Feature Likelihood Divergence: Evaluating the Generalization of Generative Models Using Samples
Authors: Marco Jiralerspong
Joey Bose
Ian Gemp
Chongli Qin
Yoram Bachrach
Gauthier Gidel
Venue: NeurIPS 2023 poster (2023)
Keywords: Generative model
FID
Evaluation
Precision
Recall
Likelihood
Overfitting
Memorization
Generalization
Diffusion
GANs

Abstract: The past few years have seen impressive progress in the development of deep generative models capable of producing high-dimensional, complex, and photo-realistic data. However, current methods for evaluating such models remain incomplete: standard likelihood-based metrics do not always apply and rarely correlate with perceptual fidelity, while sample-based metrics, such as FID, are insensitive to overfitting, i.e., inability to generalize beyond the training set. To address these limitations, we propose a new metric called the Feature Likelihood Divergence (FLD), a parametric sample-based score that uses density estimation to provide a comprehensive trichotomic evaluation accounting for novelty (i.e., different from the training samples), fidelity, and diversity of generated samples.  We empirically demonstrate the ability of FLD to identify specific overfitting problem cases, where previously proposed metrics fail. We also extensively evaluate FLD on various image datasets and model classes, demonstrating its ability to match intuitions of previous metrics like FID while offering a more comprehensive evaluation of generative models.

Feature Likelihood Divergence: Evaluating the Generalization of Generative Models Using Samples
Marco Jiralerspong" Université de Montréal and Mila
Avishek (Joey) Bose McGill University and Mila
Ian Gemp Google Deepmind
Chongli Qin Google Deepmind
Yoram Bachrach
Gauthier Gidel
Google Deepmind
Université de Montréal and Mila
Abstract
The past few years have seen impressive progress in the development of deep gener- ative models capable of producing high-dimensional, complex, and photo-realistic data. However, current methods for evaluating such models remain incomplete: standard likelihood-based metrics do not always apply and rarely correlate with perceptual fidelity, while sample-based metrics, such as FID, are insensitive to overfitting, i.c., inability to generalize beyond the training set. To address these limitations, we propose a new metric called the Feature Likelihood Divergence (FLD), parametric sample-based metric that uses density estimation to provide comprehensive trichotomic evaluation accounting for novelty (i.e., different from the training samples), fidelity, and diversity of generated samples. We empirically demonstrate the ability of FLD to identify overfitting problem cases, even when previously proposed metrics fail. We also extensively evaluate FLD various image datasets and model classes, demonstrating its ability to match intuitions of previous metrics like FID while offering more comprehensive evaluation of generative models. Code is available at http:///luub..oom/marrrrrr//u/
Introduction
Generative modeling one of the fastest-growing areas of deep learning, with success stories spanning the artificial intelligence spectrum [Karras et al., 2020, Brown et al., 2020, Wu et al., 2021, Rombach et al., 2022]. Despite the growth of applications- and unlike supervised or reinforcement tearning-there is lack of a clear consensus regimes which these models excel. In particular, the standard metric of evaluating -igllikeddood of held-out test data [Bishop and Nasrabadi, 2006, Goodfellow et al., 2016, Murphy, 2022] fails to provide meaningful evaluation signal due to its large variability between repetitions [Nowozin et al., 2016] and lack of direct correlation with sample fidelity [Theis al., 2015].
evaluation protocol in high-dimensional data
Departing from pure likelihood-hased evaluation, sample-based metrics offer appealing benefits such being able to evaluate any generative model family via their generated samples. Furthermore, sample-based metrics such as Inception score (IS) [Salimans et al., 2016], Fréchet Inception distance (FID) [Heusel al., 2017], precision, and recall [Lucie al., 2018, Sajjadi et al., 2018] have been shown to correlate with sample quality, i.e. the perceptual visual quality of a generated sample, and the perceptual sample diversity. Despite being the current defacto gold standard, sample-based
"e-mail correspondence marco. wtt.oojrrraaeerppgemmiii.. quebee
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OOur metric (FLD) named Feature Likelihood Score (FLS) in the previous versions of this paper.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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Futstee 2016
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FLD (our)
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AuthPct Nintanetel, 2010
ReACGAN
MIKGAN
wuess ooer
Spied.
PPGMM
Capyos
IS
Sitrammen
FID Dinodaa
10036+ X6
Mode coverage
Novelty Cy Score
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Like 2001
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Cappost
RoADDAN
RiggaN Deep
EnyleaNN ad
motst DRM
The generative model evaluation "tri- chotomy": fidelity, diversity and novelty. Each metric maps to a color delimiting its criteria for evaluation.
Figure 1:
Figure 2: Test FID/FLD of various models CI- FARIO. FID values are higher than usual we use the DINOv2 feature space with the test set as refer- ence (10k samples) instead of the usual 50k.
metrics miss important facets of evaluation [Xu et al., 2018, Esteban et al., 2017, Meehan et al., 2020]. For example, on CIFAR1O, the current standard FID computation uses 50k generated samples and 50k training samples from the dataset, practice that does not take into account overfitting. Consider the worst-case scenario of a model called copycat which simply outputs copies of the training samples. Using the standard evaluation protocol, such model would obtain FID of 0 (as we compare distances of two identical Gaussians) perfect score for useless model. Instead, we could try looking at the FID of the copycat relative to the test set. While this improves the situation somewhat (see Fig. 2), copycat still obtains better than SOTA test FID, demonstrating that aiming for the lowest FID is highly vulnerable overfitting.
While old generative models struggled to produce good quality samples, recent models have demonstrated the ability to memorize [Somepalli et al., 2022] and overfit [Yazici et al., 2020]. With the widespread adoption of deep generative models high-stakes and industrial production environments, important concerns regarding data privacy [Carlini et al., 2023, Arora et al., 2018, Hitaj et al., 2017] should be raised. For instance, in safety-critical application domains such as precision medicine, data leakage in the form of memorization is unacceptable and severely limits the adoption of generative models-a a few of which have been empirically shown to be guilty of "digital forgery" [Somepalli al., 2022]. These concerns highlight the limitations with current evaluation metrics:
There are currently no sample-based evaluation metrics accounting for the trichotomy between sample fidelity, diversity, and novelty (Figure 1).
We believe this trichotomy encompasses what is required for a generative model have the desired generalization properties, i.e., a "good" generative model should generate samples that are diverse and perceptually indistinguishable from the training data distribution, but at the same time, different from that training data. In other words, the more generated data looks like unseen test samples, the better. Additionally, by assessing the novelty of generated samples relation to the training set, we can better identify potential privacy and copyright risks.
Main Contribution. We propose the feature likelihood divergence (FLD): a novel sample-based metric that captures sample fidelity, diversity, and novelty. FLD enjoys the same scalability as popular sample-based metrics such as FID and IS but crucially also assesses sample novelty, overfitting, and memorization. Evaluation using FLD has many consequential benefits:
1. Explainability: Samples that contribute the most (and the least) to the performance identified. Diagnosing Overfitting: As overfitting begins (i.c., copying of the training set) FLD identifies the copies and reports an inferior value despite no drop in sample fidelity and diversity.
3. Holistic Evaluation: FL.D simultaneously is the only metric proposed in the literature that simultaneously evaluates the fidelity, diversity, and novelty of the samples (Fig. 1).
4. Universal Applicability: FLD applies to all generative models, including VAES, Normalizing Flows, GANs, and Diffusion models with minimal overhead as it computed only using samples.
5. Flexibility: Because of its connection with likelihood, FLD can be naturally extended to condi- tional and multi-n modil generative modeling.

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Intuitively, FLD achieves these goals by first mapping samples to a perceptually meaningful feature space such as a pre-trained Inception-v3 [Szegedy et al., 2016] or DINOy2 [Oquab al., 2023]. Then, FLD is derived from the likelihood evaluation protocol that assesses the generalization performance of generative models similar manner to supervised learning setups. As most models lack explicit densities, we model the density of the generative model in our chosen feature space by using a mixture of isotropic Gaussians (MoG), whose means are the mapped features of the generated samples. We then fit the variances of the Gaussians to the train set in such way that memorized samples obtain vanishingly small variances and thus worsen the density estimation of the MoG. Finally, we use the MoG and estimate the perceptual likelihood of some held-out test set.
2 Background and Related Work
Given a training dataset Dirain = {x,} Vi=1 drawn from a distribution Pdata one of the key objectives of generative modeling is to train parametric model g that is able to generate novel synthetic yet high-quality samples -ie.. the distribution Py induced by the generator is close to Piata
Likelihood Evaluation. The most common metric, and perhaps most natural, is the negative log- likelihood (NLL) of the test set, whenever it is easily computable. While appealing theoretically, generative models typically do not provide a density (e.g. GANs) or it is only possible to compute a lower bound of the test NLL (e.g. VAEs, continuous diffusion models, etc.) [Burda et al., 2015, Song et al., 2021, Huang et al.. 2021]. Even when possible, NLL-based evaluation suffers from variety of pitfalls in high dimensions [Theis et al., 2015, Nowozin et al., 2016] and may often not correlate with higher sample quality [Nalisnick et al., 2018, Le Lan and Dinh, 2021]. Indeed many practitioners have empirically witnessed phenomena such as mode-dropping, mode-collapse, and overfitting (Yazici et al., 2020], all of which are not easily captured simply through the NL...
hample-based Metrics As all deep generative models are capable of producing samples, an effective way to evaluate these models is via their samples. Such strategy has the benefit of bypassing the need to compute the exact model density of a sample
for unified evaluation setting. More precisely, given D gen {xx=m) Ji=1 generated samples, where each x8e ~ Py and Dest = {x Ji=1 drawn from Pdsta. the goal is to evaluate how "good" the generated samples are with respect to the real data distribution. Historically, sample-based metries for evaluating deep generative models have been based on two ideas: 1. using an Inception network [Szegedy et al., 2016] backbone yo as feature extractor to 2. compute notion of distance (or similarity) between the generated and the real distribution. The Inception Score (IS) and the Fréchet Inception Distance (FID) are the two most popular examples and can be computed follows:
IS: m. Em, (odd"oo
FID:
|| uppll + Tr,,, + Σ, - 2(2 2p)
where Po is the probability of each class given by the Inception network s. pa(y) is the 1 - (xx((t)) are the
ratio of each class in the real data, Bg := 1 Σ (x( (00) Mp empirical means of each distribution, and Eg m EE ( 1gg
(x
gen Mgg Ep
n (x (x! pp) are the empirical covariances.
The popularity of IS and FID as metrics for generative models is motivated by their correlation with perceptual quality, diversity, and ease of use. More recently, other metrics such KID [Bińkkoski et al., 201 8) (an unbiased version of FID) and precision/recall (which disentangles sample quality and distribution coverage) [Sajjadi et al., 2018] have added nuance generative model evaluation.
Several approaches seek to provide metrics to detect overfitting and can be categorized based whether one can extract exact likelihood [van den Burg and Williams, 2021] or a lower bound to it via annealed importance sampling [Wu et al., 2016]. For GANs, popular approaches include training additional discriminator in Wasserstein GAN [Adlam et al., 2019] and adding a memorization score to the FID [Bai et al., 2021]. Alternate approaches include finding real data samples that closest to generated samples via membership attacks [Liu et al., 2018, Webster et al., 2019]. Non-parametric tests have also been employed to detect memorization or exact data copying in generative models [Xu et al., 2018, Esteban et al., 2017, Meehan et al., 2020]. Parametric approaches to detect data copying have also been explored such as using neural network divergences [Gulrajani et al., 2020] or using latent recovery [Webster al., 2019]. Finally, the Cr
Overfitting Evaluation.
By close we mean either divergence between distributions (e.g. KL. JSD) distance like Wasserstein.
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Gen
Train
Test
Step.1: I. Gather samples and map them to the feature space.
Step 2: Instantiate isotropic Causslans centred at the generated samples
Step 3: Find the variance that maximiass the log-llkellhood the train samples.
Siep.d: d: Use the learned density to evaluate the test samples likelihood.
Figure 3: Steps involved in our overfit mixture of Gaussians illustrated on a 2D example
[Meehan et al., 2020] test statistic and Alaa et al. [2022] proposes multi-faceted metric with binary sample-wise test to determine whether sample is authentic (i.e., overfit).
Feature Likelihood Divergence
We now introduce our Feature Likelihood Score (FLD) which is predicated on the belief that a proper evaluation measure for generative models should go beyond sample quality and also inform practitioners of the generalization capabilities of their trained models. While previous sample-bases methods have foregone density estimation in favor of computing distances between sample statistics, we seek to bring back a likelihood-basee approach to evaluating generative models. To do so, we first propose our method for fitting a mixture of Gaussians (MoGs) to estimate the perceptual density of high-dimensions samples a way that accounts for overfitting. Specifically, our method aims at attributing good NLL high-quality, non-overfit images and 2) poor NLL in cases of overfitting.
Intuitively. we say a generative model is overfitting on average the distribution of generated samples is closer to the training set than the test set in feature space. We seek to characterize this exact behavior with FLD by MoG where the variance parameter o² of each Gaussian approaches zero around generated samples that responsible for overfitting. In section 3.1 deepen this intuition by outlining how the MoG used in FLD can be tuned to assess the perceptual likelihood of samples while punishing memorized samples, before giving precise definition for memorization and overfitting under FLD in section 3.2.
3.1 Overfitting Mixtures of Gaussians
Our method consists of a simple sample-base based density estimator amenable to variety of data domains inspired by traditional mixture of Gaussians (MoG) density estimator with a few key distinctions. Figure 3 summarizes the key steps in computing FLD using MoG density estimator (also detailed in Algorithm in Appendix $D.1) which we now describe below. While it is indeed possible to train any other density model, a MoG offers favorable tradeoff in being simple use use-while still being universal density estimator [Nguyen et al., 2020] and enjoying efficient scalability to large datasets.
Step I: Map to the feature space. The first change we make is to some map yp to map inputs to some perceptually meaningful feature space. Natural choices for this include the representation space of Inception-v?s vv3 and DINOv2. While still high-dimensional, we ensure that larger proportion of dimensions are useful and that the resulting le distances between images are more meaningful.
Step 2: Model the density using MoG. As in kernel density estimation (KDE), to estimate a density from some set of points D Dgm = [xgen) } we center an isotropic Gaussian around each point-i.e the mean of the Gaussian is the coordinates of the point. This means that j-th data point has Gaussian (x((xx="") of Ia). Then, to compute the likelihood of new point x, we simply calculate the mean likelihood assigned to that point by all Gaussians in the mixture:
m Σ (((x()))((xx)) 13 Po (xx = m ==1
1 m
exp (1lx(xx")) -(x)) (1)
= m j=1 (V2TT,))
20²
with the convention that (x((x) (p(x80n), 0z) is dirac (x(x8on). Henceforth, denote this MoG estimator which has fixed centers initialized to dataset (e.g. train set, generated set) as N (x=(D); E), where E is diagonal matrix of bandwidths parameters-i.e. o² where o² is a vector.
Step Use the train set select o3. An important question in kernel density estimation selecting an appropriate bandwidth o3. Overwhelmingly, a single bandwidth is selected which can either be derived statistically or by minimizing some loss through cross validation [Murphy, 2012]. We depart from this single bandwidth philosophy in favor of separate of values for each Gaussian. To select
of. instead of performing standard cross validatio samples from Pg. we fit the bandwidths using

