# Rethinking Common Assumptions to Mitigate Racial Bias in Face Recognition
Datasets

Matthew Gwilliam1 Srinidhi Hegde1 Lade Tinubu1,2 Alex Hanson1
1University of Maryland 2University of Chicago
{mgwillia<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Many existing works have made great strides towards reducing racial bias in
face recognition. However, most of these methods attempt to rectify bias that
manifests in models during training instead of directly addressing a major
source of the bias, the dataset itself. Exceptions to this are BUPT-
Balancedface/RFW [34] and Fairface [14], but these works assume that primarily
training on a single race or not racially balancing the dataset are inherently
disadvantageous. We demonstrate that these assumptions are not necessarily
valid. In our experiments, training on only African faces induced less bias
than training on a balanced distribution of faces and distributions skewed to
include more African faces produced more equitable models. We additionally
notice that adding more images of existing identities to a dataset in place of
adding new identities can lead to accuracy boosts across racial categories.
Our code is available at https://github.com/j-alex-hanson/rethinking-race-
face-datasets.

##  1 Introduction

Figure 1: Balanced vs skewed training dataset. A training distribution
balancing race (all races: $25\%$; top) produces a larger variance on the test
accuracies of different races than a skewed distribution (African: $30\%$,
other races: $23.3\%$; bottom). The overall accuracy remains at $75\%$,
demonstrating that the skewed training distribution is the preferable choice.
Data for this figure is from Section 4.2. See Figure 8 in the Appendix for
additional distribution variations.

Since before the advent of deep learning, face recognition has been one of the
most popular human-centric applications of computer vision. The introduction
of deep learning has only accelerated progress in this area. The effectiveness
of face recognition algorithms makes them a compelling candidate for real-
world, industry-level applications. In fact, face recognition is currently
used to unlock phones, validate identities at ATMs, verify drivers licenses,
and aid in forensic investigations.

We consider two key attributes of face recognition applications: performance,
which is often measured in the literature in terms of accuracy or a similar
metric, and fairness, which is sometimes not measured at all. The computer
vision community has a responsibility to ensure it delivers research,
algorithms, and solutions which are not only highly performant, but also very
fair. Therefore, given that existing face recognition applications have high
accuracy, we ought to ensure that they are also fair. One key fairness issue
is equitable performance across race. Inconsistent performance across race may
lead to further disparagement of certain groups, motivating the need for
research in this space.

Numerous recent works have attempted to address this concern and show progress
towards equability, but this is usually done by modifying the model
architecture or data sampling strategy to explicitly induce fairness. It is
understood that these modifications are necessary because the data used for
training these models will inherently cause the models to be biased without
some intervention. What is it about this data that is causing the models to
become biased? Two datasets specially created to address racial bias, BUPT-
Balancedface (BUPT) / Racial Faces in the Wild (RFW) [34] and Fairface [14],
posit that the reason bias is introduced is because racial categories are not
evenly represented in the data. This stems from the observation that most
commonly used face datasets are primarily comprised of Caucasian individuals.

But does training on a single race necessarily lead to a biased model? And is
balancing the dataset across race an optimal choice for mitigating racial
bias? Figure 1 offers evidence that, contrary to popular assumption, balanced
datasets do not always lead to the most balanced results. The figure shows
that for the BUPT dataset – which groups images into race categories African,
Asian, Caucasian, and Indian – results can be better balanced in terms of
accuracy variance, without affecting the mean accuracy, by allocating slightly
more images to African that to other race categories. This example is merely
the tip of the iceberg; more thorough analysis follows in the rest of this
paper.

There are many more assumptions that are necessary for achieving fairness in
face recognition datasets that we do not address in this work. We do not
address the issue of the actual composition and labelling of these datasets.
BUPT, which we use in this work, has 4 racial categories and every image is
placed into one of those 4 categories, sometimes in spite of the represented
individual’s actual or self-determined race/ethnicity. Another issue that we
do not cover is ensuring that the dataset is capturing a true distribution of
faces. Face datasets are typically sourced from online images, and images
which appear on the internet are in some sense curated – they were uploaded
because the subject or a third party wanted the content to exist online.

Instead, in this work, we analyze assumptions about racial bias in face
recognition by analyzing results from the BUPT training dataset and RFW
testing dataset. Our contributions are the following:

  * •

We show that, when training state-of-the-art models on a single race from
BUPT, some races generalize across all races differently than others (Section
4.1).

  * •

We sample subset training datasets of BUPT from many possible race
distributions, and demonstrate that some racially skewed datasets mitigate
racial bias better than racially balanced datasets; sometimes by a wide margin
(Section 4.2).

  * •

We observe that adding more images of existing subjects in face datasets over
adding new subjects can lead to greater accuracy boosts across racial
categories (Section 4.3).

For completeness, we define a few terms used throughout our work here. We
define face recognition as the mapping of an image of a face to an identity.
The implementation of face recognition we perform in this work, pair matching,
requires a trained model (BUPT) to identify if a pair of face images (RFW)
belong to the same individual or different individuals; these individuals and
their corresponding images are typically unavailable during training. We
define and quantify racial bias in face recognition systems as the variance of
test race accuracies. For our work, this is the variance in accuracy on the
African, Asian, Caucasian, and Indian test splits of RFW.

##  2 Related Work

##### Face Recognition

Face recognition models have benefited from the inclusion of deep learning
techniques [28]. In a typical deep face recognition model, the backbone
architecture, discriminative loss function, and a deep feature based face
matching method form the three critical components of the system [33]. A
single DNN backbone architecture is the most common choice for the extraction
of face features [23, 25]. To exploit these features, face recognition models
employ a variety of discriminative loss functions for training face
recognition models such as contrastive loss [30], triplet loss [25],
angular/cosine loss [7, 31], and softmax loss variants [18, 24]. Deep feature
based matching methods vary with the requirements of the face recognition
application. Face verification can require a more fine-grained approach [8]
and face identification necessitates discriminative features [37]. In our
work, we analyze recent generic face recognition models that are trained with
several loss functions and have a single backbone network.

##### Bias in Computer Vision for Faces

In recent years, many studies have confirmed the presence of bias in deep
neural networks [4, 20] which may result in undesired consequences especially
for face recognition [22, 27]. Several works focus on specifically mitigating
bias, either by explicitly changing the models or incorporating data sampling
strategies distinctly for this purpose. Regarding model modifications, [34]
propound a model to balance representations of face data from other datasets,
in addition to introducing a balance dataset. [10] use adaptive convolution
kernels and attention mechanisms to mitigate bias in the model. [26]
incorporate triplet loss to prevent discriminatory effects. And [21] propose
privacy preserving and learning agnostic representation to mask sensitive
information, including race and gender. Regarding sampling strategies, [32]
uses reinforcement learning dataset sampling to mitigate bias. [3] introduces
the use of sampling strategies to mitigate geographic performance differences
on photo ID documents. [9] uses demographic classifiers and adversarial
learning to make face representations more robust. [39] adapts Cycle-GAN to
racially balance training per individual in the dataset. And [35] implements a
balanced sampling strategy while training to mitigate bias, though the focus
of their work is primarily on gender.

##### Datasets

Face recognition is a problem that has garnered a sustained interest and as
such many datasets exist for various subproblems in this space. A few popular
datasets include: CASIA-WebFace [38], VGGFace2 [6], WebFace260M [41], MS-
Celeb-1M [11], LFW [13], and IJB-C [19]. Naturally, works also exist that
explore potential biases these datasets exhibit [36, 5]. Addressing racial
bias, datasets such as RFW [34] and FairFace [14] point out that most of the
popular face datasets primarily consist of Caucasian face images and propose a
racially balanced face dataset. In addition to RFW and FairFace, CASIA-SURF
CeFA [17] is another racially balanced face dataset that addresses racial bias
in anti-spoofing. Data augmentation techniques used to mitigate bias are
discussed by [29, 32, 39]. Related to our analysis are [15], which finds no
evidence that darker skin tone causes higher false match rate on the pair
matching problem, and [1], which identifies accuracy differences between
genders persist after balancing the dataset. [40] specifically argues for
racially balancing datasets to mitigate bias. Our work refutes this claim by
demonstrating that some racially skewed training datasets can result in less
racially biased models than racially balanced training datasets.

Figure 2: Face size statistics for training data (top) and test data (bottom).
Both plots show the ratio of the face bounding box area to the area of the
containing image, sorted for each race category according to the ratio. Figure
3: Results for single race distributions. Models from ArcFace (ARC),
CenterLoss (CNT), SphereFace (SPH), and VGGFace2 (VGG) were trained on each
race, and then tested on each race. Results were averaged across 5 trials.
Tabular results in Table 2.

