The Frkhet Distance between Multivariate Normal Distributions

By D. Dowson, B. Landau. 2003

The Frichet distance between two multivariate normal distributions having means ,ux, ,+ and covariance matrices Lx, Z, is shown to be given by d* = Ifix -,+(’ + tr(C, + Zr - 2(Z,Z,)“*). The quantity d, given by di = tr(Z, + Z, 1 2(C,Z,)“*) is a natural metric on the space of real covariance matrices of given order.

Subfields: Statistics and Probability

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Citations in the corpus

Overcoming catastrophic forgetting in neural networks (2016, methodology): “Precisely, we computed two tasks’ Fishers, F 1 and F 2 , normalized these to each have unit trace, ˆ F 1 and ˆ F 2 , then computed their Fréchet distance, a metric on the space of positive-semideﬁnite matrices [Dowson and Landau, 1982]: which is bounded between zero and one.”
Spectral Normalization for Generative Adversarial Networks (2018, background): “On its own, the Fre´chet distance Dowson & Landau (1982) is 2-Wasserstein distance between two distribution p 1 and p 2 assuming they are both multivariate Gaussian distributions: where { µ p 1 , C p 1 } , { µ p 2 , C p 2 } are the mean and covariance of samples from q and p , respectively.”
Projected GANs Converge Faster (2021, methodology): “To measure how well G fits a particular feature space, we employ the Fréchet Distance (FD) [12] on the spatially pooled features denoted as FDi for layer i.”
GANs Trained by a Two Time-Scale Update Rule Converge to a Nash Equilibrium (2017, background): “) the “Fréchet Inception Distance” (FID), which is given by [14]:”
Implicit Generation and Generalization in Energy-Based Models (2019, methodology): “To compute the value, we ﬁt a multivariate Gaussian on all different states at each speciﬁc time-step t and compute Frechet Distance (Dowson and Landau, 1982) between predicted and ground distributions, similar to the FID score proposed in (Heusel et al., 2017).”
Attributing Fake Images to GANs: Learning and Analyzing GAN Fingerprints (2018, methodology): “In addition, we use the ratio of inter-class and intra-class Fréchet Distance [25], denoted as FD ratio, to evaluate the distinguishability of a feature representation across classes.”
Unsupervised segmentation of natural images via lossy data compression (2008, background): “In this paper, since texture segments are modeled by Gaussian distributions, their EMD/ Mallows similarity distance has a closed-form expression [5]:”
The Role of ImageNet Classes in Fréchet Inception Distance (2022, background): “(equivalently, the 2-Wasserstein or earth mover’s) distance (Dowson & Landau, 1982) where ( µ r , Σ r ), and ( µ g , Σ g ) denote the sample mean and covariance of the embeddings of the real and generated data, respectively, and Tr ( · ) indicates the matrix trace.”
Fréchet means and Procrustes analysis in Wasserstein space (2017, background): “The optimal maps are known to be linear and admit the explicit formula (Dowson & Landau [29]; Olkin & Pukelsheim [60]) tji = S 1/2 j [S 1/2 j SiS 1/2 j ] −1/2S 1/2 j .”
Implicit Generation and Modeling with Energy Based Models (2019, methodology): “We evaluate multi-step trajectories by computing Frechet Distance [Dowson and Landau, 1982] between predicted and ground distributions across all states at timestep t .”
MGAN: Training Generative Adversarial Nets with Multiple Generators (2018, background): “Assuming that p and q are both multivariate Gaussian distributions, FID measures the Fr´echet distance (Dowson & Landau, 1982), which is also the 2-Wasserstein distance, between the two distributions.”
GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium (2017, background): “) the “Fréchet Inception Distance” (FID), which is given by [13]: d (cid:107) − (cid:107) − Next we show that the FID is consistent with increasing disturbances and human judgment.”
Fréchet Audio Distance: A Metric for Evaluating Music Enhancement Algorithms (2018, methodology): “the music enhancement model that we wish to evaluate. We then compute multivariate Gaussians on both the evaluation set embeddings N e( e; e) and the background embeddings N b( b; b). Dowson et al. [3] show that the Frechet distance between two Gaussians is: F(N b;N e) = k b ek2 + tr( b+ e 2 p b e)”
On the Bures–Wasserstein distance between positive definite matrices (2017, result): “[14] D. Dowson and B. Landau, The Fréchet distance between multivariate normal distribu- tions, J. Math.”
Reliable Fidelity and Diversity Metrics for Generative Models (2020, methodology): “The distance between the two Gaussians is computed by the Fr´echet distance (Dowson & Landau, 1982), also known as the Wasserstein-2 distance (Vaserstein, 1969).”