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a subset of training examples f } by minimizing their negative log-likelihood (NLL). Specifically, we solve the following optimization problem:
8² arg max O
Σ 72
1
T log
m m
1
exp -1 (-10(x"))-0x("""
-=0xx"))) + Li
(2)
(V2TT,) j=1
203
where L; is a base likelihood given to each sample (see Appendix $D for details). In particular, by centering each Gaussian on each x²" Dgen and fitting each of the train set, we aim to have memorized samples obtain an overly small oî. worsening the quality of the MoG estimator (and thus penalizing memorization). The following proposition (proof in $F) formalizes this intuition.
Proposition 1. Lei Dij :- 111x(xx))) (x((rri)) be the distance between generated sample and a train sample. Assume j Dij I D with 6j == min; Dij. Then, for any E (1....,m). we have that ō = ((t) where o² a solution of Eq.
Proposition 1 implies that each element of the training set that has been memorized induces Dirac in the MoG density Eq. 1. Thus, one can identify copies of training samples with the learned density. More generally, if one of the generated samples unreasonably close associated x² will be very small as this maximizes the likelihood the training sample. We illustrate this phenomenon with the Two-Moons dataset [Pedregosa et al., 201 1] in Figure 4. Note that since this dataset is low-dimensional, we do not need to use a feature extractor (Step 1). In Figure 4 we can see that the more approximate copies of the training set appear in the generated set, the more the estimated density (using Eq. 2) contains high values around approximate copies of the training set. As such, overfitted generated samples yield an overfitted MoG that does not model the distribution of real data Paata and will yield poor (i.e., low) log- likelihood on the test set Dest.
training sample, its
== approximate copies
k= 50 apprxxmmate copies
k 100 approximate copies
k 150 approximate capies
Figure 4: Estimated density (in purple) of the generated distribution using an MoG centered at the generated
samples x;" (in blue) Eq. The selection of is done Eq. 2. The training points xmi ~ pa, sampled from the two-moons dataset, are represented in orange. The generated points correspond to k approximates copies of the training set x!" = =x, jam + (0, 10 10-4), i i= 1,.... k and 200 k independent samples from the 200. The dark areas correspond to high-density values.
data distribution x;
ghh Pa. = h
Step 4: Evaluate MoG density. A foundational concept used by FLD is to evaluate the perceptual negative log-likelihood of a held-out test set using an MoG density estimator. To quantitatively evaluate the density obtained in Step 3, we evaluate the negative log-likelihood of Diea under pa (x). As demonstrated in Figure 4, in settings with A >> O, the generated samples are too close to the training set, meaning that all test samples will have high negative log-likelihood (as they are far from the center of Gaussians with low variances). Evaluation of the test set provides a succinct way of measuring the generalization performance of our generative model, which is key aspect that is lost in metrics such as IS and FID. Our final FLD score is thus given by the following expression:
FLD(Diest Dgen) = 100 d
log Pa (Dues (Dgen) C,
where d is the dimension of the feature space yo and is equivalent looking the th root of the likelihood, and C' is a dataset dependant constant, As a result of this adjustment by a constant, FLD is essentially estimating the forward Kullback-Leibler (KL) divergence (up constant factor) between the learned MoG distributions of the true data and the generated data. Higher FLD score values are indicative of problems in some of the three areas evaluated by FLD. Poor sample fidelity
We set C' for the FLD to be zero with an ideal generator (i.e. a generator that could sample images from the true data distribution). In practice we can estimate such a C by splitting the train set in 2. one part to represent the perfect gen and the other compute ö. Note that C does not change the relative score between models.
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leads to Gaussian centers that are far from the test set and thus higher NLL. Similarly, a failure to sufficiently cover the data manifold will lead some test samples yielding very high NLL. Finally, overfitting to the training set will yield a MoG density estimator that overfits and yield a poor NLLI value on the test set.
Computational Complexity of FLD. To quantify the computational complexity of FLD we plot, in Figure 12 in $A.2, the computation time of various metrics with the number of samples. As depicted, we observe linear scaling in the number of train samples for FLD which is in line with the computational cost of popular metrics like FID. Finally, the cost of computing FLD and other metrics- is dwarfed by the cost of generating samples and then mapping them to an appropriate feature space which is one-time cost and can be done prior to any metric computation.
3.2 Detecting Memorization and Overfitting
Memorization. One of the key advantages of FLD over other metrics like FID is that it can be used to precisely characterize memorization at the sample level. Intuitively, memorization occurs when a generated sample x is approximate copy of training sample x! By Proposition 1, such phenomenon encourages the optimization of Eq. 2 select a 03 < 1 to achieve high training NLL. As a result, such samples will assign a disproportionately high likelihood to that x!!i.. To quantify this phenomenon, we compute the train likelihood assigned by each fitted Gaussian.
Definition 3.1. Let 8 > 0. The sample z
gen is said to be 6-memorized if
Oj max mriin
(1 20n oj 2.21) > 8.
The quantity Oj is appealing because it is efficient to compute, and the distribution of {O₂} allows us to quantify what a "large" value for ō and identify the generated samples that are the most-likely copies of the training set. In $..1.3, we explore this method to assess sample novelty.
Overfitting, Intuitively, a generative model is overfitting when it is more likely to generate samples closer to the train set than to the (unseen) test set. Thus, overfitting for deep generative models can be precisely defined using the standard tools for likelihood estimation.
[x8"") Fi=1 from a generative model G trained on Dirai and unseen test samples Dies. We say that is overfitting if log pa (Dreal Dgen) < log pa (Dinn (D (Dgen), ie if:
Definition 3.2. Given samples 'D
Generalization Gap FLD == (Dtaiii Dgen) - FLD Dgen) <0
(5)
For FLD, this effect is particularly noticeable due to the MoG being fit to the training set. In fact, samples that are too close to the train set relative to the test set have two effects: they worsen the density estimation of the MoG (increasing FLD(Drest Pest, D gen) and assign higher likelihood to the train set (lowering (LD(D p.. In section 1.3 and Tab. I experiments we empirically validate the overfitting behavior of popular generative models using our above definition and also visualize samples that are most overfit.
Evaluating individual sample fidelity. While FLD focuses on estimating the density learned by the generative model, is also possible to estimate the density of the data and use that to evaluate the likelihood of the generated samples". In particular, instead of centering the Gaussians at the generated samples, we can instead center them at the test set. Then, after fitting this MoG to the train set, we compute the likelihood it assigns to generated samples and use that as measure of sample quality. More formally:
2j : = |/p (Pres)) E).
(6)
As is the case for O,, 2, is easy to compute once the Mog is fit and can be used to rank and potentially filter out poor fidelity samples.
4 Experiments
We investigate the application of FLD on generative models that span broad category of model families, including popular GAN and diffusion models. For datasets, we evaluate a variety of popular
'This is somewhat analoguous to the difference between Recall and Precision [Sajjadi et al., 2018, Kynkään- niemi et al., 2019] where FLD can be seen smoother version of these metrics.