##  3 Experimental Setup

###  3.1 Models

We run our experiments on four recent face recognition models: VGGFace2,
CenterLoss, SphereFace, and ArcFace. VGGFace2 [6] is simply an implementation
of a squeeze-and-excitation network (a modified ResNet architecture) [12],
trained with a standard cross-entropy loss where the classes are the training
dataset identities. The outputs of the penultimate layer are used as feature
embeddings at test time for pair-matching (see below for more details).
Because VGGFace2 is a standard network implementation, we view this as a
baseline method. In addition to cross-entropy loss, CenterLoss [37] maintains
an additional center vector per identity and imposes a squared L2 distance
loss between the feature vectors of the training samples and their
corresponding centers. The centers are initialized randomly and are
incrementally updated by the mean features of each identity as training
progresses. SphereFace [18] modifies the softmax function on the outputs of
the network to impose a multiplicative margin loss on the angle between L2
normalized feature embeddings. ArcFace [7] extends SphereFace by including an
additive margin loss on this angle between L2 normalized feature embeddings.

Additionally, for fair comparisons between methods we used a ResNet50 as the
backbone of each model and trained for 50 epochs. All inputs are resized to
$128\times 128$.

###  3.2 Datasets

Dataset | African | Asian | Caucasian | Indian
---|---|---|---|---
BUPT (train) | # sbjct | 7000 | 7000 | 7000 | 7000
# img | 324376 | 325475 | 326484 | 275095
RFW (test) | # pair | 6000 | 6000 | 6000 | 6000
Table 1: Data Sources. Relevant statistics pertaining to the composition of
BUPT-Balancedface (BUPT), which is the source for our training datasets, and
Racial Faces in the Wild (RFW), which is the source for our testing datasets.

BUPT-Balancedface (BUPT) [34], also known as EqualizedFace in the literature,
is the source of our training data. This dataset, in addition to offering
subject/class labels for each image, also groups images into 1 of 4 possible
race categories. Unlike standard datasets, the distribution of these races in
BUPT is balanced, with each race having the same number of subjects, and
roughly the same number of images. For testing, we do a pair-matching task on
the pairs in Racial Faces in the Wild (RFW)[34], which is set up such that
each race has the same number of corresponding pairs. Relevant statistics for
both datasets can be found in Table 1.

We perform additional analysis regarding the contents of these two datasets to
produce additional statistics. In Figure 2, we show the size distributions of
the faces of both datasets, grouped by race. We calculated the ratios for each
image by dividing the bounding box area of the face, calculated via OpenCV’s
pre-trained face detector [2], by the area of the containing image. Notice
that while the African and Caucasian size statistics of BUPT are similar to
those of all 4 races for RFW, the sizes of the Indian and Asian faces are
distributed quite differently. Specifically, both races are represented by an
abundance of larger faces (relative to the images) in the training data. This
could contribute to discrepancies in accuracy and representational power that
we observe in Section 7. Moreover, if the balance hypothesis was valid but
perturbed by these inconsistent ratios, we would expect an optimally fair
distribution to contain an even split between African and Caucasian. However,
our experiments demonstrate this is not the case.

Figure 4: Effect of noise injection in training data on model performance.
$0.1$, $0.2$, $0.3$, $0.4$, and $0.5$ are the different noise percentage
probabilities.

####  3.2.1 Single Race Setup

For the experiments in Section 4.1, we construct our training and testing
subsets from the two datasets described above. Specifically, we take training
data from BUPT-Balancedface and testing data from RFW. Rather than training on
the entire BUPT dataset, for the Single Race experiments we train a given
model only on a single race. We repeat this for each race, such that we have
different models corresponding to the training data associated with each of
the 4 race groups in BUPT. We evaluate each model on all 4 races in the
testing data, separately, such that each model is associated with 4 different
test accuracy results, 1 per race.

####  3.2.2 Race Distribution Setup

Figure 5: Cluster vs. accuracy for the single race experiments using ArcFace.

Prior work on these datasets deals primarily with the uniform (balanced)
dataset which consists of equal amounts of persons representing each racial
category. In Section 4.2, we expand our analysis from Section 4.1 to a
representative collection of non-uniform distributions of data. Thus, we
explore the space of imbalanced datasets.

In order to conduct meaningful exploration, we impose the following
constraints:

  * •

Each dataset must have the same number of images.

  * •

Each dataset must have the same number of subjects.

To abide by these constraints, we only consider the 5000 persons corresponding
to each race for whom the most data is available. Since the least-represented
of these persons has 18 corresponding images, we choose to use 18 images to
represent each person. Furthermore, since we only have 5000 persons available
for each race, every dataset considered has exactly 5000 persons, selected
from the different races as described below.

We describe each dataset as a tuple consisting of 4 values,
$(w_{1},x_{1},y_{1},z_{1})$, where $w_{1}$ gives the portion of the dataset
comprised of African persons (as a percent), $x_{1}$ gives the portion of the
dataset that represent Asian persons, and so forth. Since the portions are
given as percents, the values of each tuple always sum to 100. From the set of
all possible tuples, we consider the uniform distribution, $(25,25,25,25)$ and
88 other points. To choose the 88 points, we consider 4 nested 3-simplexes.
The corners of the outermost simplex are given by permutations of
$(100,0,0,0)$. The 3 inner simplexes have corners given by permutations of
$(60,13.3,13.3,13.3)$, $(40,20,20,20)$, $(30,23.3,23.3,23.3)$ such that we
examine more distributions that lie closer to the center (closer to uniform).
Note that the corners only account for 16 points (4 corners each for 4
simplexes). The remaining points are derived by taking 3 equidistant points
from each edge on each simplex. With 6 edges per simplex, and 4 simplexes,
this accounts for the remaining 72 points. We believe this method gives
reasonable coverage of non-uniform race distributions.

###  3.3 Evaluation

####  3.3.1 Test Accuracy

We compute accuracy for the pair-matching task on RFW. This is a binary task,
a pair is either a match or not a match. Test accuracy is thus the percent of
all pairs for a race that are correctly identified as matches or non-matches.

####  3.3.2 Cluster Analysis

In order to better understand why different racial image distributions produce
different results, we analyze the relationships between features. We consider
the 4 races as clusters, and assess the image representations by computing
metrics that approximate the tightness and spread of the clusters. We compute
2 metrics to accomplish this, Intra-race Cosine Distance and Race Cluster
Membership.

Intra-race Cosine Distance The first form of “clustering analysis” we perform
attempts to measure the compactness of the representations corresponding to
each race. We represent images from RFW using the same features used in the
pair matching task. We first compute the mean vector of all the features for
the images of a given race. Then, we compute the cosine distances between the
image features for a race and that race’s mean vector. We thus report the
average of the cosine distances (cluster compactness) from the mean vector
(cluster center).

Figure 6: Race distribution results of ArcFace models on the 89 distributions
identified in Section 3.2.2. Each plot gives percent accuracy values for a
single test race. These plots roughly correspond to flattened versions of the
4 nested 3-simplexes; note that each plot consists of 4 connected equilateral
triangles, where each triangle contains 4 parallel simplex faces projected
into 2 dimensions, one from each simplex, as well as the center point. The
corners of the outer simplex, which correspond to training distributions
containing only data from a single race, and are labeled accordingly, i.e.,
the point labeled “Afn” is (100, 0, 0, 0). For readability, some points are
therefore represented multiple times, such as the center (25, 25, 25, 25),
which appears in the middle of all triangles. Thus, each plot contains 181
points, of which only 89 are unique. In addition to per-race accuracy, overall
average accuracy of the races is represented, as well as the accuracy variance
between the 4 per-race accuracy results. The variance plot thus represents how
“balanced” the results are. For reference, the highest mean race accuracy is
(75, 0, 25, 0) and the lowest race accuracy variance is (25, 75, 0, 0).
Tabular results in Tables 3 and 4.

Race Cluster Membership To complement the cosine distance metric, which is an
absolute measure of cluster compactness (treating races as clusters), we
perform clustering by taking the 20 nearest neighbors in feature space for
each of 5000 randomly selected images from each race of RFW (20000 images in
total). The neighbors then vote, with the vote weighted according to the
inverse of the inner product distance between the image’s feature vector and
the neighbor’s feature vector. Each images is then assigned the race that
receives the maximum votes as the “cluster” label. We repeat this process for
each trained model and report the average percent of the RFW images that are
assigned to each race. The resulting cluster membership statistics serve as a
relative measure of cluster compactness.

##  4 Experiments and Analysis

As Figure 1 establishes, a balanced dataset doesn’t actually lend itself to
the most balanced results. In this section, we explore how training set
composition affects results. In Section 4.1, we begin by examining the
accuracy results that models trained on a single race achieve when transferred
to other races. Then, in Section 4.2, we compare models which are trained
exclusively on a single race to models which are trained on an even
distribution of data to every race, along with 84 distributions in between
those extremes, as explained in Section 6. Finally, in Section 4.3, we compare
the two different ways to increase the size of a dataset (overall number of
images)- gathering more images from existing subjects, and acquiring new
subjects.

###  4.1 Generalizing from Single Race

We train each model (ArcFace, CenterLoss, SphereFace, and VGGFace2) on each
race of BUPT (African, Asian, Caucasian, Indian) for 5 trials. This gives 16
models per trial and 80 models in total. We test each model on each race of
RFW. Results are averaged across trials.

