---
kind: concept
slug: frechet-inception-distance
title: Fréchet Inception Distance (FID)
summary: >-
  A set-level score for generated images: the Fréchet distance between Gaussian fits to pretrained Inception features of real and generated images. Covers the base score and its variants for spatial detail, classes, memorization, small samples, preprocessing, video and feature choice. The score depends on sample count, feature network and preprocessing.
tags: [evaluation, generative-models, image-generation, evaluation-metric, fid]
papers: ["2103.09396"]
related: [generative-model-evaluation, generative-adversarial-network, diffusion-models, self-supervised-learning, algorithmic-fairness, human-evaluation, benchmark-validity]
status: draft
updated_at: 2026-11-01
---

## Overview

FID compares two sets of images with no pairing between them. Features from a
pretrained Inception-v3 network are computed for the real set and the generated
set. Each set is summarised by a Gaussian fit, and the score is the Fréchet
distance between the two Gaussians, which the survey describes as a Wasserstein-2
distance [arxiv:2103.09396 §1.1]. Because the Gaussian fits use the full feature
covariance, the score reflects both the centre and the spread of the two sets.

The score has known limits. It is biased, it needs many samples (the survey
gives 50K as a usual size, not a rule, and says small samples over-estimate it)
[arxiv:2103.09396 §1.1]. It combines quality and diversity into one number, so
it does not say which of the two has failed [arxiv:2103.09396 §1.1]. It can be
raised by memorising training images [arxiv:2103.09396 §4.1]. Its result also
depends on the feature network and on image preprocessing, as the variants below
show.

The survey is the only source on this page. It is a review that runs no
experiments, so the numbers below are the ones it quotes from cited works and
have not been checked here. Each variant is linked to the same card.

## Variants

### Standard FID

The base score as described above: pooled Inception-v3 features, one Gaussian
per set, and the Fréchet distance between them [arxiv:2103.09396 §1.1]. The
survey gives neither the formula in full, nor the feature layer, nor the image
preprocessing. Card: [Pros and Cons of GAN Evaluation Measures: New
Developments](/papers/2103.09396).

### Spatial FID (sFID)

Adds intermediate features to the pooled ones. The survey lists the features as
the pooled pool3 layer plus the first seven channels of the mixed 6/conv layer
[arxiv:2103.09396 §2.1.1]. The survey does not give the reason for the choice of
layers. [Card](/papers/2103.09396).

### Class-aware FID (CAFD)

Computes a Fréchet distance for each class and averages them over K classes
[arxiv:2103.09396 §2.1.2]. It needs class labels for the generated and real
images. The value of K depends on the dataset and is not stated in the survey.
[Card](/papers/2103.09396).

### FastFID

An optimisation of FID that the survey describes as cheap enough to use as a
training loss [arxiv:2103.09396 §2.1.3]. The cited work reports the computation
as 25 to 500 times faster than standard FID, depending on the number of generated
samples [arxiv:2103.09396 §2.1.3]. [Card](/papers/2103.09396).

### Memorization-informed FID (MiFID)

Multiplies FID by a penalty that is large when generated images are close to
training images [arxiv:2103.09396 §2.1.4]. The closeness measure is a
cosine-based memorization distance, and the penalty switches on below a
threshold τ. The survey does not state τ, the small constant ε, or the feature
space used for the cosine similarity [arxiv:2103.09396 §2.1.4]. [Card](/papers/2103.09396).

### Bias-corrected FID and IS

Removes the finite-sample bias of FID and Inception Score by extrapolation. The
survey notes that the bias depends on both the number of images and the generator
[arxiv:2103.09396 §2.1.5]. Use this when the sample count is fixed and below the
usual size. [Card](/papers/2103.09396).

### Clean FID

A standard preprocessing protocol. The survey reports that the choice of resize
library, interpolation kernel and image encoding can change FID substantially
[arxiv:2103.09396 §2.1.6]. Using the same protocol removes one source of difference
between two scores. [Card](/papers/2103.09396).