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200
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Nade
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Figure 5: Starting from a set of SOTA samples produced by PFGM++, we replace each sample with transformed copy. Left: Effect of nearly imperceptible transformations FLD and FID (with corresponding values for various models as reference). Right: Effect of large transformations on FLD and FID.
natural image benchmarks in CIFAR 10 [Krizhevsky et al., 2014], FFHQ [Karras et al., 2019] and ImageNet [Deng al., 2009] Through our experiments, seek to validate the correlation between FLD and sample fidelity, diversity, and novelty.
Stein et al. [2023] find that the DINOv2 feature space allows for more comprehensive evaluation of generative models (relative to Inception-V33 V33 and correlates better with human judgement. As such, for experiments, unless indicated otherwise,
map samples to the DINOv2 feature space [Oquab et al., 2023]. We do so even for other metrics (e.g. FID, Precision, Recall, etc.) which have typically used other feature spaces (comparisons with vanilla FID provided in Appendix SC).
4.1 Trichotomic evaluation of FLD
We now experimentally validate FLD's ability to evaluate the samples of generative models along the three axes of fidelity, diversity and novelty.
4.1.1 Sample Fidelity
We evaluate the effect of 2 types of transformations FLD and plot the results in Fig. 5, The first consists of minor, almost imperceptible transformations (very minor gaussian blur, posterizing and converting to a high quality JPG) that are problematic for FID. As described in [Parmar et al., 2022] small changes to images such as compression or the wrong form of anti-aliasing increase FID substantially despite yielding essentially indistinguishable samples. These transformations affect FLD but noticeably less than FID. This phenomenon occurs even when we use DINOv2 for both FLD and FID (though the effect is more drastic using the original Inception- V3 feature space, see Appendix $C).
Specifically, when all the samples are transformed, FID rates the imperceptibly transformed samples PFGM++ samples as worse than those produced by StyleGAN-XL (or even worse than StyleGAN2- ada). On the other hand, while the imperceptible transforms yield slightly worse FLD values (in part due to the feature space being sensitive to them), the FLD values are barely changed for the "Posterize" and "Light Blur" transforms and only somewhat worse for "JPG 90" The second type of transformations are larger transformations that affect the structure of the image (e.g. cropping, rotation, etc.) and have a significant negative impact on both FLD and FID.
4.1.2 Sample Diversity
We consider two experiments on CIFARIO evaluate the effect of mode coverage and diversity on FLD. For the first, we vary the number of classes included in the set of generated samples for various conditional generative models. For the second, instead the original n generated samples, we look at set comprised of samples from the original set combined with (k: 1) approximate copies of each those samples (i.e. for total of n samples). In both cases, we find a strong and consistent relationship (across all models) indicating that both sample diversity and mode coverage are important to get good FLD values and report our findings in Fig. 6.
4.1.3 Sample Novelty
We now study the effect of data copying on CIFAR10 on evaluation metrics for generative models. As overfitting can be more subtle than direct copying of the training set, also consider transformed
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Figure 6: Left: FLD for various models as we increase the number classes included in the generated samples. Right: FLD as the set of generated samples includes increasing amounts of copies of itself.
100.00
Comer Cre
JPG.25
Conter Chali Trucsfer
Overti lie
-23
capled
Figure 7: Comparison of various overfitting metrics when Doe consists combination of transformed generated samples and transformed copies of the train set.
copies of the train set. For each transform in Fig. 7, we start with PFGM++ samples that have had the transform applied and gradually replace them with transformed copies of the training set. As such, sample fidelity/dieersity in this "pseudo-generater" set remains roughly constant while overfitting increases.
We then evaluate this set of samples using variety of metrics designed to detect overfitting. For FLD, we look at the generalization gap FLD FLD gen) FLD D pen))). For FID, we consider the difference FID FID rest (where FID is FID using the test set). We use the same amount of samples (10k) for both as FID is biased. Cr [Meehan et al., 2020] is the result of a Mann- Whitney test on the distribution of distances between generated and train samples compared to the distribution of distances between train samples and test samples (negative implies overfit, positive implies underfit). The AuthPet j0, 100) is derived from authenticity described in [Alaa et al., 2022] and is simply the percentage of generated samples deemed authentic by their metric.
We find that FL.D is the only metric consistently capable of detecting overfitting for all transforms (though the effect is less pronounced for large transforms). The generalization gap of FID oscillates though generally trends in the right direction. However, the magnitude of the gap is small (considering that the FID values using DINOv2 are considerably larger, e.g. the difference between StyleGAN2- ADA and StyleGAN-XL. is >60). AuthPet trends in the right direction but seems to only be able to detect a subset of the memorized samples (as it detects none of the sample sets as overfit). Finally, C is capable of detecting overfitting for all but the largest transforms (though it does require significant proportion of copied samples).
4.1.4 Interaction effects
We now examine the relative effect of these three axes of evaluation on FLD. For fidelity, we use an increasing blur intensity. For diversity, we take subset of the generated samples and replace the rest of the sample with copies of this subset. For novelty, we replace generated samples with copies the training set. Then, in Fig. 8, starting from high quality generated samples, we vary each combination of the axes and observe the effect on FLD through a heatmap.
In summary, Fig. 8 indicates that for FLD. fidelity matters more than diversity which matters more than novelty (with the notable exception of all samples being copied resulting in very high FLD). We argue this ordering is very much aligned with the potential usefulness of generative model. If samples have poor fidelity, then regardless of their diversity and novelty, they will not useful. With poor diversity but good fidelity and novelty, removing duplicates yiclds useful generative model.
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25
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Figure 8: Heatmap of FLD values for all pairs of the three axes of generalization.
Finally, if the model generates copies of the training set, it can still be useful as long it generates novel samples as well.
4.2 Comparison of Evaluation of State-of-the-Art Models
Using samples provided by [Stein al., 2023], perform large-scale evaluation of various generative models Tab. using different metrics on CIFARIO, FFHQ and ImageNet. We find that FLD yields a similar model ranking as FID with some exceptions (for example MHGAN vs BigGAN-Deep on CIFAR 10). In addition, we observe that modern generative models are overfitting in benign way CIFAR, i.e., there is noticeable gap between train and test FLD even though the test FLD reasonably low (indicating good generalization).
Dataset
Model
FLD
FID
Gen Gap FLD
CT
AuthPet
Precision
Recall
ACGAN-Mod
24.22
1143.07
0.10
26.11
72.09
0.76
0.00
LOGAN
18.94
753.34
0.18
55.66
84.10
0.63
0.13
CIFARIO
BigGAN-Deep
9,28
203.90
-0.05
55.70
88.10
0.50
0.26
MHGAN
8,84
231.38
47.87
86.69
0.55
0.26
StyleGAN2-ades
6,86
178.64
-0.09
45.31
86.40
0.54
0.33
DODPM-DDIL
5.63
128.57
-0.33
39.65
84.60
0.57
0.55
StyleGAN-XI.
5.58
109.42
-0.23
36.79
85.29
0.57
0.13
PFGMPP
4.58
80.47
-0.35
32.79
83.54
0.60
0.62
Efficient- VDVAE
9.40
465.34
-0.10
32.22
86.48
0.66
0.07
Projected-GAN
8.61
339.72
36.77
93.46
0.11
StyleGAN2-ADA
7.24
296.93
-0.08
27.47
90.95
0.49
0.06
FFHQ256
Unlcashing
7.23
287.38
-0.06
38.98
90.01
0.57
0.19
InsGen
6.20
249.91
-0.08
26.06
89.74
0.52
0.12
StyleGAN-XL
5.94
155.88
-0.04
39.41
88.17
0.56
0.30
StyleSwin
5.77
200.80
-0.29
32.19
87.68
0.60
0.20
StyleNAT
4.69
156.38
-0.12
29.38
86.35
0.62
0.30
LDM
4.63
162.45
-0.23
28.28
84.84
0.66
0.34
RQ- Trannffooreer
11.55
212.99
-0.33
125,48
86.10
0.39
0.55
ImageNer255
StyleGAN-XI.
8.46
150.27
-0.40
98.69
84.10
0.43
0.26
GigaGAN
8.34
156.40
-0.42
98.78
82.48
0.47
0.32
Mask-GIT
6.74
144.23
78.97
80.02
0.44
LDM
3.41
82.42
-0.74
33.63
69.23
0.63
0.45
DiT-XL-2
1.98
62.42
-0.99
22.57
65.79
0.69
0.55
Table 1: Summary of performance metrics for various generative models.
4.3 Applications of FLD
Identifying memorized samples. As discussed in 3.2, we sort the samples generated by various models trained on CIFAR 10 according to their respective O, and plot samples for different percentiles. As demonstrated Fig. all models produce neg-negl amount of very near copies (especially specific car of which there are multiple copies in the train set). For PFGM+ and StyleGAN-Xi. near copies exist even at the percentile of O, (roughly in line with the findings of [Carlini et al., 2023, Stein et al., 2023]).

---

Top
Top
Top
PRONPE
StyloCAN-XL
MHCAAN SkylaCAN2-aak
Figure 9: Ranked generated samples for different models from CIFAR10 according O3. The left sample
generated, the right one the nearest sample in the train set (using distances in DINOv2 feature space).
Best
Top 10%
Top 20%
Top 30%
Top 40%
Median
Bol, 400%
Bot 30%
Bot 20%
Bot 10%
Worsi
IDOPM SigleOLN- PHOMP?
MEIGAN
ACDAAN-e LOCAN
Figure 10: Ranked generated samples for different models from CIFAR10 according to Qj.
Evaluating individual sample fidelity. We repeat the process looking instead at the 2, of the models. From Fig. 10, the progression in fidelity of generative models is quite striking with older models such as LOGAN and ACGAN-Mou producing implausible images for all but their best samples. For recent. SOTA generative models, the top half of samples in terms of Q, generally of high quality. However, the bottom half has many examples of poor quality samples that would easily be identified by most humans being fake. Consequently, even if the ranking is not perfect, filtering generated samples using 2, could potentially be beneficial for downstream applications.
Conclusion
We introduce FLD, a new holistic evaluation metric for deep generative models. FLD is easy to compute, broadly applicable all generative models, and evaluates generation quality, diversity, and generalization. Moreover, we show that, unlike previous approaches, FLD provides more explainable insights into the overfitting and memorization behavior trained generative models. We empirically demonstrate both on synthetic and real-world datasets that FLD can diagnose important failure modes such as memorization//verfittn models that generate photo-realistic images. While we focused on the domain of natural images, a fertile direction for future work is to extend FLD to other data modalities such as text, audio, or time series and also evaluate conditional generative models.
informing practitioners on the potential limitations of generative
10

---

Acknowledgements
The authors acknowledge the material support of NVIDIA in the form of computational resources. AJB was generously supported by an IVADO Ph.D. fellowship. Finally, the authors would like to thank Quentin Bertrand, David Dobre, and Alexia Salicoeur-Martnna for helpful feedback on drafts of this work.

---

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14

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A Additional Results
A.1 Number of samples
From Fig. 11, FLD appears to be reliable even when using less reference set samples (similarly to KID Binkowski al. [2018]). This is of particular importance for applications low data regimes or even conditional generation (c.g. the test set of CIFARIO 10 only contains 1K images for each class), On the other hand, FID highly recommends a minimum of 10k Heusel et al. [2017]. Generally however, the whole training set is used as the value is still biased at 10k and noticeably lower score can be obtained by using more samples.
As for generated samples, for FID, most papers use 50k as gives lower FID than with 10k. We find that using 10k samples for FLD is sufficient for robust evaluation. This is benefit is particularly important given the computational complexity of sampling from modern diffusion models (for example, generating 50k samples often takes hours).
600
0
330
025
020
0.15
2000
50.0
10000
2000
4030
5:00
10000
2000
0000
1000
1050
15000
Size
5iee
Figure 11: FLD, FID and KID for various sizes of the test set (here, for FID/KID, the test as reference set). Both FLD and KID appear to be relatively consistent, even for smaller test set sizes, while FID is clearly biased.
A.2 Computational Complexity of FLD
We include the plot of computation time of our score relative to other metrics and find that while it scales linearly with the number of train samples at a faster rate than many metrics, it is still roughly within the same order of magnitude to other metrics and insignificant compared to sample generation time.
FID
12.5
Erediis Wor
7.5
2.5
20000
40000
60000
80000
100000
Numbet of traising sumplos
vary the number of train samples. We solely include the time to compute the metric value once the samples have been generated and mapped to the appropriate feature space (as these steps necessary for each metric and significantly more computationally expensive than the metric computation).
Figure 12: Time taken (on 1xRTX8000) for different metries
A.3 Synthetic Experiments
We include synthetic 2D experiment analyzing how FLD/FID varies the generated samples go from underfit to overfit. Specifically, we generate 3000 points from the Two Moons dataset using scikit-learn [Pedregosa et al., 2011] with noise value of 0.1. The first 2000 points are used as
15