####  4.1.1 Test Accuracy

Figure 3 gives average accuracy results. Note the major role played by the
train race, as it dominates the range of possible accuracy scores. Also note
that, for a given train race, results tend to be grouped not by method, but by
train race. For a given train race and test race, the performance of the
various models are relatively similar, with ArcFace tending to have the best
results, followed by SphereFace, VGGFace2, and CenterLoss, respectively. Data
is clearly the determining factor in terms of both mean accuracy (across 4
races) as well as fairness (variance in test race accuracies).

Robust Augmentation For understanding the effects of data augmentation on
model performance we inject noise in the BUPT train images. To introduce noise
in the image samples, we, first, localize the face using template matching
method with Haar features [16]. Then we subdivide the localized face bounding
box into a $4\times 4$ grid and pick a square patch uniformly at random.
Finally, we apply Gaussian blur on the selected square patch. For the Gaussian
blur filter, we pick a Gaussian kernel whose size varies randomly between $11$
and $21$ pixels and with a variance of $1.5$. Throughout this experiment we
maintain the original number of training identities, i.e, $7000$. Figure 4
represents the model performance under different noise injection probabilities
across the different races which means that each image sample is assigned
noise with the specified noise injection probability. We see that the
introduction of noise indeed results in slight boost in accuracy for African
and Caucasian races which have stronger representations compared to other
races.

####  4.1.2 Cluster Analysis

We attempt to understand how test accuracy is related to the representations
learned during training. As described in Section 3.3.2, we do this both by
considering the races as clusters and obtaining a cosine distance measure for
each, as well as clustering via a nearest neighbor voting method. Figure 5
shows the results of these methods, in addition to test accuracy, for ArcFace.

We observe reasonable correspondence between the values in the clustering
matrices and the accuracy matrix. Caucasian, from the clustering plots,
appears to have the representations which are the most spread out in embedding
space, featuring the highest cosine distance as well as the lowest number of
assigned image for its own cluster. African also is fairly spread out
according to the cosine distance metric, although it is worth considering that
it is somewhat unique in the race cluster membership analysis. Specifically,
African-African is the highest value on the main diagonal, and African
training seems to induce more balanced clusters than the other races. This
suggests that while the African embeddings spread out in absolute terms
(cosine distance), they also maintain a distinct cluster while also keeping
other races clusters distinct. Overall this seems to empirically suggest that
training on races that spread out more in feature space may lead to higher
overall accuracy and that more distinct clusters in those spread out
embeddings may yield better transfer performance.

###  4.2 Analyzing Race Distributions

It is clear that the distribution of subjects and images among the races
affects per-race accuracy. Furthermore, it is apparent that training on a
single race can achieve, in some cases, reasonable performance on other races.
Nevertheless, neither training on a single race nor training on equal amounts
of data from all races achieves totally balanced performance across all races.
While, as discussed previously, some researchers attempt to address the
remaining imbalances by adopting specific data sampling strategies during
training or altering the model embeddings, we study how the dataset itself
might be used to balance results. To more aptly investigate this, we carefully
vary the distribution of training data among races, as explained in Section
3.2.2. In this way we gain an understanding, not only of the performance of
the perfectly balanced dataset, but of a representative portion of the
possible distributions of training data among the 4 race supercategories.

Figure 6 shows the results from the selected data distributions with ArcFace.
Notice the importance of training race. For all races, distributions
containing a mix of Caucasian and African images tend to be the highest
performing. This matches some of the findings from Figure 3, where Asian
accuracy is higher when trained on African or Caucasian images than when
trained on Asian images. Further, the data shown here confirms that the
equally balanced dataset, the central point in each plot, is not the best for
either performance (mean accuracy) or fairness (variance). Instead, as stated
before, datasets containing African and Caucasian images tend to give the
highest accuracy scores, while various blends that contain African images are
the most “fair” (lowest variance). Refer to the Supplementary Material for the
race distribution analysis of VGGFace2.

###  4.3 Understanding Increasing Dataset Size

Figure 7: Increasing dataset size by adding new subjects vs adding data to
existing subjects, for a given race.

This section continues our exploration of the importance of data distribution
by investigating how the addition of new data affects results. For these
experiments, we establish a base dataset, consisting of 10 images per subject
for 2500 subjects from each race. We examine a special case of data addition
where data is added to only one race, and consider two ways this data can be
added. The first is the introduction of new images for existing subjects; we
add 5 images to each of the existing subjects for the selected race. The
second way is adding new subjects; we introduce 1250 new subjects, and add 10
images to each. Both methods for adding data result in the same number of
total images, enabling us to compare the two methods for adding to an existing
dataset in terms of performance and fairness.

Figure 7 shows the results of this experiment. Note that, with the exception
of a single train-test pair, accuracy improvements are much higher when images
are added to the existing subjects. Furthermore, adding Caucasian and Indian
images in this manner increases accuracy results in a far more balanced
fashion than one might expect, with nearly identical improvement for African
and Caucasian when adding Caucasian images, and very similar improvement for
African, Caucasian, and Indian when adding Indian images. This reinforces the
idea that balancing the distribution of training data can contribute to, but
is neither necessary nor sufficient, for balanced results.

##  5 Conclusion

We have shown that common assumptions to mitigate racial bias in datasets do
not necessarily hold. The African BUPT data split produces more equitable
models on RFW than a balanced BUPT data split. Training set distributions
skewed to include more African faces also mitigate racial bias better than
balanced training sets. Data augmentations appear to benefit more robust
racial categories and adding more images to the base identities of a dataset
can boost performance across race. We have demonstrated some ways to improve
mitigation of racial bias on existing datasets, but we hope that illuminating
these erroneous assumptions will ultimately assist the face recognition
community in building more equitable systems.

Acknowledgments: Alex Hanson was supported by the NDSEG fellowship.

