}}$ . (i) $d_{\mathsf{FID}}$: The error bound for computing non-negative eigenvalues of diagonalizable $\mathbf S_1 \mathbf S_2$ is reduced to $\mathcal{O}(\varepsilon) \|\mathbf S_1 \| \|\mathbf S_1 \mathbf S_2 \|$, along with reducing the run time by $\sim25\%$. (ii) $d_{\mathsf{Eig}}$: The error bound for computing non-negative eigenvalues of sample covariance $\mathbf S_i$ is further tightened to $\mathcal{O}(\varepsilon) \|\mathbf S_i \|$, with reducing $\sim90\%$ run time. Last, we discuss limitations and future work for $d_{\mathsf{Eig}}$.