---

the training set and the last 1000 as test set. We fit KDE to the train set using bandwidth values varying from 10-4 to 10 and sample 1000 points as generated samples before computing our score. As expected, FLD decreases initially as the samples begin to match the distribution. However, unlike Test FID, as the samples begin getting too close the train samples, FL.D increases, punishing the model for overfitting.
291
125
12
390
-30
-55
10"
16'
10
20"
1/ generator KDC (from underfit to overfit)
Figure 13: FLD vs. FID for samples from a KDE centered at the train points. As we decrease the bandwidth of the KDE, the generated samples go from A) (underfit e.g. low fidelity resulting in high FID/FLD) to B) (just right, the samples match the distribution but are not overfit to the train samples, yielding low FID/FLD) C) (overfit, the samples are almost identical the training samples, yielding low FID but high FL.D).
16

---

B Additional Sample Evaluations
Below, we provide additional results of sample evaluations for FFHQ and ImageNet using both O and jj.
Top
Top
Top
DIXLE
LDM
Mask Gft
StyleCaN GigaGANN
BQ
Figure 14: Ranked generated samples for different models ImageNet according O,. The left sample generated, the right one the nearest sample the train set (using distances in DINOv2 feature space).
Best
Top 10%
Top 20%
Top 30%
Top 40%
Median
Bot 40%
Bot 30%
Bot 20%
Bot 10%
Worst
LDM
StyleNii
StyleSwin
ProGAN SQIleGAN2 ADA InoOen
Figure 15: Ranked generated samples for different models on FFHQ according to Q
17

---

Best
10%
Top 30%
40%
Median
10%
Worst
DIXXL2
LDM
Mask GET
StyleGAN XL GIgSBAN
BQ
Figure 16: Ranked generated samples different models on ImageNet according 2,.
18

---

C
Inception Results
As per the recommendation of Stein al. [2023], we have updated our experiments use the DINOv2 feature space instead of Inception- V3. We include below some of the same experimental results with all metries using the Inception- V3 feature space.
160
N0ow
Lipht meve
Postertie
Nerie
Hiter
Raserrae
JG
Color Dister
Cinter Crop
Elstic thosiio Bur
Color Didert
Bldiie Thoasfon Rear Hour
Conter Crop
Figure 17: Starting from a set of SOTA samples produced by PFGM++ we replace each sample with transformed copy. Left: Effect of nearly imperceptible transformations on FLD and FID (with corresponding values for various models as reference). Right: Effect of large transformations on FLD and FID.
Sez
20030
ComarCop
JNG
Chacc Trcenfrn
Ocetie lie
Groeestnette FID
20
80
-10
-40
1.0
S.ogleedccog
Figure 18: Comparison of various overfitting metries when Dpm consists of combination of transformed generated samples and transformed copies of the train set.
Dataset
Model
FLD
FID
Gen Gap FLD
Cr
AuthPet
Precision
Recall
ACGAN-Mod
9.85
35.45
0.31
48.38
94.50
0.49
0.14
LOGAN
6.07
17.90
0.32
42.03
92.72
0.53
0.52
CIFARIO
MHGAN
1.63
4.23
0.20
8.89
89.39
0.64
0.52
BigGAN-Deep
1.53
3.88
0.14
5.48
87.35
0.65
0.51
StyleGANN-ada
1.13
2.52
-0.02
6.54
87.71
0.64
0.55
iDDPM-DDIM
1.03
3.30
-0.03
4.00
87.31
0.68
0.58
StyleGAN-XL
0.85
1.88
0.13
1.80
86.30
0.67
0.41
PFGMPP
0.79
1.81
0.06
4.19
87.09
0.68
0.61
Efficient- VDVAE
3.73
34.70
-0.82
2.75
85.75
0.73
0.10
Unleashing
1.62
8.92
-0.51
7.31
87.94
0.66
0.46
Projected-GAN
1.08
4.27
-0.59
3,47
88.14
0.65
0.47
FFHQ256
LDM
1.00
8.00
-0.60
5.07
87.95
0.70
0.40
StyleGAN2-ADA
5.27
-0.69
-1.16
86.26
0.69
0.35
StyleGAN-XI.
0.92
2.19
-0.56
4.42
88.55
0.66
0.51
StyleNAT
0.86
2.04
-0.80
3.10
87.57
0.67
0.54
InsGen
0.75
3.37
-0.53
0.70
85.83
0.68
0.45
StyleSwin
0.67
2.85
-0.50
3.46
87.62
0.69
0.47
ImageNer256
RO-Transformer
2.79
9.41
-1.18
8.26
81.78
0.63
0.56
GigaGAN
0.04
3.83
-1.48
-9.10
72.48
0.75
0.45
StyleGAN-X XL
-0.04
2.51
-1.51
-3,52
72.47
0.72
0.45
Mask-GIT
-0.34
5.36
-1.95
-9.13
68.31
0.78
0,46
DiT-XL-2
-0.50
2.53
-1.94
-10.79
66.00
0.76
0.56
LDM
-0.92
3.95
-1.95
-21.52
63.85
0.80
0.47
Table 2: Summary of performance metrics for various generative models using Inception-V3s V3 features. Interestingly, many models obtain better NLL than the train set on ImageNet. We hypothesize that this is due Inception- V3 being trained ImageNet, resulting in noticeably different activations for the train set.
19

---

D Method Details
D.1 Algorithm for MoG
Algorithm Fitting MoGs for FLD
Inputs: Duiin agen S, a
Train. Fit MoG on Dusin with 1 gD(gon)
1: o²² ==
/// Initialize all bandwidths of 1]
2: d(p((anm)) (DDn))
(// Pre-compute distance matrix. )
3: while 0² not converged do
4:
L= - log (D
5:
x² f- 0² o2
(W/ Gradient descent on NLL
6: end while
return o
(// Return Trained MoG)
D.2 Optimization
Even with the simplifying assumption of diagonal covariance matrix, to solve Eq. 2 we still need to learn one parameter per Gaussian (of which there are 10000). As the likelihood of train sample is the result of the log of the sum of individual Gaussians, the σj cannot be optimized independently. Thus, we turn to full batch gradient descent using the Adam optimizer with the following hyperparameters:
10000 generated samples
50 epochs
Ir = 0.5
Initial value for the variance vector: 0
We modified the original version of Eq. to include a per sample term L; intended to provided a minimum amount of likelihood each train sample. In practice, set L. be equivalent the likelihood of the point (p relative to a Gaussian with identity covariance existing at distance (2)mn)))) 112 from pp ). We find this helps the optimization process recover low variances in the edge cases where most of the generated samples are copied. Otherwise, non-copied train samples have an exponentially small likelihood assigned to them.
In addition, getting the likelihood assigned by each Gaussian to each point requires computing the distance between each pair gen xttain) which (n²) and time-consuming. Also, as the distances and dimensions are large (at least for our high-dimensional experiments), the exponentiation and summations were often numerically unstable. To address this, we:
Compute the (nn²) distance matrix once and store it not recompute for each step the optimization procedure.
Optimize the log variances instead of the variances themselves
Convert o7 to xxp(-dlog(a)) to be able to take advantage of a numerically stable logsumexp.
From plotting the loss, the variances almost always converged in short order (usually less than 10 steps). While we have no guarantee that these were global minima, when there were exact copies or close to exact copies, would recover very low log variances, as shown in 1.
20

---

E
Experimental Setup
E.1
Transforms
To assess both quality and overfitting, we use standard torchvision [maintainers and contributors, 2016] image transforms described below:
Posterize: Posterize the image to 5 bits (we found was the lowest we could go while the difference still being mostly imperceptible).
Resize: Resize the image the size accepted by Inception using bicubic interpolation which has been shown to be problematic [Parmar et al., 2022].
Color Distort: Apply the color jitter transform described [maintainers and contributors, 2016] with default parameters.
Elastic transform: Apply the elastic transform described in [maintainers and contributors, 2016] with default parameters.
of 0.5. For the heavy blur we use = 1.4. For the heatmap, the blur intensity corresponds to the value of a
Blur: Gaussian blur with (5, kernel. For the light blur,
Center crop x: Center crop the image to (x, x) and fill the rest with black.
JPG x: Convert image to a JPG of quality x.
E.2
Generated samples
For all experiments, we have updated our paper to use the samples provided by Stein et al. [2023] who have generously open-sourced excellent set of samples from wide variety of SOTA models on various datasets.
21

---

Proof of Proposition 1
We now prove our Proposition from the main paper.
Proposition 1. Let Dij 11(((x="") (x(("ri)) be the distance between generated sample and
train sample. Assume Vi,j : Dis D with ój == min; Dig. Then, for any E {I..... m we
have that 8j = ( where 8² a solution of Eq.
Proof. We begin by showing that of (6f) for ót small enough.
We define the variance vector ā to the maximizer of the following equation
8² E arg max a²
Σ log
(V2To,)d exp (-1le(x; (-1le(x2")))) (pxx !1?
(7)
2
==1
Without loss of generality, we consider the following shifted and rescaled objective function with the same argmax Eq.
[(a) = Σ log
1 Σ
exp
20
Di
(8)
m
iml
Then, for any i = I,..., n1, we have
D
1
D2 -
-D
max oj
exp
20
exp
20,
m max of exp
20
(9)
j=1
and using the fact that log is an increasing function, we have that for any
(-dlogo,
D2 S f(o) S g(o) + n log(m)
g(0)
max
(10)
20²
Now, let o² E arg maxxlo and take E {l...., m.. We now show that there exists a ô such that for 6, < 8, oj = va and appears exactly once in the sum g(o").
Suppose of appears in l terms (WLOG we can assume they are the first terms). We can then break down g as follows,
gx(o")) Σ
dlogoī
D3
+
12
max
-dlogo
ij
(11)
200²
37
20
and define
C = sup
max jz1
dlog a,
400.
(12)
20
We then examine the following three cases:
A = 0:: The sum in g(a*) only consists of j + L.
D3
go(
max 341
-dlogo
20
(13)
which bounded by Eq. 12.
A 1: Eq. 11 maximized with respect to of for of
Dy Vd
Plugging in to Eq. 11
gi(0") -dlog(D1.) - (log(d) + +1) + Σ max -dlogo
d
n
D2
(14)
20
22

---

Specifically, if the only term appears in is the one with D1 = dj
d (i(d") - (log(d) (d) + 1) _ max ollogo
"
D 20,
(15)
where the term -dlog(ot) and thus 1 contrast, if of appears for Du > 5j we get bounded J1 (independently of 61).
be made arbitrarily large 6; 0. In
k > 1: Eq. 11 is maximized with respect to of for of
D ldd
Plugging in to Eq. 11
(V
i== D
D
(x(a))
Σ
-d
(16)
kd
2i= i= kd
D
Σ -==
D2
kd
log
(17)
kid
2
(k: 11)11) kid
kd
< ked log
81
(18)
and is thus bounded independently of ōj.
Thus, we conclude that, for 6 small enough, of = 6r since 01(0²) > (90(0").91(0") (o)"))))
Va
Now, consider such that ō; of pt/v and oj oo otherwise. We now look at g(ō). From the above
g = g00g0(0))))))))r 2 (log(1 - c) c)
(19)
as long as on is found single term of the summation gō5) (i.e. k 1). not, with the same argument as above, g is bounded above by constant U independent of 5, and c. Thus 30 < ő < such that vo; < 6, we have + n log m S g(o"). Thus, in both cases V6, < and for all such that (log(1 c) + c) < -n log m, we have
((ō) < ((ō) logm
(20)
d (log(1 c) c)] + n logm
maxxUU,gor 2
(21)
max((U,g(ft log m] n log m
(22)
$ g(a")
(23)
f(0).
(24)
Since 4 (log(1 c) + c) is decreasing in (for > 0) and ōn is increasing in must have Ō = TvTcc where 1 - exp(-200 2n+1 log(m)). Thus, ô O(ū;) for the case d,
For the case 5, 2 8, first show that o; is bounded (ō: & M). Denote the individual likelihood terms of f(a) as follows
D2
Li((o,) = exp 0
(25)
20g
Then Vi, we have that
La(or) <
+ 0
(26)
04 01-200
Thus, since f() is increasing in the (i)(o), it suffices to show there is 0; S M such that all L (
are positive. Taking 1, we have that Vi La(1) 2 e- 62/2 > 0. Thus we can take M = 23 Hence, for the case d; >
p²
23