## References

  * [1] Vitor Albiero, Krishnapriya KS, Kushal Vangara, Kai Zhang, Michael C King, and Kevin W Bowyer.  Analysis of gender inequality in face recognition accuracy.  In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision Workshops, pages 81–89, 2020.
  * [2] G. Bradski.  The OpenCV Library.  Dr. Dobb’s Journal of Software Tools, 2000.
  * [3] Martins Bruveris, Jochem Gietema, Pouria Mortazavian, and Mohan Mahadevan.  Reducing geographic performance differentials for face recognition.  In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision Workshops, pages 98–106, 2020.
  * [4] Joy Buolamwini and Timnit Gebru.  Gender shades: Intersectional accuracy disparities in commercial gender classification.  In Conference on fairness, accountability and transparency, pages 77–91. PMLR, 2018.
  * [5] Dong Cao, Xiangyu Zhu, Xingyu Huang, Jianzhu Guo, and Zhen Lei.  Domain balancing: Face recognition on long-tailed domains.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 5671–5679, 2020.
  * [6] Qiong Cao, Li Shen, Weidi Xie, Omkar M Parkhi, and Andrew Zisserman.  Vggface2: A dataset for recognising faces across pose and age.  In 2018 13th IEEE international conference on automatic face & gesture recognition (FG 2018), pages 67–74. IEEE, 2018.
  * [7] Jiankang Deng, Jia Guo, Niannan Xue, and Stefanos Zafeiriou.  Arcface: Additive angular margin loss for deep face recognition.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4690–4699, 2019.
  * [8] Weihong Deng, Jiani Hu, Nanhai Zhang, Binghui Chen, and Jun Guo.  Fine-grained face verification: Fglfw database, baselines, and human-dcmn partnership.  Pattern Recognition, 66:63–73, 2017.
  * [9] Sixue Gong, Xiaoming Liu, and Anil K Jain.  Jointly de-biasing face recognition and demographic attribute estimation.  In European Conference on Computer Vision, pages 330–347. Springer, 2020.
  * [10] Sixue Gong, Xiaoming Liu, and Anil K Jain.  Mitigating face recognition bias via group adaptive classifier.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3414–3424, 2021.
  * [11] Yandong Guo, Lei Zhang, Yuxiao Hu, Xiaodong He, and Jianfeng Gao.  Ms-celeb-1m: A dataset and benchmark for large-scale face recognition.  In European conference on computer vision, pages 87–102. Springer, 2016.
  * [12] Jie Hu, Li Shen, and Gang Sun.  Squeeze-and-excitation networks.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7132–7141, 2018.
  * [13] Gary B. Huang, Manu Ramesh, Tamara Berg, and Erik Learned-Miller.  Labeled faces in the wild: A database for studying face recognition in unconstrained environments.  Number 07-49, October 2007.
  * [14] Kimmo Karkkainen and Jungseock Joo.  Fairface: Face attribute dataset for balanced race, gender, and age for bias measurement and mitigation.  In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 1548–1558, 2021.
  * [15] KS Krishnapriya, Vítor Albiero, Kushal Vangara, Michael C King, and Kevin W Bowyer.  Issues related to face recognition accuracy varying based on race and skin tone.  IEEE Transactions on Technology and Society, 1(1):8–20, 2020.
  * [16] Rainer Lienhart and Jochen Maydt.  An extended set of haar-like features for rapid object detection.  In Proceedings. international conference on image processing, volume 1, pages I–I. IEEE, 2002.
  * [17] Ajian Liu, Zichang Tan, Jun Wan, Sergio Escalera, Guodong Guo, and Stan Z. Li.  Casia-surf cefa: A benchmark for multi-modal cross-ethnicity face anti-spoofing.  In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), pages 1179–1187, January 2021.
  * [18] Weiyang Liu, Yandong Wen, Zhiding Yu, Ming Li, Bhiksha Raj, and Le Song.  Sphereface: Deep hypersphere embedding for face recognition.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 212–220, 2017.
  * [19] Brianna Maze, Jocelyn Adams, James A Duncan, Nathan Kalka, Tim Miller, Charles Otto, Anil K Jain, W Tyler Niggel, Janet Anderson, Jordan Cheney, et al.  Iarpa janus benchmark-c: Face dataset and protocol.  In 2018 International Conference on Biometrics (ICB), pages 158–165. IEEE, 2018.
  * [20] Ninareh Mehrabi, Fred Morstatter, Nripsuta Saxena, Kristina Lerman, and Aram Galstyan.  A survey on bias and fairness in machine learning.  ACM Computing Surveys (CSUR), 54(6):1–35, 2021.
  * [21] Aythami Morales, Julian Fierrez, Ruben Vera-Rodriguez, and Ruben Tolosana.  Sensitivenets: Learning agnostic representations with application to face images.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 43(6):2158–2164, 2020.
  * [22] Shruti Nagpal, Maneet Singh, Richa Singh, and Mayank Vatsa.  Deep learning for face recognition: Pride or prejudiced?  arXiv preprint arXiv:1904.01219, 2019.
  * [23] Omkar M Parkhi, Andrea Vedaldi, and Andrew Zisserman.  Deep face recognition.  2015\.
  * [24] Rajeev Ranjan, Carlos D Castillo, and Rama Chellappa.  L2-constrained softmax loss for discriminative face verification.  arXiv preprint arXiv:1703.09507, 2017.
  * [25] Florian Schroff, Dmitry Kalenichenko, and James Philbin.  Facenet: A unified embedding for face recognition and clustering.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 815–823, 2015.
  * [26] Ignacio Serna, Aythami Morales, Julian Fierrez, Manuel Cebrian, Nick Obradovich, and Iyad Rahwan.  Sensitiveloss: Improving accuracy and fairness of face representations with discrimination-aware deep learning.  arXiv preprint arXiv:2004.11246, 2020.
  * [27] Harini Suresh and John V Guttag.  A framework for understanding unintended consequences of machine learning.  arXiv preprint arXiv:1901.10002, 2019.
  * [28] Yaniv Taigman, Ming Yang, Marc’Aurelio Ranzato, and Lior Wolf.  Deepface: Closing the gap to human-level performance in face verification.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1701–1708, 2014.
  * [29] Valeska Uchôa, Kelson Aires, Rodrigo Veras, Anselmo Paiva, and Laurindo Britto.  Data augmentation for face recognition with cnn transfer learning.  In 2020 International Conference on Systems, Signals and Image Processing (IWSSIP), pages 143–148. IEEE, 2020.
  * [30] Feng Wang, Xiang Xiang, Jian Cheng, and Alan Loddon Yuille.  Normface: L2 hypersphere embedding for face verification.  In Proceedings of the 25th ACM international conference on Multimedia, pages 1041–1049, 2017.
  * [31] Hao Wang, Yitong Wang, Zheng Zhou, Xing Ji, Dihong Gong, Jingchao Zhou, Zhifeng Li, and Wei Liu.  Cosface: Large margin cosine loss for deep face recognition.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5265–5274, 2018.
  * [32] Mei Wang and Weihong Deng.  Mitigating bias in face recognition using skewness-aware reinforcement learning.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9322–9331, 2020.
  * [33] Mei Wang and Weihong Deng.  Deep face recognition: A survey.  Neurocomputing, 429:215–244, 2021.
  * [34] Mei Wang, Weihong Deng, Jiani Hu, Xunqiang Tao, and Yaohai Huang.  Racial faces in the wild: Reducing racial bias by information maximization adaptation network.  In The IEEE International Conference on Computer Vision (ICCV), October 2019.
  * [35] Zeyu Wang, Klint Qinami, Ioannis Christos Karakozis, Kyle Genova, Prem Nair, Kenji Hata, and Olga Russakovsky.  Towards fairness in visual recognition: Effective strategies for bias mitigation.  In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 8919–8928, 2020.
  * [36] Xin Wen, Biying Li, Haiyun Guo, Zhiwei Liu, Guosheng Hu, Ming Tang, and Jinqiao Wang.  Adaptive variance based label distribution learning for facial age estimation.  In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XXIII 16, pages 379–395. Springer, 2020.
  * [37] Yandong Wen, Kaipeng Zhang, Zhifeng Li, and Yu Qiao.  A discriminative feature learning approach for deep face recognition.  In European conference on computer vision, pages 499–515. Springer, 2016.
  * [38] Dong Yi, Zhen Lei, Shengcai Liao, and Stan Z Li.  Learning face representation from scratch.  arXiv preprint arXiv:1411.7923, 2014.
  * [39] Seyma Yucer, Samet Akçay, Noura Al-Moubayed, and Toby P Breckon.  Exploring racial bias within face recognition via per-subject adversarially-enabled data augmentation.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pages 18–19, 2020.
  * [40] Yaobin Zhang and Weihong Deng.  Class-balanced training for deep face recognition.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pages 824–825, 2020.
  * [41] Zheng Zhu, Guan Huang, Jiankang Deng, Yun Ye, Junjie Huang, Xinze Chen, Jiagang Zhu, Tian Yang, Jiwen Lu, Dalong Du, et al.  Webface260m: A benchmark unveiling the power of million-scale deep face recognition.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 10492–10502, 2021.

Appendix

##  Appendix A Additional Single Race Analysis

In Table 2 we provide the results of our single race experiments in tabular
form for completeness.

##  Appendix B Additional Race Distribution Analysis

In Figure 8 we provide an extended view of Figure 1, highlighting additional
distribution points of interest. This view reinforces that more heavily skewed
distributions sometimes do lead to more fairness (less variance) across races.

In Figure 9, we show VGGFace2 [6] race distribution results are comparable to
those shown for ArcFace [7] in Figure 6. The trends they demonstrate are
similar in terms of both accuracy and variance (fairness).

In Table 3, Table 4, Table 5, and Table 6 we provide the results of our race
distribution experiments in tabular form for ArcFace and VGGFace2 respectively
(2 tables each) again for completeness.

ArcFace
---
Train Race | Test Race
African | Asian | Caucasian | Indian
African | 85.5 $\pm$ 0.3 | 75.0 $\pm$ 0.2 | 82.4 $\pm$ 0.2 | 80.9 $\pm$ 0.7
Asian | 58.6 $\pm$ 0.5 | 71.8 $\pm$ 0.6 | 68.7 $\pm$ 0.5 | 66.s5 $\pm$ 0.4
Caucasian | 71.5 $\pm$ 0.5 | 76.4 $\pm$ 0.4 | 89.1 $\pm$ 0.2 | 81.2 $\pm$ 0.3
Indian | 61.2 $\pm$ 0.6 | 68.5 $\pm$ 0.5 | 73.7 $\pm$ 0.4 | 75.8 $\pm$ 0.4
CenterLoss
Train Race | Test Race
African | Asian | Caucasian | Indian
African | 73.7 $\pm$ 0.8 | 70.0 $\pm$ 0.6 | 74.8 $\pm$ 0.5 | 73.9 $\pm$ 0.4
Asian | 56.1 $\pm$ 0.4 | 66.6 $\pm$ 0.6 | 63.7 $\pm$ 0.8 | 63.1 $\pm$ 0.9
Caucasian | 66.7 $\pm$ 0.9 | 72.3 $\pm$ 1.0 | 81.4 $\pm$ 1.1 | 75.2 $\pm$ 0.8
Indian | 58.6 $\pm$ 1.3 | 66.5 $\pm$ 1.5 | 71.1 $\pm$ 1.2 | 71.6 $\pm$ 0.8
SphereFace
Train Race | Test Race
African | Asian | Caucasian | Indian
African | 82.2 $\pm$ 0.5 | 74.3 $\pm$ 0.4 | 80.7 $\pm$ 0.4 | 80.0 $\pm$ 0.5
Asian | 58.6 $\pm$ 0.5 | 69.2 $\pm$ 0.4 | 67.5 $\pm$ 0.4 | 65.7 $\pm$ 0.5
Caucasian | 68.4 $\pm$ 0.3 | 75.4 $\pm$ 0.2 | 87.2 $\pm$ 0.3 | 79.0 $\pm$ 0.4
Indian | 62.5 $\pm$ 0.3 | 70.6 $\pm$ 0.3 | 75.3 $\pm$ 0.5 | 78.3 $\pm$ 0.3
VGGFace2
Train Race | Test Race
African | Asian | Caucasian | Indian
African | 79.1 $\pm$ 0.4 | 72.3 $\pm$ 0.4 | 78.6 $\pm$ 0.3 | 76.5 $\pm$ 0.5
Asian | 57.7 $\pm$ 0.6 | 69.5 $\pm$ 0.7 | 67.0 $\pm$ 0.7 | 65.6 $\pm$ 0.5
Caucasian | 68.4 $\pm$ 0.3 | 73.5 $\pm$ 0.5 | 85.3 $\pm$ 0.4 | 77.1 $\pm$ 0.3
Indian | 59.9 $\pm$ 0.3 | 68.2 $\pm$ 0.5 | 73.2 $\pm$ 0.2 | 75.1 $\pm$ 0.4
Table 2: Results for single race experiments. We train on BUPT-Balancedface
data corresponding to the race in the Train Race row, and test on RFW data
corresponding to the race in the Test Race column. These results correspond to
those reported in Figure 3; they are provided as a table here for convenience.
Figure 8: Balanced vs skewed training datasets. An extension of Figure 1, with
additional distributions. Data for this figure is described in Section 4.2
provided in Table 3 and Table 4 Figure 9: Race distribution results of
VGGFace2 on the 89 distributions identified in the main work. Each plot gives
percent accuracy values for a single test race. These plots roughly correspond
to flattened versions of the 4 nested 3-simplexes; note that each plot
consists of 4 connected equilateral triangles, where each triangle contains 4
parallel simplex faces projected into 2 dimensions, one from each simplex, as
well as the center point. The corners of the outer simplex, which correspond
to training distributions containing only data from a single race, are labeled
accordingly, i.e., the point labeled “Afn” is (100, 0, 0, 0). For readability,
some points are therefore represented multiple times, such as the center (25,
25, 25, 25), which appears in the middle of all triangles. Thus, each plot
contains 181 points, of which only 89 are unique. In addition to per-race
accuracy, overall average accuracy of the races is represented, as well as the
accuracy variance between the 4 per-race accuracy results. Tabular results in
Tables 5 and 6.