### Fréchet Video Distance (FVD)

The video counterpart, with features from a video network. The survey gives the
feature network as I3D, pretrained on Kinetics-400 and Kinetics-600
[arxiv:2103.09396 §2.1.7]. For the video setting and its human study, see
[generative-model-evaluation](/concepts/generative-model-evaluation). [Card](/papers/2103.09396).

### Self-supervised features

Replaces the ImageNet classifier features with self-supervised features in FID
and related scores [arxiv:2103.09396 §2.2]. The survey lists this as an option
for the bias that Inception-based scores inherit from ImageNet. It gives no
head-to-head comparison between this option and the others [arxiv:2103.09396
§1.1] [§2.2]. [Card](/papers/2103.09396).

## When to use which

| Situation | Use | Why |
|---|---|---|
| Comparing with a published FID number | Clean FID protocol [arxiv:2103.09396 §2.1.6] | Resize and encoding choices can move the score substantially |
| Fewer generated images than the usual size | Bias-corrected FID [arxiv:2103.09396 §2.1.5] | Finite-sample bias depends on the image count |
| Suspected copying of training images | MiFID [arxiv:2103.09396 §2.1.4] | Penalises sets close to the training images; the threshold is not stated |
| FID needed inside a training loop | FastFID [arxiv:2103.09396 §2.1.3] | Reported 25 to 500 times faster in the cited work |
| Class-conditional model, per-class quality matters | Class-aware FID [arxiv:2103.09396 §2.1.2] | Averages a Fréchet distance over classes |
| Spatial detail matters more than global statistics | Spatial FID [arxiv:2103.09396 §2.1.1] | Uses intermediate features as well as pooled ones |
| Generated video | FVD [arxiv:2103.09396 §2.1.7] | Uses a video feature network |
| Concern about ImageNet bias in the features | Self-supervised features [arxiv:2103.09396 §2.2] | Option named by the survey; not compared head-to-head there |

Whatever the variant, report the sample count, the feature network and the
preprocessing. The survey treats all three as changing the score
[arxiv:2103.09396 §1.1] [§2.1.6].

## Implementation sketch

The core is the same for every variant: fit two Gaussians to two feature sets
and take the distance. The variants change which features are used, how many
there are, or what is multiplied or averaged afterwards.

```pseudocode
# Fréchet distance between two feature sets  [arxiv:2103.09396 §1.1]
# Input:  real features x_real (n x d), generated features x_gen (m x d)
# Output: scalar; the survey describes it as W2 between Gaussians

def frechet_distance(x_real, x_gen):
    # Fit one Gaussian to each feature set  [arxiv:2103.09396 §1.1]
    mu_r, cov_r = mean(x_real), covariance(x_real)
    mu_g, cov_g = mean(x_gen), covariance(x_gen)
    # Squared distance between the means  [arxiv:2103.09396 §1.1]
    mean_term = sum((mu_r - mu_g) ** 2)
    # Covariance term; the survey does not say how to compute the
    # matrix square root  [arxiv:2103.09396 §1.1]
    cov_term = trace(cov_r + cov_g - 2 * sqrtm(cov_r @ cov_g))
    return mean_term + cov_term
```

Where the variants differ: sFID uses intermediate features as well as the pooled
ones [arxiv:2103.09396 §2.1.1]; CAFD calls the function once per class and
averages [arxiv:2103.09396 §2.1.2]; MiFID multiplies the result by a
memorization penalty [arxiv:2103.09396 §2.1.4]. The feature network and the
image preprocessing are not part of this sketch and are not stated in the survey
[arxiv:2103.09396 §1.1] [§2.1.6].

## Contradictions

None known between sources; the page has one. Within the survey, the
class-aware formula prints the mean term as an unsquared norm, and the survey
does not say whether that is intended [arxiv:2103.09396 §2.1.2]. The sketch above
uses the squared form implied by the Wasserstein-2 description. Check the
original work before implementing CAFD.

## Papers

- [Pros and Cons of GAN Evaluation Measures: New Developments](/papers/2103.09396): a survey of FID variants and other GAN evaluation measures, with no new experiments.