---

81
6, -
8
M
8 6/.
Finally 6j is O()) by taking C = max
}.
dy1-6'
5
24

---

G
Feature spaces
Feature spaces play very important role in our metric and similar metries (FID, Precision/Recall. etc.). While those produced by Inception V3 have proved useful, there is work illustrating some potential issues, motivating our switch to DINOv2. proved adequate for our experiments, it is likely that FLD could provide an even better assessment of generated samples with better feature embeddings. Additionally, while this work has solely focused on the evaluation of generative models in the image domain, there is nothing about FLD indicating it could not be applied other domains (using appropriate feature embeddings) and we leave this as fruitful future area of research.
such, while the resulting feature psace has
H Broader Impact
Generative modeling has the potential to create copies of people's identities or artistic styles without their consent, which can have negative impacts individuals and communities. While evaluation metrics can ensure the generalizability of generative models, they should not be employed to justify copying artist's style without their permission. The concept of copyrighting is intricate, and while some may argue its challenges and ambiguities, it is vital to acknowledge the impact on the artist whose style is being reproduced. Respecting the rights of individuals and communities whose data is used in creating generative models is crucial to avoid harm or exploitation.
artistic style
As the field of generative modeling progresses. regarding the ethical implications of these technologies and establish guidelines for their responsible application. Additionally, if evaluation metries fail consider the generation harmful or unethical content, negative societal consequences can emerge. Therefore, it is crucial to consider diverse range of evaluation metrics that encompass various aspects of model performance while aligning with ethical and societal considerations.
imperative to engage ongoing discussions
25