# Images per Race | Per-race Accuracy | Overall
---|---|---
African | Asian | Caucasian | Indian | African | Asian | Caucasian | Indian | Accuracy | Variance
1250 | 1250 | 1250 | 1250 | 71.68 | 71.70 | 80.68 | 75.25 | 74.83 | 13.53
1167 | 1167 | 1166 | 1500 | 71.83 | 71.70 | 79.68 | 75.08 | 74.58 | 10.53
1250 | 1167 | 1167 | 1416 | 71.87 | 71.57 | 80.60 | 75.40 | 74.86 | 13.26
1333 | 1167 | 1167 | 1333 | 72.87 | 71.98 | 80.28 | 74.52 | 74.91 | 10.44
1417 | 1167 | 1166 | 1250 | 72.60 | 71.40 | 80.47 | 74.38 | 74.71 | 12.16
1500 | 1167 | 1167 | 1166 | 73.00 | 71.77 | 79.32 | 75.15 | 74.81 | 8.24
1417 | 1250 | 1167 | 1166 | 73.83 | 71.08 | 80.57 | 74.82 | 75.07 | 11.93
1333 | 1333 | 1167 | 1167 | 71.98 | 71.25 | 79.22 | 74.57 | 74.25 | 9.73
1250 | 1417 | 1167 | 1166 | 72.18 | 71.55 | 79.38 | 74.55 | 74.42 | 9.47
1167 | 1250 | 1167 | 1416 | 71.62 | 70.73 | 79.13 | 74.75 | 74.06 | 10.81
1167 | 1333 | 1167 | 1333 | 71.83 | 72.10 | 80.53 | 73.53 | 74.50 | 12.55
1167 | 1417 | 1166 | 1250 | 72.45 | 71.13 | 79.83 | 74.72 | 74.53 | 11.01
1167 | 1500 | 1167 | 1166 | 72.08 | 71.00 | 79.67 | 74.65 | 74.35 | 11.18
1167 | 1417 | 1250 | 1166 | 71.48 | 71.55 | 79.85 | 74.02 | 74.22 | 11.59
1167 | 1333 | 1333 | 1167 | 71.25 | 71.15 | 79.43 | 73.97 | 73.95 | 11.30
1167 | 1250 | 1417 | 1166 | 72.02 | 71.52 | 81.02 | 74.65 | 74.80 | 14.30
1167 | 1167 | 1250 | 1416 | 72.05 | 71.42 | 79.72 | 73.70 | 74.22 | 10.76
1167 | 1167 | 1333 | 1333 | 72.43 | 71.58 | 80.60 | 75.40 | 75.00 | 12.45
1167 | 1167 | 1416 | 1250 | 72.67 | 71.53 | 80.53 | 74.83 | 74.89 | 12.02
1167 | 1167 | 1500 | 1166 | 71.88 | 71.62 | 80.00 | 75.00 | 74.62 | 11.40
1250 | 1167 | 1417 | 1166 | 71.93 | 71.50 | 80.65 | 73.68 | 74.44 | 13.52
1333 | 1167 | 1333 | 1167 | 73.62 | 71.30 | 80.75 | 75.23 | 75.22 | 12.13
1417 | 1167 | 1250 | 1166 | 73.00 | 71.55 | 79.65 | 74.70 | 74.73 | 9.33

# Images per Race | Per-race Accuracy | Overall
---|---|---
African | Asian | Caucasian | Indian | African | Asian | Caucasian | Indian | Accuracy | Variance
1250 | 1250 | 1250 | 1250 | 71.68 | 71.70 | 80.68 | 75.25 | 74.83 | 13.53
1000 | 1000 | 1000 | 2000 | 71.08 | 71.78 | 79.65 | 74.33 | 74.21 | 11.32
1250 | 1000 | 1000 | 1750 | 72.02 | 72.40 | 78.62 | 74.38 | 74.35 | 6.86
1500 | 1000 | 1000 | 1500 | 72.68 | 71.22 | 79.90 | 74.37 | 74.54 | 10.81
1750 | 1000 | 1000 | 1250 | 72.92 | 72.15 | 79.53 | 74.82 | 74.85 | 8.24
2000 | 1000 | 1000 | 1000 | 74.53 | 71.85 | 80.33 | 75.57 | 75.57 | 9.40
1750 | 1250 | 1000 | 1000 | 72.17 | 72.28 | 79.30 | 74.12 | 74.47 | 8.39
1500 | 1500 | 1000 | 1000 | 72.98 | 71.65 | 80.33 | 74.32 | 74.82 | 11.02
1250 | 1750 | 1000 | 1000 | 72.27 | 71.20 | 79.72 | 73.98 | 74.29 | 10.80
1000 | 1250 | 1000 | 1750 | 70.83 | 71.15 | 78.83 | 73.47 | 73.57 | 10.26
1000 | 1500 | 1000 | 1500 | 71.80 | 71.00 | 79.42 | 73.68 | 73.98 | 10.82
1000 | 1750 | 1000 | 1250 | 70.48 | 70.70 | 79.20 | 73.28 | 73.42 | 12.36
1000 | 2000 | 1000 | 1000 | 71.18 | 71.62 | 78.80 | 73.57 | 73.79 | 9.17
1000 | 1750 | 1250 | 1000 | 72.02 | 71.47 | 79.42 | 74.08 | 74.25 | 9.86
1000 | 1500 | 1500 | 1000 | 72.28 | 72.42 | 81.07 | 73.58 | 74.84 | 13.19
1000 | 1250 | 1750 | 1000 | 71.80 | 72.47 | 81.73 | 74.72 | 75.18 | 15.49
1000 | 1000 | 1250 | 1750 | 71.88 | 70.85 | 80.00 | 74.47 | 74.30 | 12.57
1000 | 1000 | 1500 | 1500 | 71.62 | 71.05 | 80.67 | 74.12 | 74.36 | 14.58
1000 | 1000 | 1750 | 1250 | 71.80 | 71.78 | 81.40 | 74.78 | 74.94 | 15.40
1000 | 1000 | 2000 | 1000 | 71.55 | 71.28 | 81.37 | 74.78 | 74.75 | 16.51
1250 | 1000 | 1750 | 1000 | 72.35 | 71.78 | 80.52 | 74.85 | 74.88 | 11.94
1500 | 1000 | 1500 | 1000 | 72.60 | 71.68 | 80.63 | 75.82 | 75.18 | 12.26
1750 | 1000 | 1250 | 1000 | 73.82 | 71.48 | 80.35 | 74.58 | 75.06 | 10.64

Table 3: Results for ArcFace distribution experiments for the two innermost
simplexes. See Table 4 for results for the outermost simplexes. See Figure 6
for these same results in plot form.

