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

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

[ Abstract ] [ Project Page ]

[ OpenReview]

Abstract: For

i = 1, 2

$i = 1, 2$ , let

S_{i}

$\mathbf{S}_i$ be the sample covariance of

Z_{i}

$\mathbf{Z}_i$ with

n_{i}

$n_i$

p

$p$ -dimensional vectors. First, we theoretically justify an improved Fréchet Inception Distance (

d_{F I D}

$d_{\mathsf{FID}}$ ) algorithm that replaces np.trace(sqrtm(

S_{1} S_{2}

$\mathbf{S}_1 \mathbf{S}_2$ )) with np.sqrt(eigvals(

S_{1} S_{2}

$\mathbf{S}_1 \mathbf{S}_2$ )).sum(). With the appearance of unsorted eigenvalues in the improved

d_{F I D}

$d_{\mathsf{FID}}$ , we are then motivated to propose sorted eigenvalue comparison (

d_{E i g}

$d_{\mathsf{Eig}}$ ) as a simple alternative:

d_{E i g} (S_{1}, S_{2})^{2} = \sum_{j = 1}^{p} (\sqrt{λ_{j}^{1}} - \sqrt{λ_{j}^{2}})^{2}

$d_{\mathsf{Eig}}(\mathbf{S}_1, \mathbf{S}_2)^2=\sum_{j=1}^p (\sqrt{\lambda_j^1} - \sqrt{\lambda_j^2})^2$ , and

λ_{j}^{i}

$\lambda_j^i$ is the

j

$j$ -th largest eigenvalue of

S_{i}

$\mathbf{S}_i$ . Second, we present two main takeaways for the improved

d_{F I D}

$d_{\mathsf{FID}}$ and proposed

d_{E i g}

$d_{\mathsf{Eig}}$ . (i)

d_{F I D}

$d_{\mathsf{FID}}$ : The error bound for computing non-negative eigenvalues of diagonalizable

S_{1} S_{2}

$\mathbf S_1 \mathbf S_2$ is reduced to

O (ε) ‖ S_{1} ‖ ‖ S_{1} S_{2} ‖

$\mathcal{O}(\varepsilon) \|\mathbf S_1 \| \|\mathbf S_1 \mathbf S_2 \|$ , along with reducing the run time by

\sim 25 %

$\sim25\%$ . (ii)

d_{E i g}

$d_{\mathsf{Eig}}$ : The error bound for computing non-negative eigenvalues of sample covariance

S_{i}

$\mathbf S_i$ is further tightened to

O (ε) ‖ S_{i} ‖

$\mathcal{O}(\varepsilon) \|\mathbf S_i \|$ , with reducing

\sim 90 %

$\sim90\%$ run time. Last, we discuss limitations and future work for

d_{E i g}

$d_{\mathsf{Eig}}$ .

Chat is not available.