Reviews:
[{"review_id": "jS625hBxrj", "reviewer": "Reviewer_euCm", "summary": "Limitation of existing evaluation metrics\n* Likelihood-based metrics rarely correlated with perceptual fidelity.\n* Sample-based metrics are insensitive to overfitting. E.g., FID\n* Copycat (=a model randomly outputs training set) outperforms SOTA generators in $\\text{FID}_\\text{test}$\n\nProposed evaluation metric: feature likelihood score (FLS)\n1. Map generated / train / test images to an embedding (Inception-v3 or CLIP).\n1. Initialize isotropic Gaussians centered at the generated samples.\n1. Update the variance of above Gaussians to maximize the likelihood of training samples.\n1. Compute the NLL of test samples on the Gaussians.\n\nIt measures\n* Novelty = opposite of Memorization\n  * a generated sample is $\\delta-$ memorized if NLL of the training set is $\\delta-$ lower than the test set on the Gaussian of the generated sample.\n* Diversity\n  * Empirically shown\n* Fidelity\n  * Empirically shown\n\nFLS is high when\n* The generated samples have poor quality -> Gaussian centers are far from the test set -> high test NLL\n* The generated samples do not cover the data manifold -> high test NLL\n* The generated samples overfit to the training set -> high test NLL\n", "questions": "1. Why should the SOTA generators be better than the copycat?\n1. Could you clarify the difference between definition 3.1 and 3.2 other than the margin $\\delta$?\n1. How do diversity, fidelity, and memorization affect the metric? Is there a metric affects more than another?\n\nTo me, Q3 is the most important.", "limitations": "Use cases on non-natural images are not shown (mentioned as future work).\n\n", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* It measures novelty, fidelity, and diversity of generated samples. (not fully explained below)\n* FLS correlates to sample fidelity (actually, corruption) as FID does.\n* It is a one-value metric that reflects three aspects. It is easy to rank different models.\n* The authors provide the benchmark table of SOTA models.\n", "weaknesses": "1. A super-close literature is missing: [rarity score]\n1. Some important definitions are missing. E.g., training/test split, L189 data manifold\n1. FLS correlates to sample fidelity not as linearly as FID\n1. The mechanism how FLS correlates with sample fidelity/diversity is not described (although straightforward).\n1. It is a one-value metric that reflects three aspects. It is difficult to compare different aspects with same value.\n1. FLSs for different k and C are not provided.\n1. Train FLS and test FLS are not defined.\n\n\nminor\n1. Figure 5 middle shows FID but caption describes it differently.\n1. Figure 5 right shows FLS versus Precision but caption describes it FLS versus Recall.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We want to thank Reviewer euCm for their feedback. We are glad that Reviewer euCm highlighted that a strength of FLS is that it “is a one-value metric that reflects three aspects [Fidelity, Diversity, Novelty],” making it “easy to rank different models.” We now address the specific comments points raised by the reviewer:\n\n### How do diversity, fidelity, and memorization affect the metric?\nWe provide a detailed answer to this question in the global response to the reviewers. We also ran additional experiments that are presented in the 1-page pdf attached. \n\nIn summary, Fig.1 indicates that for FLS, fidelity matters more than diversity, which matters more than novelty, with a notable exception when almost all examples are copied where FLS produces among the worst scores. We argue this order is very much aligned with the usefulness of the generative model in a potential downstream task.\n\n### SOTA generators vs. copycat?\nThe “copycat” generative model corresponds to a model that exactly copies the empirical data distribution. Such a model is useless as it cannot generate new samples from the data distribution and is unusable in most downstream tasks—e.g., data augmentation. Conversely, we know that data augmentation using diffusion models provides an improvement in terms of test accuracy [1,2]. A performance metric for generative modeling is useful if it is also a good proxy for the performance of downstream tasks. Hence, SOTA generators should be ranked better than the copycat model.\n\n### Clarifications on Def 3.1 and 3.2\nIn Def. 3.1 we consider the (normal) distribution induced by a **single example** $x_j^{gen}$ and look at the difference between the likelihood of the train set $D_{train}$ and the test set $D_{test}$. \n\n$$ p_{\\hat \\sigma}(x| x^{gen}_j) := \\mathcal{N}( \\varphi(x) | \\varphi(x_j^{\\text{gen}}),\\hat \\sigma_j^2 I_d)$$\nIt corresponds to whether this **individual** point ranks the train set more likely than the test set. (Thus it is a proxy for $x^{gen}_j$ being a memorized point) \n\nIn Def. 3.2 we consider the (MoG) distribution $p_{\\hat \\sigma}$ induced by our sampled generated points $D_{gen}$ (defined in Eq.1). \n\n$$p_{\\hat \\sigma}(x| D_{\\text{gen}}) := \\frac{1}{m} \\sum_{j=1}^m \\mathcal{N}( \\varphi(x) | \\varphi(x_j^{\\text{gen}}),\\hat \\sigma_j^2 I_d)$$\n\nIt corresponds to whether the generated distribution ranks the training set as more likely than the test set—which is a useful measure of detecting the degree of overfitting.\n### Related work: rarity score\n\nWe thank the reviewer for pointing out this recent and relevant reference which we will include in the updated manuscript. Our work complements [3] with a critical difference: FLS evaluates models along the three axes we mention. FLS punishes models that overly memorize, while the rarity score [3] allows models that produce very novel/rare samples to be adequately recognized.\n\n### Definitions of training/test split, data manifold, train/test FLS\n\nIn the definition section, we mention that $D_{train}$ were the samples used to train the generative model, and $D_{test}$ were not used at the training stage. We will clarify this in the revision of the paper. \n\nRegarding “data manifold” L189, we will replace this ill-defined term with “data distribution”. \n\nTrain/test FLS refers to the score associated with the likelihood of the training/test set over our density model $p_{hat \\sigma}$ (See Eq. 1 and 2 in the submission). Formally, \n\n$$FLS_{train} =  \\text{FLS}(D_{\\text{train}},D_{\\text{gen}}) := - \\tfrac{100}{d}\\log p_{\\hat \\sigma}(D_{\\text{train}} | D_{\\text{gen}}) - C,$$\n\n$$FLS_{test} =  \\text{FLS}(D_{\\text{test}},D_{\\text{gen}}) := - \\tfrac{100}{d}\\log p_{\\hat \\sigma}(D_{\\text{test}} | D_{\\text{gen}}) - C,$$\nWe will clarify that point in the revision of the paper. \n\n\n### FLS correlates with sample fidelity not as linearly as FID\nWe thank the reviewer for this critical question. We address it in the global response. \n\n### Mechanism on FLS and its correlates with sample fidelity/diversity\n\nWe describe in L188-191 of the main paper how FLS correlates with sample fidelity and diversity, “Poor sample quality leads to Gaussian centers that are far from the test set and thus a higher NLL. Similarly, a failure to sufficiently cover the data manifold will lead to some test samples yielding very high NLL.” We will clarify the phrasing in the final version.\n\n### Using FLS (a one-value) metric to reflect three aspects\n\nThe field of deep generative modeling has been driven by the use of single metrics for evaluation as it allows ranking models and measuring the progress of the field. For example, FID (fidelity and diversity) has been used to assess the progress on image generation tasks from early GANs such as WGAN-GP to recent diffusion models such as EDM++. As shown in Fig.2 and Tab.1 of our pdf rebuttal, we see the capabilities of these models to memorize and overfit, a facet that is missing current evaluation metrics. That is why we believe FLS can be seen as a holistic score extending the purpose of FID to a third aspect—i.e., novelty.\n\n\n### FLSs for different k and C are not provided\n\nThe $k$s in Fig. 4. are illustrative to show a visual example of density overfitting and are not used to compute FLS. The $C$, on the other hand, is a dataset-dependent constant used to make our score positive (see footnote 2 in main) and does not affect the relative values of FLS between models. \n\nWe appreciate the reviewer's feedback on our paper. We believe we have answered all the great points raised by the reviewer in our rebuttal and kindly request a reconsideration of the paper's score. We are also happy to answer any additional questions.\n\n[1] Wang, Zekai, et al. \"Better diffusion models further improve adversarial training.\" 2023\n\n[2] Azizi, Shekoofeh, et al. \"Synthetic data from diffusion models improves imagenet classification.\" 2023\n\n[3] Han, Jiyeon, et al. \"Rarity score: A new metric to evaluate the uncommonness of synthesized images.\" 2022\n"}, {"review_id": "2rKQb91rUV", "reviewer": "Reviewer_cRFc", "summary": "This paper addresses the problem that there are currently no sample-based evaluation metrics accounting for the trichotomy between sample fidelity, diversity, and novelty. Likelihood based metrics are not particularly interpretable and sample quality based metrics do not take into account novelty -- they are easy to cheat just by copying the training data. To address these issues, the paper proposes the feature likelihood score (FLS) -- which assesses sample novelty, overfitting, and memorization.", "questions": "* The paper should motivate in more detail why the CIFAR10, ImageNet, LSUN and AFHQ datasets were choosen as the evaluation platform.\n\n* Is the FLS score applicable to text to image models? Can the FLS model detect duplicates in case of text to image models?\n\n* Does the FLS show the same behavior on very large datasets such as LAION as on smaller datasets such as CIFAR10 or ImageNet?", "limitations": "The paper does not discuss the limitations of the FLS score in detail. The paper should ideally also failure cases to highlight limitations -- where copied training examples do not degrade the FLS score -- if observed.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The paper deals with an important problem and proposes an innovative solution.\n* The paper provides theoretical guarantees that the proposed FLS score can detect overfitting in Proposition 1.  \n* The shows evaluations on a variety of datasets -- CIFAR10, ImageNet, LSUN and AFHQ and on a variety of models -- StyleGAN-XL, SNGAN, LOGAN, LSGAN etc.\n* The paper is well written and easy to understand.", "weaknesses": "* Can the proposed FLS score be extended to Text to image models such as Stable Diffusion or Latent Diffusion.\n\n* The datasets considered such as CIFAR10, ImageNet, LSUN and AFHQ are quite limited in size compared to popular datasets such as LAION. The paper should include an analysis of the effect of dataset size on the ability of FLS to detect generated copies of the training set.\n\n* The paper should also include an analysis of the effect of image resolution on the FLS score -- as the image resolution increases will the FLS score be able to detect copies of the training set images?\n\n* In Figure 7 (right) why does the FID increase for some models when the % copied samples increases.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time, feedback, and positive appraisal of our work. We are heartened that the reviewer feels that FLS tackles an “important problem and proposes an innovative solution”. We also appreciate that the reviewer finds our evaluation extensive in the number of datasets and models while stating our manuscript is “well written” and “easy to understand”.\n\n### Extending FLS to Text-to-image\n\nThis is a great suggestion! Evaluating multi-modal generative models is an exciting direction for extending FLS. However, multi-modal models present several technical challenges as we need to estimate $P(x|y)$ for unseen $y$ (unrestricted text prompts) of which usually a single $(x,y)$ is available. This is in stark contrast with evaluating class conditional image generative models as we do in Table 2 in our manuscript, as the classes $y$ are known $\\textit{apriori}$. Thus while interesting, we believe extending FLS to multi-modal models like latent diffusion deserves its own paper as it requires non-trivial considerations whose solutions are not an immediate application of the conditional FLS.\n\n### Datasets Considered for Evaluation\n\nWe appreciate the reviewer’s concern regarding the datasets used for evaluation. We believe our selection of datasets is relatively standard for image-generative models and is consistent with the primary ones used in the literature of non-multi-modal generative models [1, 2, 3]. Furthermore, such datasets already have pre-trained models, which enable us to perform a large-scale evaluation on a variety of popular generative models (e.g., GANS, Diffusion). Using pre-trained models on standard datasets also ensures fairness and consistency with models evaluated using other measures in the literature. We understand that this motivation may not have been initially evident in the manuscript, and we will update it to highlight this aspect. As for dataset size, in Figure 10  and Appendix B.3, we demonstrate the applicability of FLS in low-data regimes. The effect of FLS versus dataset size. For larger datasets, while we have experiments for ImageNet, LAION is typically not used for non-multi-modal generation. Nonetheless, we believe the aforementioned non-trivial extension of FLS could be used to evaluate generative models trained on LAION and datasets of similar size.\n\n\n### FLS vs. Image Resolution\n\nWe value the reviewer's comment regarding the ability of FLS to detect copies as the image resolution increases. We believe this is a non-issue primarily because FLS operates on the representation space of a pre-trained encoder (e.g., Inception, CLIP, Dino-V2) with a fixed dimensionality regardless of the original input image dimension. Consequently, computing FLS is unaffected by the input image resolution, which we demonstrate on a range of datasets with small resolutions 32x32 for CIFAR10 to AFHQ, consisting of high-quality images of resolution 512x512. For each dataset, we effectively detect overfitting (see Appendix B.2). We hope this allays the reviewer's concern as we argue FLS is independent of image resolution provided a sufficiently rich representation space.\n\n### Fig 7 \n\nThe increase in FID is due to the copied samples consisting of transformed versions of the trained samples. For sufficiently strong transformations, the decrease in quality has a more substantial effect than overfitting (considered beneficial by FID). Please see our global response for a detailed discussion on how fidelity, diversity, and novelty affect FLS.\n\n### Discussion of Limitations\n\nWe acknowledge the reviewer's comment on adding a more detailed discussion on the limitations of FLS. We agree with the reviewer that such a discussion can add valuable nuance to our score's proper use and interpretation. \n\nWe find that FLS is less impacted if there is a combination of memorization of a subset of training examples and generalization (according to definitions 3.1 and 3.2). This can partially be seen in Figure 7, where a model producing great novel samples 10% of the time and producing copies 90% of the time still gets a reasonnable FLS for a heavy case of memorization. Nonetheless, in such a case, the overfitting is still detected by looking at the gap between the train and test FLS. \nAs discussed in Appendix G, our score is heavily influenced by the embedding we consider. We require good embeddings to assess fidelity, diversity, and novelty. For image domains, many efficient pre-trained models exist (CLIP, DINOv2), but this could be a challenge for other modalities such as audio or time series. \nAs discussed in Appendix F, our score scales reasonably with the dataset size (i.e., has a similar complexity to its competitors, such as FID). However, it may still be challenging to scale it to very large datasets with billions of training examples. \n\nWe will update the paper to highlight such scenarios and include a dedicated limitations section.\n\n\n### Conclusion and references\n\nWe thank the reviewer for their valuable feedback and great questions. We hope that our rebuttal fully addresses all the important salient points raised by the reviewer. We kindly ask the reviewer to consider updating their score if the reviewer is satisfied with our responses. We are also more than happy to answer any further questions that arise. Please do let us know.\n\n[1] Karras, Tero, et al. \"Elucidating the design space of diffusion-based generative models.\" 2022.\n\n[2] Sauer, Axel, Katja Schwarz, and Andreas Geiger. \"Stylegan-xl: Scaling stylegan to large diverse datasets.\" 2022.\n\n[3] Parmar, Gaurav, Richard Zhang, and Jun-Yan Zhu. \"On aliased resizing and surprising subtleties in gan evaluation.\" 2022.\n"}, {"review_id": "lkfya3etzy", "reviewer": "Reviewer_SDnT", "summary": "The paper proposes an approach for evaluating generate images by fitting a mixture of Gaussians (MoG) to feature embeddings extracted from the generated and real images. For this, images are first mapped to a feature space (e.g. via an Inception or CLIP image encoder) and then map a Gaussian distribution to each image feature with the mean being centered at the specific image feature. The variance of each Gaussian is then optimized via NLL such that the MoG fits the training examples or a subset thereof.\nThe MoG can then be used to evaluate image memorization (any Gaussian with near-zero variance will be a memorized sample) and overfitting (if the MoG assigns higher likelihood to the train set than to an unseen test set), as well as sample fidelity and diversity.\nThe experiments show that the new metric is more robust to small transformations in image space than FID and generally correlates well with FID, precision, and recall.", "questions": "I also wonder about the relationship between this approach and precision+recall. Couldn't precision+recall also be used for detecting memorization and overfitting by applying a similar approach as this one here, applied to the distances already calculated by precision+recall? Would that be simpler? Would the results agree with the results of this metric?", "limitations": "My main concern is that the most obvious improvement of the new evaluation method, namely easily detecting memorization and overfitting, may not work as expected since it relies on image embeddings from feature extractors that were not trained for this (neither Inception nor CLIP were trained to detect memorization and instead focus more on high-level features). I would not define image in Fig 8 as memorizations.\nAside from that, other improvements/differences seem marginal compared to existing metrics such as FID and precision+recall and it's not clear to me what exactly makes the new metric better than, e.g., FID.\n\n---\n\nI have adjusted my score to WA based on the author's rebuttal.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The approach aims to improve upon existing and well-used metrics such as FID and precision+recall. As such, it is easy to calculate, needs minimal human intervention, and scales to large numbers of images.\nIt also aims to improve the evaluation by detecting overfitting/memorization which many current metrics can't detect. Finally, it can provide more detailed insights into models by evaluating the quality of specific classes individually. The approach also seems to be more robust to using smaller sample sizes (e.g., <5K samples), whereas other metrics such as FID need tens ouf thousands of samples.", "weaknesses": "While the approach promises some improvements over other metrics the main improvement seems to be being able to detect overfitting and memorization. However, I am not convinced that feature embeddings are the best way to detect memorization. E.g., looking at the memorized results in Fig 8, those results don't look like memorization to me. Were these obtained based on Inception or CLIP embeddings?\n\nMore detailed insights into the performance of specific classes is also already possible with other metrics, e.g., conditional FID. \n\nAside from that the main other advantage seems to be that the approach is more sample efficient but that is a minor difference since calculating FID and other metrics is usually not the computationally intensive part of model training and evaluation.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and feedback on our manuscript. We are happy to hear that the reviewer views FLS as “easy to calculate” and that it can scale to a “large number of images”. We also are pleased by the reviewer stating that FLS aims to evaluate the overfitting/memorization behavior of generative models, a fact that “many current metrics can’t detect”. Finally, we thank the reviewer for acknowledging that FLS is “more robust” to smaller sample sizes compared to FID.  We now address the main comments and questions raised by the reviewer.\n### Detecting memorization using FLS\n> I am not convinced that feature embeddings are the best way to detect memorization […]\n\nWe acknowledge the reviewer's healthy skepticism regarding the memorized examples shown in Fig. 8. Our current Fig 8 was obtained using Inception features. We agree that these features may not be ideal and many works have indeed revealed their limitations [1, 2, 3]. While raw L2 distances have shown some promise at detecting memorized examples [4] they are also unreasonably sensitive to imperceptible transformation [5].\n\nWe investigate the effect of using a more modern feature extractor in DINO-V2 [3,6] combined with a slightly modified ranking. These results can be found in our 1-page rebuttal PDF in Fig 2. We believe the presented samples are undeniable evidence of copies. We provide more details about this additional experiment in the global response. Specifically, many of these copies (e.g. the car that appears many times in the CIFAR10 training set) exhibit a slightly modified scale/lighting or background shading than the training samples which would lead to a higher pixel-wise distance than their distance in a sufficiently strong feature space. Given this compelling evidence, we believe FLS using DINO-V2 as a feature space is capable of effectively detecting copies.\n### Using precision+recall to detect memorization\n> Couldn't precision+recall also be used for detecting memorization […] Would the results agree with the results of this metric?\n\nThat is an excellent question. FLS bears similarities to a continuous/smooth version of recall.  Unfortunately, directly applying recall as in [8] to the test set (i.e. using the generated samples to get nearest-neighbor (NN) distances) yields the same problem as FID: a model that generates exact copies outperforms SOTA models in recall (0.74 vs 0.70 for EDM).\n\nIf we understand correctly, the reviewer is instead referring to using the train samples to get NN distances (similarly to FLS) for the manifold of generated samples.\n\nWhile this form of recall would be simpler to compute, it presents a key issue addressed by FLS. First, fitting on the train set unfairly rewards bad models: poor quality samples are far from the train set resulting in the construction of balls with larger radii and the engulfing of more test samples. As a result, many models obtain very similar recalls for $k=1$:\n- EDM: 0.41\n- StyleGAN2: 0.4\n- BigGAN-CR: 0.4 \n- SNGAN: 0.39\n\nA similar phenomenon occurs for $k > 1$, for example at $k=4$, used in [6]. The binary nature of the recall metric (a sample is either inside or outside the ball) is problematic here. Contrastingly, the likelihood assigned by the MoG in FLS smoothly takes into account the distance of that sample to the MoG. As for precision, it is not immediately clear to us how it could be used to detect overfitting but we are happy to discuss and test other suggestions.