# Images per Race | Per-race Accuracy | Overall
---|---|---
African | Asian | Caucasian | Indian | African | Asian | Caucasian | Indian | Accuracy | Variance
1250 | 1250 | 1250 | 1250 | 71.68 | 71.70 | 80.68 | 75.25 | 74.83 | 13.53
667 | 667 | 666 | 3000 | 70.42 | 70.68 | 78.32 | 73.17 | 73.15 | 10.06
1250 | 667 | 666 | 2417 | 72.03 | 71.22 | 79.13 | 75.17 | 74.39 | 9.68
1833 | 667 | 667 | 1833 | 73.52 | 71.60 | 79.68 | 75.27 | 75.02 | 8.94
2417 | 667 | 666 | 1250 | 75.15 | 71.53 | 79.87 | 75.38 | 75.48 | 8.73
3000 | 667 | 667 | 666 | 75.97 | 72.05 | 80.73 | 75.82 | 76.14 | 9.49
2417 | 1250 | 667 | 666 | 75.88 | 71.70 | 79.42 | 75.50 | 75.62 | 7.47
1833 | 1833 | 667 | 667 | 72.72 | 71.63 | 79.57 | 75.25 | 74.79 | 9.32
1250 | 2417 | 667 | 666 | 72.05 | 71.30 | 77.97 | 72.98 | 73.58 | 6.78
667 | 1250 | 666 | 2417 | 70.82 | 71.70 | 78.17 | 73.53 | 73.55 | 8.05
667 | 1833 | 667 | 1833 | 69.80 | 71.10 | 77.90 | 73.17 | 72.99 | 9.47
667 | 2417 | 666 | 1250 | 68.77 | 70.08 | 77.15 | 72.20 | 72.05 | 10.17
667 | 3000 | 667 | 666 | 68.78 | 69.87 | 76.97 | 72.32 | 71.98 | 9.92
667 | 2417 | 1250 | 666 | 70.12 | 71.07 | 79.55 | 73.02 | 73.44 | 13.55
667 | 1833 | 1833 | 667 | 69.68 | 71.32 | 80.33 | 74.08 | 73.85 | 16.47
667 | 1250 | 2417 | 666 | 69.93 | 71.93 | 81.73 | 74.95 | 74.64 | 19.97
667 | 666 | 1250 | 2417 | 70.62 | 70.27 | 79.93 | 73.88 | 73.67 | 15.05
667 | 667 | 1833 | 1833 | 70.17 | 71.67 | 80.93 | 75.00 | 74.44 | 17.11
667 | 666 | 2417 | 1250 | 71.80 | 71.88 | 81.63 | 73.68 | 74.75 | 16.36
667 | 667 | 3000 | 666 | 70.78 | 72.17 | 82.45 | 75.57 | 75.24 | 20.35
1250 | 667 | 2417 | 666 | 72.60 | 72.83 | 81.78 | 76.10 | 75.83 | 13.73
1833 | 667 | 1833 | 667 | 74.52 | 72.47 | 81.58 | 76.12 | 76.17 | 11.44
2417 | 667 | 1250 | 666 | 75.75 | 71.92 | 81.05 | 75.58 | 76.07 | 10.60

# Images per Race | Per-race Accuracy | Overall
---|---|---
African | Asian | Caucasian | Indian | African | Asian | Caucasian | Indian | Accuracy | Variance
1250 | 1250 | 1250 | 1250 | 71.68 | 71.70 | 80.68 | 75.25 | 74.83 | 13.53
0 | 0 | 0 | 5000 | 58.38 | 66.17 | 70.57 | 74.32 | 67.36 | 35.17
1250 | 0 | 0 | 3750 | 71.33 | 67.37 | 73.18 | 73.60 | 71.37 | 6.07
2500 | 0 | 0 | 2500 | 74.85 | 69.60 | 74.32 | 74.50 | 73.32 | 4.64
3750 | 0 | 0 | 1250 | 78.03 | 69.87 | 77.10 | 76.58 | 75.40 | 10.46
5000 | 0 | 0 | 0 | 78.92 | 71.05 | 77.28 | 76.65 | 75.97 | 8.77
3750 | 1250 | 0 | 0 | 76.78 | 70.40 | 76.40 | 76.00 | 74.90 | 6.81
2500 | 2500 | 0 | 0 | 72.80 | 68.87 | 74.02 | 72.75 | 72.11 | 3.76
1250 | 3750 | 0 | 0 | 71.52 | 70.15 | 72.05 | 70.43 | 71.04 | 0.60
0 | 1250 | 0 | 3750 | 56.57 | 64.72 | 63.62 | 65.02 | 62.48 | 11.92
0 | 2500 | 0 | 2500 | 55.47 | 65.80 | 63.97 | 64.58 | 62.45 | 16.71
0 | 3750 | 0 | 1250 | 56.12 | 65.03 | 61.52 | 63.20 | 61.47 | 11.09
0 | 5000 | 0 | 0 | 55.73 | 66.03 | 62.98 | 62.60 | 61.84 | 14.19
0 | 3750 | 1250 | 0 | 62.55 | 71.00 | 78.43 | 72.10 | 71.02 | 31.97
0 | 2500 | 2500 | 0 | 63.67 | 72.03 | 81.35 | 73.30 | 72.59 | 39.29
0 | 1250 | 3750 | 0 | 64.23 | 71.50 | 82.80 | 74.70 | 73.31 | 44.41
0 | 0 | 1250 | 3750 | 63.23 | 69.72 | 79.02 | 74.00 | 71.49 | 33.57
0 | 0 | 2500 | 2500 | 64.83 | 70.78 | 81.98 | 74.82 | 73.10 | 38.89
0 | 0 | 3750 | 1250 | 64.13 | 72.58 | 83.27 | 75.25 | 73.81 | 46.66
0 | 0 | 5000 | 0 | 66.48 | 73.05 | 84.15 | 76.92 | 75.15 | 40.91
1250 | 0 | 3750 | 0 | 74.78 | 72.50 | 83.67 | 76.65 | 76.90 | 17.42
2500 | 0 | 2500 | 0 | 76.38 | 72.90 | 83.68 | 77.10 | 77.52 | 15.20
3750 | 0 | 1250 | 0 | 78.68 | 73.58 | 82.30 | 77.83 | 78.10 | 9.61

Table 4: Results for ArcFace distribution experiments for the two outermost
simplexes. See Table 3 for results for the innermost simplexes. See Figure 6
for these same results in plot form.

# Images per Race | Per-race Accuracy | Overall
---|---|---
African | Asian | Caucasian | Indian | African | Asian | Caucasian | Indian | Accuracy | Variance
1250 | 1250 | 1250 | 1250 | 66.95 | 68.97 | 75.82 | 69.73 | 70.37 | 10.93
1167 | 1167 | 1166 | 1500 | 67.60 | 69.32 | 76.25 | 70.00 | 70.79 | 10.70
1250 | 1167 | 1167 | 1416 | 67.33 | 69.67 | 76.03 | 70.62 | 70.91 | 10.17
1333 | 1167 | 1167 | 1333 | 66.70 | 68.63 | 76.35 | 70.72 | 70.60 | 13.04
1417 | 1167 | 1166 | 1250 | 68.72 | 69.23 | 76.58 | 71.65 | 71.55 | 9.68
1500 | 1167 | 1167 | 1166 | 67.25 | 68.73 | 75.98 | 71.22 | 70.80 | 10.98
1417 | 1250 | 1167 | 1166 | 67.73 | 69.50 | 76.00 | 70.93 | 71.04 | 9.48
1333 | 1333 | 1167 | 1167 | 67.55 | 69.28 | 75.73 | 71.45 | 71.00 | 9.36
1250 | 1417 | 1167 | 1166 | 67.62 | 69.22 | 75.57 | 71.43 | 70.96 | 8.92
1167 | 1250 | 1167 | 1416 | 66.98 | 70.00 | 76.08 | 71.20 | 71.07 | 10.75
1167 | 1333 | 1167 | 1333 | 66.47 | 68.98 | 76.42 | 70.53 | 70.60 | 13.38
1167 | 1417 | 1166 | 1250 | 67.18 | 69.12 | 75.88 | 70.45 | 70.66 | 10.45
1167 | 1500 | 1167 | 1166 | 68.15 | 68.90 | 76.03 | 70.13 | 70.80 | 9.62
1167 | 1417 | 1250 | 1166 | 66.18 | 67.95 | 74.50 | 70.00 | 69.66 | 9.64
1167 | 1333 | 1333 | 1167 | 67.02 | 68.73 | 75.45 | 70.27 | 70.37 | 9.94
1167 | 1250 | 1417 | 1166 | 67.30 | 69.28 | 76.08 | 70.13 | 70.70 | 10.72
1167 | 1167 | 1250 | 1416 | 67.18 | 69.00 | 77.12 | 70.48 | 70.95 | 14.06
1167 | 1167 | 1333 | 1333 | 66.85 | 68.40 | 76.67 | 70.20 | 70.53 | 13.96
1167 | 1167 | 1416 | 1250 | 68.02 | 68.82 | 76.25 | 70.92 | 71.00 | 10.31
1167 | 1167 | 1500 | 1166 | 67.90 | 69.68 | 76.50 | 70.87 | 71.24 | 10.35
1250 | 1167 | 1417 | 1166 | 67.78 | 69.82 | 76.43 | 70.82 | 71.21 | 10.28
1333 | 1167 | 1333 | 1167 | 68.08 | 69.28 | 76.58 | 70.43 | 71.10 | 10.73
1417 | 1167 | 1250 | 1166 | 67.37 | 68.92 | 76.08 | 71.13 | 70.88 | 10.83

# Images per Race | Per-race Accuracy | Overall
---|---|---
African | Asian | Caucasian | Indian | African | Asian | Caucasian | Indian | Accuracy | Variance
1250 | 1250 | 1250 | 1250 | 66.95 | 68.97 | 75.82 | 69.73 | 70.37 | 10.93
1000 | 1000 | 1000 | 2000 | 66.32 | 69.25 | 75.75 | 70.33 | 70.41 | 11.66
1250 | 1000 | 1000 | 1750 | 67.52 | 69.25 | 75.47 | 70.68 | 70.73 | 8.74
1500 | 1000 | 1000 | 1500 | 67.58 | 68.45 | 75.20 | 70.98 | 70.55 | 8.76
1750 | 1000 | 1000 | 1250 | 67.75 | 69.52 | 75.52 | 70.70 | 70.87 | 8.30
2000 | 1000 | 1000 | 1000 | 68.73 | 70.03 | 76.82 | 70.78 | 71.59 | 9.64
1750 | 1250 | 1000 | 1000 | 67.33 | 68.88 | 75.65 | 69.92 | 70.45 | 9.87
1500 | 1500 | 1000 | 1000 | 67.80 | 68.52 | 75.30 | 70.33 | 70.49 | 8.57
1250 | 1750 | 1000 | 1000 | 66.67 | 67.80 | 75.60 | 69.75 | 69.95 | 11.84
1000 | 1250 | 1000 | 1750 | 66.52 | 68.98 | 75.32 | 71.12 | 70.48 | 10.44
1000 | 1500 | 1000 | 1500 | 66.33 | 69.05 | 75.78 | 70.08 | 70.31 | 11.85
1000 | 1750 | 1000 | 1250 | 66.88 | 69.65 | 75.10 | 70.30 | 70.48 | 8.75
1000 | 2000 | 1000 | 1000 | 65.00 | 68.50 | 75.60 | 70.75 | 69.96 | 14.79
1000 | 1750 | 1250 | 1000 | 66.82 | 68.52 | 75.68 | 70.07 | 70.27 | 11.09
1000 | 1500 | 1500 | 1000 | 66.45 | 69.15 | 76.08 | 69.97 | 70.41 | 12.41
1000 | 1250 | 1750 | 1000 | 67.38 | 69.25 | 77.18 | 70.15 | 70.99 | 13.77
1000 | 1000 | 1250 | 1750 | 65.88 | 69.27 | 74.83 | 69.72 | 69.93 | 10.23
1000 | 1000 | 1500 | 1500 | 67.33 | 69.20 | 75.98 | 70.68 | 70.80 | 10.36
1000 | 1000 | 1750 | 1250 | 66.67 | 69.63 | 76.72 | 70.63 | 70.91 | 13.36
1000 | 1000 | 2000 | 1000 | 66.50 | 69.58 | 77.07 | 70.93 | 71.02 | 14.77
1250 | 1000 | 1750 | 1000 | 68.67 | 68.32 | 76.55 | 69.65 | 70.80 | 11.28
1500 | 1000 | 1500 | 1000 | 67.57 | 69.87 | 76.90 | 70.57 | 71.23 | 11.97
1750 | 1000 | 1250 | 1000 | 68.55 | 69.93 | 76.87 | 70.43 | 71.45 | 10.27

Table 5: Results for VGGFace2 distribution experiments for the two innermost
simplexes. See Table 6 for results for the outermost simplexes. See Figure 9
for these same results in plot form.

# Images per Race | Per-race Accuracy | Overall
---|---|---
African | Asian | Caucasian | Indian | African | Asian | Caucasian | Indian | Accuracy | Variance
1250 | 1250 | 1250 | 1250 | 66.95 | 68.97 | 75.82 | 69.73 | 70.37 | 10.93
667 | 667 | 666 | 3000 | 66.77 | 68.33 | 74.28 | 70.43 | 69.95 | 7.94
1250 | 667 | 666 | 2417 | 66.65 | 68.88 | 75.15 | 71.20 | 70.47 | 9.89
1833 | 667 | 667 | 1833 | 68.32 | 68.92 | 75.22 | 71.52 | 70.99 | 7.40
2417 | 667 | 666 | 1250 | 68.90 | 69.23 | 75.55 | 71.43 | 71.28 | 7.03
3000 | 667 | 667 | 666 | 68.68 | 69.98 | 75.63 | 70.07 | 71.09 | 7.18
2417 | 1250 | 667 | 666 | 68.70 | 69.88 | 75.63 | 71.55 | 71.44 | 6.88
1833 | 1833 | 667 | 667 | 67.68 | 69.97 | 75.45 | 71.52 | 71.15 | 8.01
1250 | 2417 | 667 | 666 | 67.65 | 67.75 | 74.27 | 70.07 | 69.93 | 7.19
667 | 1250 | 666 | 2417 | 66.22 | 68.93 | 73.73 | 70.67 | 69.89 | 7.45
667 | 1833 | 667 | 1833 | 66.28 | 68.62 | 74.20 | 69.30 | 69.60 | 8.30
667 | 2417 | 666 | 1250 | 66.98 | 68.43 | 74.37 | 70.48 | 70.07 | 7.71
667 | 3000 | 667 | 666 | 65.35 | 68.53 | 73.28 | 68.48 | 68.91 | 8.03
667 | 2417 | 1250 | 666 | 65.70 | 69.47 | 75.95 | 70.35 | 70.37 | 13.44
667 | 1833 | 1833 | 667 | 64.92 | 69.12 | 75.75 | 70.27 | 70.01 | 14.94
667 | 1250 | 2417 | 666 | 65.13 | 70.48 | 77.32 | 70.73 | 70.92 | 18.66
667 | 666 | 1250 | 2417 | 66.70 | 68.73 | 75.87 | 70.47 | 70.44 | 11.59
667 | 667 | 1833 | 1833 | 66.43 | 68.60 | 76.33 | 70.42 | 70.45 | 13.54
667 | 666 | 2417 | 1250 | 65.58 | 69.78 | 77.60 | 70.63 | 70.90 | 18.62
667 | 667 | 3000 | 666 | 65.88 | 70.45 | 77.45 | 70.70 | 71.12 | 17.03
1250 | 667 | 2417 | 666 | 67.25 | 69.85 | 77.05 | 71.20 | 71.34 | 12.89
1833 | 667 | 1833 | 667 | 68.77 | 70.15 | 76.67 | 70.75 | 71.58 | 9.13
2417 | 667 | 1250 | 666 | 69.55 | 69.57 | 77.13 | 71.40 | 71.91 | 9.65

# Images per Race | Per-race Accuracy | Overall
---|---|---
African | Asian | Caucasian | Indian | African | Asian | Caucasian | Indian | Accuracy | Variance
1250 | 1250 | 1250 | 1250 | 66.95 | 68.97 | 75.82 | 69.73 | 70.37 | 10.93
0 | 0 | 0 | 5000 | 56.75 | 65.33 | 67.72 | 69.63 | 64.86 | 24.24
1250 | 0 | 0 | 3750 | 66.83 | 65.90 | 70.07 | 69.08 | 67.97 | 2.80
2500 | 0 | 0 | 2500 | 68.82 | 67.72 | 72.15 | 70.62 | 69.83 | 2.87
3750 | 0 | 0 | 1250 | 69.53 | 69.20 | 71.73 | 70.98 | 70.36 | 1.08
5000 | 0 | 0 | 0 | 70.80 | 69.68 | 73.13 | 70.82 | 71.11 | 1.58
3750 | 1250 | 0 | 0 | 70.28 | 69.60 | 72.20 | 69.35 | 70.36 | 1.25
2500 | 2500 | 0 | 0 | 68.10 | 68.40 | 69.85 | 68.80 | 68.79 | 0.44
1250 | 3750 | 0 | 0 | 66.60 | 67.45 | 68.97 | 68.00 | 67.75 | 0.74
0 | 1250 | 0 | 3750 | 54.90 | 63.37 | 62.20 | 63.20 | 60.92 | 12.27
0 | 2500 | 0 | 2500 | 54.98 | 63.48 | 61.10 | 62.87 | 60.61 | 11.31
0 | 3750 | 0 | 1250 | 54.92 | 62.97 | 61.18 | 61.37 | 60.11 | 9.47
0 | 5000 | 0 | 0 | 56.02 | 64.97 | 62.75 | 61.88 | 61.40 | 10.94
0 | 3750 | 1250 | 0 | 60.98 | 67.83 | 74.58 | 68.12 | 67.88 | 23.14
0 | 2500 | 2500 | 0 | 63.07 | 68.23 | 77.50 | 70.08 | 69.72 | 26.78
0 | 1250 | 3750 | 0 | 63.25 | 71.12 | 78.28 | 70.65 | 70.82 | 28.28
0 | 0 | 1250 | 3750 | 59.98 | 67.72 | 75.08 | 70.32 | 68.27 | 29.90
0 | 0 | 2500 | 2500 | 62.42 | 67.78 | 76.88 | 69.83 | 69.23 | 26.86
0 | 0 | 3750 | 1250 | 62.87 | 69.23 | 77.57 | 71.85 | 70.38 | 27.89
0 | 0 | 5000 | 0 | 62.83 | 70.90 | 78.52 | 70.93 | 70.80 | 30.76
1250 | 0 | 3750 | 0 | 69.05 | 70.70 | 78.75 | 70.98 | 72.37 | 14.11
2500 | 0 | 2500 | 0 | 69.17 | 70.05 | 78.42 | 71.20 | 72.21 | 13.37
3750 | 0 | 1250 | 0 | 70.65 | 70.18 | 77.45 | 71.32 | 72.40 | 8.66

Table 6: Results for VGGFace2 distribution experiments for the two outermost
simplexes. See Table 5 for results for the innermost simplexes. See Figure 9
for these same results in plot form.