\n### Other Questions \n> More detailed insights into the performance of specific classes is also already possible with other metrics, e.g., conditional FID […] not the computationally intensive part of model training and evaluation.\n\nIn certain situations (e.g. small test set for class conditional evaluation), the main bottleneck is not compute, but rather dataset set size. FID is biased and thus requires a large number of samples. For instance, for CIFAR10, practitioners usually consider the FID between 50k generated samples and the training set. Borji [7] mentions that ``A major drawback with FID is its high bias. The sample size to calculate FID has to be large enough (usually above 50K)’’. This is particularly problematic for conditional FID (e.g. CIFAR10 has 5k train + 1k test samples per class while ImageNet has ~1k train + ~150 test), conditional FID would yield a highly biased estimate of the full FID. \n### Main improvement\nWe believe that Fig. 2 in the 1-page PDF which uses the amended memorization metric combined with the DINOv2 feature space convincingly demonstrates the ability to detect memorized samples. In addition, even in Inception space, we provide solid evidence that FLS is capable of detecting overfitting. Fig. 7 shows that FLS is highly affected by the addition of copies and these copies are detected by the percent of overfit gaussians. In App B.1, we show it does so better **even when compared to metrics designed specifically to detect overfitting**.\n\nWe thank the reviewer for their time and effort in reviewing our work and we hope the reviewer would kindly consider a fresh evaluation of our work given the main clarifying points outlined above.\n\n[1] Zhou, Sharon, et al. \"Hype: A benchmark for human eye perceptual evaluation of generative models.\" 2019.\n\n[2] Kynkäänniemi, Tuomas, et al. \"The Role of ImageNet Classes in Fr\\'echet Inception Distance.\" 2022.\n\n[3] Stein, George, et al. \"Exposing flaws of generative model evaluation metrics and their unfair treatment of diffusion models.\" 2023.\n\n[4] Carlini, Nicholas, et al. \"Extracting training data from diffusion models.\" 2023.\n\n[5] Schäfer,F, et al \"Implicit competitive regularization in GANs.\" 2019.\n\n[6] Oquab, Maxime, et al. \"Dinov2: Learning robust visual features without supervision.\" 2023.\n\n[7] Borji, Ali. \"Pros and cons of GAN evaluation measures: New developments.\" 2022\n\n[8] Kynkäänniemi, Tuomas, et al. \"Improved precision and recall metric for assessing generative models.\" 2019."}, {"review_id": "VLrPD9ycHm", "reviewer": "Reviewer_HK8L", "summary": "The work proposes a new metric (FLS) for image generation tasks that is motivated by the observation that current metrics consider the quality and the diversity of the samples, but not their novelty, and therefore are not penalized by memorization of the training set.\nThe proposed method instead encompasses all three aspects in a single evaluation.\nTo compute the score, they extract features from the generated, training, and test images. They then build a mixture of isotropic Gaussians centered around the embeddings of the generated samples. The variances are optimized to maximize the log-likelihood of samples of the training set. FLS is then the likelihood of the test samples.\n\nThe properties of the metrics are demonstrated by evaluating a number of well-established GAN models.", "questions": "My main concerns are, for the sake of practical usage, about the assessment of the minimum number of samples needed:\n- Can the authors provide a reliability assessment for overfitting detection in few-sample settings?\n- Can the authors update Fig.10 with KID and standard deviations?\n- Accordingly, can the authors provide recommendations in terms of the minimal number of samples for reliable assessment?\n\nRelated to Fig5 and the related discussion in Section 4.1:\n- the authors mention that “FLS is noticeably less affected”, but it is not that clear since we don't have a scale on which to compare the two metrics. Can the authors explain more explicitly how they derive this observation?\n\nAs an additional question, one aspect that can be useful in some cases is adversarial perturbation detection.\n- How do the authors anticipate the metric to behave when faced with adversarially perturbated images? Considering it is supposed to be more robust to noise, it might not be affected by such perturbation. This can be a good or bad thing depending on the task to be evaluated, but useful to know regardless. \n\nAs a final comment, because the paper is proposing a holistic metric, I believe it might be important for the paper to include, in a separate section, a discussion summarizing good practices and recommendations about when and how to use it.", "limitations": "Limitations are adequately discussed in the paper and broader impacts in the supplementary.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is well-motivated. While memorization was not much of an issue for generative models so far, some recent study indicates that this might be the case now with the introduction of new powerful generative models. A holistic evaluation including novelty would be a useful tool going forward.\n- The principles behind FLS are well-grounded in already established methods (FID, Precision/Recall) and the changes are sound and motivated.\n- FLS is not biased by the number of samples and can therefore be compute on a smaller number of points than FID.\n- Experiments are extensive, covering many common use-cases including truncation trick, and show promising results. In particular, FLS correlates well with other metrics when memorization is not an issue and the overfitting detection experiments in the main paper and the supplementary are convincing enough.", "weaknesses": "A very important aspect is the reliability of FLS w.r.t the number of evaluation samples. As with FID, in practice, people will use the metric in diverse settings and it is important to know when the metric is reliable and when it isn't. The paper touches upon these questions in Fig. 10 but could be more complete. In particular:\n- Since the models we compare display very little overfitting, Fig 10 only evaluates the robustness of FLS on the Quality and Diversity aspects, not on the Novelty aspect. As novelty evaluation is one central selling point of the metric, information about the reliability of the overfitting detection aspects is necessary.\n- For the Quality and Diversity aspects, FLS should also be compared to KID, which tackles this same issue of sample size bias.  When evaluating with a smaller number of samples, standard deviation should also be provided to better assess the test size that would be needed in practice. Some intervals are depicted in Fig10 but they are barely visible and not commented on at all.\n\nFLS might be harder to interpret intuitively than FID. There is a chance that it could lead to misuse or misunderstandings down the road.\n- In the results presented Fig5, FLS is less linear than FID w.r.t. the % of perturbed data. I would guess this effect is caused by the fact that the variance is adaptive, leading to complex interactions between the samples. This could make the interpretation of the values even less intuitive, as we are more used to linear scales.\n- In the same figure, until larger percentages of perturbed data, random rotation, elastic transforms, center crops and bicubic resizes all seem to yield the same FLS, while FID assigns different values to different perturbations. As far as I'm concerned, FID results seem more aligned with my intuitions than FLS.\n\nOther weaknesses:\n- FLS might be unsuited to easier tasks. Most well-performing models are nearly indistinguishable by FLS in Table 2 (conditional cifar10).\n- The computation of FLS has more steps than FID, making it more cumbersome to use. It is also more compute-demanding when scaling.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time and detailed review of our work. We are glad that the reviewer finds our paper “well-motivated”, “well-grounded”, and that the holistic evaluation of generative models as we do by introducing FLS is a “useful tool going forward”. We also appreciate that the reviewer thinks that our empirical results are supported by “extensive” experiments and that the overfitting detecting experiments are “convincing enough”. We now address the key clarification points grouped by theme.\n\n### Effect of dataset size on FLS (and comparison with KID)\n\n>For the Quality and Diversity aspects, FLS should also be compared to KID, which tackles this same issue of sample size bias. \n\nWe have run the experiment and will include an updated version of Fig.10 with KID in the final version of the paper and a complete description of the figure (currently, the confidence intervals are 95% confidence intervals of the bootstrapped distribution). They are very small, essentially invisible for FID. The KID results can be summarized as follows:\n- KID exhibits little to no bias at all dataset sizes but more variance than FLS\n- FLS exhibits some bias at very small dataset sizes (<500) and more variance than FID.\nKID correctly addresses the sample size bias issue of FID but still has the same problem of FID when it comes to detecting overfitting.\n\n### FLS might be harder to interpret intuitively than FID. \n\n> In [...] Fig5, FLS is less linear than FID w.r.t. the % of perturbed data.\n\nWe agree that this is an important point and believe it a non-linear relationship makes sense given the x-axis of the figure. Specifically, a model that produces perfect samples 10% of the time and poor samples 90% of the time has learned to generalize significantly better and is more useful than one that produces poor samples 100% of the time. This argument is detailed further in the global response.\n### FLS might be unsuited to easier tasks. \n> Most well-performing models are nearly indistinguishable by FLS in Table 2 (conditional cifar10).\n\nWe thank the reviewer for pointing out this mistake on our end. The Table 2 in the paper contains unnormalized NLLs (i.e. before subtracting C and multiplying 100). When fixed, the table contains differences in value of similar scale to Table 1. We will fix this small error in the final paper.\n\n### FLS is more expensive computationally \n\n> The computation of FLS has more steps than FID, [...] It is also more compute-demanding when scaling.\n\nIn practice, we find that FLS requires slightly more time to compute than FID at large dataset sizes (as shown in a Appendix F) with computation time scaling linearly with the number of samples. Nonetheless, we believe this is a non-issue as the main bottlenecks are mapping samples to the chosen feature space (~10x slower than computing FLS) and producing samples (up to 100x slower for certain diffusion models).\n\nAs for ease of use, all code will be available on GitHub with an easy to use interface for practitioners (provide a folder of images, tensor of features or function that produces samples and the FLS is returned).\n\n> Can the authors provide a reliability assessment for overfitting detection in few-sample settings?\n\nWe agree with the reviewer that the few-sample setting is an important area for overfitting detection. While we do not currently have a full reliability assessment, in Appendix B.3. we evaluate StyleGAN2-ADA models trained on datasets of varying sizes (500-4000). There, we find that the % of overfit Gaussians identifies overfitting not detected by other methods.\n\n> Fig5 and the related discussion in Section 4.1: the authors mention that “FLS is noticeably less affected”,[...] Can the authors explain more explicitly how they derive this observation?\n\nFig. 4 of the rebuttal provides a clearer figure illustrating this behavior. For the transforms which have very little visual effect on humans, the effect on FLS is relatively small whereas for FID, it leads to a significant increase where EDM samples are rated similarly to those produced by SNGAN. \n\n> How do the authors anticipate the metric to behave when faced with adversarially perturbated images?\n\nThat is an interesting idea! While we have not tested with any adversarially perturbed image, in light of the above (i.e. the effect of imperceptible transforms), they might be less affected by these perturbations than FID. However, FLS is dependent on the feature embeddings: adversarial perturbations with respect to the Inception-v3 will have a larger effect. This could be tackled by using the features of robust models (e.g. [1]).\n\n> include, [...] a discussion summarizing good practices and recommendations about when and how to use it.\nThank you for the great suggestion. We will add the following list of good practices to the paper.\n- Keep a separate test set for evaluation. Many generative modeling datasets do not have an explicit test set (FFHQ, LSUN-Bedroom, or LAION [2]).\n  - Recommended to use **a minimum of 1k test samples**, preferably 10k.\n  - Recommended to use **a minimum of 10k generated samples** preferably 50k. \n- Look at FLS but also the generalization gap (difference between Train/Test FLS) and the percentage of overfit Gaussians to identify potentially problematic overfitting.\n- Visually inspect the maximally overfit samples as a means to qualitatively evaluate overfitting behavior.\n\nWe thank the reviewer for their valuable feedback and great questions. We hope that our rebuttal fully addresses all the important salient points raised by the reviewer and we kindly ask the reviewer to potentially upgrade their score if the reviewer is satisfied with our responses. We are also more than happy to answer any further questions that arise. \n\n[1] Wang, Zekai, et al. \"Better diffusion models further improve adversarial training.\" 2023.\n\n[2] Schuhmann, Christoph, et al. \"Laion-5b: An open large-scale dataset for training next generation image-text models.\" 2022.\n"}, {"review_id": "kgIYhDamZd", "reviewer": "Reviewer_V3Uo", "summary": "The paper suggests a sample-based method (termed FLS) for evaluating generative models. Similarly to FID and IS, the method uses a pre-trained network (InceptionV3 or CLIP) for image representation. Unlike FID, the proposed metric is based on a variant of KDE - fitting isotropic Gaussians around each generated sample and choosing variance values that maximize the likelihood of a subset of train samples. The metric is then the test likelihood. The authors claim and demonstrate that FLS is good at measuring the fidelity, novelty and diversity of generated samples and specifically has the advantages of being able to detect overfitting and memorization.", "questions": "About the correlation between FLS and Sample Fidelity (4.1). Can the authors explain what causes this correlation and the different behavior than in FID? Is it the embedding network or the different approaches to compare distributions?\n\nHave the authors tried evaluating more recent generative models (e.g. Stable Diffusion) on larger datasets (e.g. LAION)? Showing that FLS can scale to these models and dataset can strengthen the significance of the paper.\n\nThe proposed method seems to be somewhat related to the NDB evaluation method proposed in [1].\n\n[1] E Richardson, Y Weiss, On GANs and GMMs, Advances in neural information processing systems, 2018", "limitations": "A discussion about limitations of the proposed method seems to be missing. One example of such possible limitation is the ability of FLS to scale up to modern image-generation models (e.g. trained on the LAION dataset), generating a much larger variability of images (can the KDE-like approach capture this diversity ?)\n\nI could not identify a possible negative social impact.", "rating": 7, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "Significance: The ability to quantitively assess the quality of image generation models is highly important, especially due to the recent success and impact of \"generative AI\". There hasn't been much progress in this area in recent years - FID was introduced in 2017 and is still the most commonly used evaluation method and the de-facto standard. Works in this area therefore have high potential impact and should be encouraged. The proposed method seems to have advantages over FID (e.g. the ability to detect memorization and over-fitting).\n\nClarity: The paper is very clearly written and is easy to follow. The motivation is clearly stated and the method is well explained.\n(I do however have some comments about missing information, in the following sections)\n\nThe proposed method of using a KDE-like approach rather than some divergence seems to be original and well motivated.\n\n", "weaknesses": "Missing information: It is stated that FLS uses \"Inception-V3 or CLIP\" feature space, however it wasn't clearly indicated what feature space was actually used in each experiment (at least, I couldn't find this information). In addition, I would expect to see an analysis of the different behavior of these two representations (pros and cons of each). \n\nAdditional information that I could not find was the number of samples used for FLS evaluation and how it compares with the number of samples needed for FID. This specifically raised a concern because KDE methods (fitting a Gaussian around each sample) in general require a large number of samples. I would also expect that the ability to detect overfit depends on the number of used samples.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their thoughtful comments and feedback and we value the fact that the reviewer felt “the ability to quantitively assess the quality of image generation models is highly important” and that works in this area have a “high potential impact”. We also are thrilled to hear that the reviewer found our motivation as being “clearly stated”; that our paper was “well-written” and that our method was “well-explained” and has “advantages over FID”. \n### Missing Information\nWe thank the reviewer for raising the concern about which feature space was used in our experiments. For our experiments—for a fair comparison with FID which uses Inception-v3 — we also use Inception-v3 except for the ImageNet experiments which use CLIP (due to Inception-v3 being trained on ImageNet).\n\nRecent work has also demonstrated that Dino-V2 is a compelling alternative feature space for images, and we provide results on using this feature space in our 1-page PDF. We will add these results to our appendix and explicitly indicate which of Inception, CLIP or Dino-V2 is used where.\n### Number of Samples used for FLS\nWe thank the reviewer for pointing out the importance of the number of samples used for evaluation. For most of our experiments, we use 10k train/test/gen samples (except Table 1 where we use the full train set and 20k generated samples). \n\nImportantly, in Fig 10, we find that FLS is effective and unbiased even when using less test samples, of particular importance for applications in low data regimes or even conditional generation (e.g. the test set of CIFAR10 only contains 1k images for each class). On the other hand, FID highly recommends a minimum of 10k  [1]. Generally however, the whole training set is used as the value is still biased at 10k and a noticeably lower score can be obtained by using more samples.\n\nSimilarly, most papers use 50k generated samples as it gives a lower FID than with 10k. We find that using 10k samples for FLS is sufficient for robust evaluation which is in contrast with the number of generated samples used by FID.\n\n### Correlation between FLS and Sample FID\nWe thank the reviewer for their great question. The difference is not due to embedding networks as both FLS/FID are using Inception-V3 for this experiment. There are two ways the behavior is different. For imperceptible transforms, both are affected due to the imperfectness of the feature space. We posit that these transforms affect FID more as they affect the entire distribution of generated samples. As FID examines the distance between distributions, this distribution-wide shift likely has a larger effect (for example, to the covariance matrices of the Gaussians). Conversely, in FLS, the pairwise distances are only slightly increased in the case of these smaller transforms. For Figure 5, we believe the non-linear effect better matches our intuition about the ranking of models and their use in downstream tasks (more details in the global response).\n\n### Scaling to Text-Image Datasets\nWe value the reviewer's feedback on using FLS to evaluate recent generative models, such as Stable Diffusion on LAION. This is indeed an interesting direction of investigation! However, we believe such experiments are beyond the scope of this current paper as Stable Diffusion and models of this type are multi-modal generative models—i.e. text and image—and this presents fresh challenges. For example, we would need to approximate $p(x|y)$ for an unseen text prompt $y$ and it is not immediately clear how this can be accomplished. Note that this is a different setting than the image conditional experiments we considered in Tab. 2 where $y$ is from a known and finite set of class labels. Furthermore, the datasets we chose to evaluate largely are those examined by prior works on evaluation of non multi-modal generative models. Due to these complexities, we believe LAION and Stable Diffusion type models require an extended investigation which is deserving of its own paper.\n\n### Method is similar to NDB evaluation\nWe thank the reviewer for pointing out this interesting reference which we will include in our updated manuscript. Their method, instead of estimating densities like FLS, performs a statistical test on the number of samples in various Voronoi cells for the training/generated sets. The Voronoi cells are constructed by performing K-means clustering of the training set (all of this is done in the pixel space). As a whole, the process is quite different from FLS.\n\nNonetheless, it is interesting to note that they also use mixtures of Gaussians but for a completely different purpose (as the generative model). \n\n### Discussion on Limitations\nThe reviewer raises a valuable point by adding more discussion regarding the potential limitations of FLS. We agree with the reviewer that such a discussion can help practitioners effectively use FLS for evaluating their own generative models. As we outlined in our response to Reviewer cRFC we find that FLS is less sensitive to the scenario where there is both a little bit of memorization and the generative model still generalizes well (according to definitions 3.1 and 3.2). However, in cases where there is only memorization FLS still correctly reflects this in its score. \n\nWhile we acknowledge the reviewer's comment regarding a limitation of FLS to larger-scale datasets like LAION we believe this is not a pertinent limitation (as we argue in our response above). We believe that the linear computational complexity of FLS in the number of samples—i.e. $O(n)$ is in line with the best one can hope for any evaluation metric that operates purely on samples.  We will add a bigger discussion of these points in the main paper.\n\nWe thank the reviewer for their time and effort in reviewing our work, and we hope our efforts to clarify the main points and allow the reviewer to consider improving their score.\n\n[1] https://github.com/bioinf-jku/TTUR "}]

Author rebuttal:
We thank all reviewers for their thorough reviews and valuable feedback. We are encouraged that they found FLS well-motivated and that the holistic evaluation of generative models has “potential for high impact” (**HK8L, V3Uo**). We also thank the reviewers for viewing our paper as “well written and easy to understand” (**cRFc, V3Uo**). We are also pleased to hear that reviewers found our empirical investigation to contain “extensive experiments” on “SOTA models” on a variety of datasets (**euCM, cRFc, HK8L**). Finally, we appreciate that the reviewers found FLS easy to calculate (**SDnT**) and useful in ranking different models (**euCM**) while being robust to smaller sample sizes in comparison to FID (**SDnT, HK8L**). We now address the main shared concerns, grouped by theme below. 

### Additional results on how diversity, fidelity, and novelty affect FLS

To better illustrate the effect of changes in these three aspects of evaluation, we examine how they influence the FLS of 10k CIFAR10 samples generated by EDM G++. We report these results in Fig 1 of the attached 1-page PDF, which shows a plot of the FLS values as heatmaps for each combination of the different axes—i.e., Fidelity vs. Diversity, Novelty vs. Diversity, and Fidelity vs. Novelty. We change the values of fidelity, diversity, and novelty in the following concrete ways:
- **Fidelity:** Increasing severity of Gaussian blur applied to all samples (as measured by the $\sigma$ of the blur).
- **Diversity:** Decreasing diversity by duplicating several times the same sample (e.g., five duplicate samples correspond to replacing the 10k samples with 2k different samples duplicated five times).
- **Novelty:** Increasing the amount of memorized samples (as measured by the % of generated samples replaced by copies of the training set).

In summary, Fig.1 indicates that for FLS, fidelity matters more than diversity which matters more than novelty, with a notable exception when almost all examples are copied where FLS produces among the worst scores. We argue this ordering is very much aligned with the potential usefulness of the generative model in a downstream task:
- If samples have poor fidelity, then regardless of their diversity and novelty, they will not be useful. 
- If samples have poor diversity but good fidelity and novelty, removing duplicates yields a useful generative model.
- If there are some copies in the generated sample, it is serviceable as long as it is not the majority. 
- However, if the generative model only generates copies, then the generated data is useless (since we already have the training set, we do not need copies of it)

Finally, if a large enough fraction of the generated data point has high fidelity, diversity, and novelty FLS would provide a relatively good score regardless of the other fraction of the generated data. This explains why FLS does not correlates as linearly as FID in Fig 5. We develop this aspect in the following section.


### FLS correlates with sampling fidelity not as linearly as FID 

While we agree with Reviewer euCm and HK8L that linear correlation is helpful because it is intuitive, we argue that it is not clear that the most desirable behavior is a linear relationship between our score and **the fraction of perturbed data** (Fig.5 of our submission). In fact, we argue the non-linear relationship is a benefit of our score over FID. For example, consider a model that generates samples with poor fidelity 90% of the time and perfect data 10% of the time. Such a model should be considered significantly better than one that produces poor fidelity samples 100% of the time. The former has potential uses (especially if one can filter out the bad samples), whereas the latter is nearly useless. This aspect is directly captured by FLS—due to its connection to likelihood—which rates the fully copied model as significantly worse (e.g. Fig 5) whereas the more linear relationship found in FID only deems it slightly worse.

Finally, note that in Fig. 1 of the attached pdf, a linear relationship between FLS and the increasing severity of Gaussian blur applied to all samples can be noticed. Thus our score scales linearly with a global change in terms of fidelity (i.e. 100% of the data is similarly corrupted). 


### DINO-V2 improves upon Inception features 

To address Reviewer SDnT’s comment on the ability of the Inception feature to measure similarities (and thus detect copies from the training set), we conducted additional experiments using a more modern feature extractor in DINO-V2 [1,2], combined with a new ranking. These results can be found in our 1-page rebuttal PDF in Fig 2 and Table 1 (an updated version of Table 1 in the main paper). Visually inspecting the new memorized samples in Fig 2 we believe we have found undeniable evidence for problematic copies that are more visually striking than the ones initially presented in Fig 8. Given this compelling evidence, we believe FLS using DINO-V2 as a feature space is capable of effectively detecting memorized copies. 

### Figure 3. FLS is less sensitive to imperceptible perturbations (for the same feature space)
 
Another benefit of FLS is that it is less sensitive to small, nearly imperceptible perturbations relative to FID. We make this explicit in Fig 3 of our 1-page PDF with Inception-v3 features for transformations corresponding to slight JPG compression, blurring, and posterizing. While FLS is slightly worsened, the FID of EDM samples reaches that of SNGAN.

We thank the reviewers again for their valuable time and feedback. We hope that we address all their questions with this global response and individual responses. We look forward to further discussion. 

[1] Oquab, Maxime, et al. "Dinov2: Learning robust visual features without supervision." 2023.

[2] Stein, George, et al. "Exposing flaws of generative model evaluation metrics and their unfair treatment of diffusion models." 2023.


Meta-review: The authors present a novel metric for the evaluation of generative models. The main motivation behind this metric is the fact that current common metrics like FID reward memorization, which this new metric punishes. The authors also show empirically that their metric correlated with fidelity and diversity making it an important tool for comparison between generative models. The authors also addressed all the reviewer's concerns, specifically better showing memorization examples with DINO.
Decision: Accept (poster)