# The nonlocal advection diffusion equation and the two-slit experiment in
quantum mechanics

Glenn Webb  Glenn Webb
Mathematics Department, Vanderbilt University, Nashville, TN 37212 USA
<EMAIL_ADDRESS>Dedicated to the memory of John W. Neuberger

###### Abstract.

A partial differential equation model is analyzed for the two-slit experiment
of quantum mechanics. The state variable of the equation is the probability
density function of particle positions. The equation has a diffusion term
corresponding to the random movement of particles, and a nonlocal advection
term corresponding to the movement of particles in the transverse direction
perpendicular to their forward movement. The model is compared to the
Schrödinger equation model of the experiment. The model supports the ensemble
interpretation of quantum mechanics.

###### Key words and phrases:

Nonlocal, advection, diffusion, Schrödinger equation, two-slit experiment,
ensemble interpretation;

###### 2020 Mathematics Subject Classification:

35J10, 35Q40

###### Contents

  1. 1 Introduction
  2. 2 The Schrödinger equation model
    1. 2.1 The Schrödinger Equation with Step Function Initial Data
  3. 3 The nonlocal advection diffusion equation model
    1. 3.1 Analysis of Equation (3.4)
  4. 4 Numerical simulations of the models
  5. 5 Summary

##  1\. Introduction

The two-slit experiment demonstrates the fundamental probabilistic nature of
quantum mechanics. In this experiment quantum particles are projected forward
toward a screen with two parallel slits, and then observed on a detection
screen further downstream (Figure 1). An interference diffraction pattern of
regularly spaced intensities is registered on the detection screen. The
highest. density occurs in the center of the detection screen (Figure 2),
which is not the sum of the patterns observed for single slits separately
([8]). The observed fringe pattern for two slits is characteristic of wave
phenomena.

Figure 1. The two slit experiment with electrons. wikipedia.org$/$wiki/Double-
slit experiment Figure 2. The interference diffraction pattern of a two slit
experiment with high order fringe patterns. wikipedia.org$/$wiki/Double-slit
experiment

In previous work we have investigated mathematical models for the two-slit
experiment ([9], [10]): the Schrödinger equation model (SE) and the nonlocal
advection diffusion equation model (NLAD). In this work we extend these
investigations to account for higher levels in the fringe patterns images in
the experiment. We numerically simulate the SE and NLAD model outputs and
compare the simulations to experimental data.

The organization of this paper is as follows: In Section 2 we develop the SE
model, in Section 3 we develop the NLAD model, in Section 4 we provide
numerical simulations of both models and compare these outputs to experimental
images, and in Section 5 we provide a summary of out work.

##  2\. The Schrödinger equation model

The one-dimensional time-dependent complex-valued Schrödinger equation is the
foundational phenomenological model of quantum mechanics:

$\frac{\partial}{\partial
t}\psi(x,t)\,=\,i\frac{\hbar}{2m}\,\frac{\partial^{2}}{\partial
x^{2}}\psi(x,t),\,t>0,\,\,\psi(x,0)\,=\,\psi_{0}(x),\,\,-\infty<x<\infty.$
(2.1)

Here $\hbar$ is the reduced Planck’s constant and $m$ is the particle mass,
which without loss of generality can be assumed to satisfy
$\frac{\hbar}{m}=1$. The interpretation of the solution is that
$\int_{x_{1}}^{x_{2}}\rho(x,t)dx$ is the probability of finding a single
particle in the interval $(x_{1},x_{2})$ at time $t$, where
$\rho(x,t)=|Re\,\psi(x,t)|^{2}+|Im\,\psi(x,t)|^{2}$, and $\rho(x,0)$ is
normalized so that $\int_{-\infty}^{\infty}\rho(x,0)dx=1$. In this
formulation, the interpretation of the state variable $\psi$ at time $t$ is
sometimes applied to a single individual quantum particle. For this
interpretation, the following questions arise: What do the real and imaginary
parts of $\psi$ represent for an individual particle? What is time $t$ in an
experiment with randomly separated independent temporal events? What does the
initial condition $\psi_{0}$ correspond to for single particles emitted one at
a time?

The general solution of (2.1) (with scaling $\hbar/2m=1$), is as in [2]:

$\psi(x,t)=\frac{1}{\sqrt{2\pi
it}}\int_{-\infty}^{\infty}e^{{\frac{i(x-y)^{2}}{2t}}}\psi_{0}(y)dy,\,\,\,\psi_{0}\in
L^{1}((R;C),y^{2}dy)\,\cap\,L^{2}(R;C),$ (2.2)

where

$\rho(x,t)=\frac{1}{2\pi
t}\Bigg{(}\bigg{(}\int_{-\infty}^{\infty}\cos\bigg{(}\frac{(x-y)^{2}}{2t}\bigg{)}Re\psi_{0}(y)-\sin\bigg{(}\frac{(x-y)^{2}}{2t}\bigg{)}Im\psi_{0}(y)dy\bigg{)}^{2}$
(2.3)

$+\bigg{(}\int_{-\infty}^{\infty}\sin\bigg{(}\frac{(x-y)^{2}}{2t}\bigg{)}Re\psi_{0}(y)dy+\cos\bigg{(}\frac{(x-y)^{2}}{2t}\bigg{)}Im\psi_{0}(y)dy\bigg{)}^{2}\Bigg{)},\\\
$

with

$\int_{-\infty}^{\infty}\rho(x,0)dx=\int_{-\infty}^{\infty}|\psi(x,0)|^{2}dx=1,\text{
which implies }\int_{-\infty}^{\infty}\rho(x,t)dx=1,t\geq 0.$

The probability amplitude $\rho(x,t)$ in (2.3) exhibits a two-phase pattern as
$t$ advances. In the first phase, the initial information $\rho(x,0)$ evolves
to an established pattern, in which the lower peaks in the fringe pattern lie
almost on the $x$-axis. In the second phase, this established pattern
undergoes a space-time dilation as time advances. In the second phase the
profile of $\rho(x,t)$ is propagated in the spatial $x$-direction at a
constant speed. In [9] it is proved that for general initial data $\psi_{0}$
in (2.3) the probability amplitude $\rho(x,t)$ in (2.3) satisfies the
asymptotic space-time dilation property: uniformly for $x\in R,\,T>0,\,t\geq
1$

$\Bigg{|}\rho(x,t\,T)\,-\,\frac{1}{t}\rho(\frac{x}{t},T)\Bigg{|}\,\leq\,\frac{\sqrt{2}}{\pi
tT^{2}}\Bigg{(}\int_{-\infty}^{\infty}y^{2}\,|\psi_{0}(y)|dy\Bigg{)}\Bigg{(}\int_{-\infty}^{\infty}|\psi_{0}(y)|dy\Bigg{)},$
(2.4)

which implies

$\lim_{t\rightarrow\infty}t\,\rho(x,t)=\lim_{T\rightarrow\infty}tT\,\rho(x,tT)=\frac{1}{2\pi}|\int_{-\infty}^{\infty}\psi_{0}(y)dy|^{2};$
(2.5)

uniformly for $x\in R$.

###  2.1. The Schrödinger Equation with Step Function Initial Data

In [10] an example for the Schrödinger equation applied to the two-slit
experiment was given with initial data consisting of two rectangular step
functions, symmetric about the origin of the $x$-axis. In this example, the
centers of the two rectangular strips are taken as $s=\pm 1$, which can be
applied generally by scaling the spatial variable $x$. The probability density
$\rho(x,t)$ for this example evolves from the first phase to the second phase.
In Figure 3, $\rho(x,t)$ is illustrated at $t=1/\pi$, which is approximately
the value of $t$ at which the transition from the first phase to the second
phase occurs.

Figure 3. $\rho(x,1/\pi)$ for $\rho(x,(0)=$ two rectangles with centers $s=\pm
1$, width $2b,\,b=0.1$, and height scaled so that
$\int_{-s-b}^{-s+b}\rho(x,0)dx\,+\,\int_{s-b}^{s+b}\rho(x,0)dx=1$. Left:
$\rho(x,1/\pi)$, the inset is the graph of $\rho(x,0)$. The coordinates $x=\pm
10,\pm 20,\pm 30,\dots$ are local maxima for $\rho(x,1/\pi)$ and $x\approx\pm
15,\pm 25,\pm 35,\dots$ are local minima for $\rho(x,1/\pi)$. Right:
$\log\rho(x,1/\pi)$.

######  Remark 2.1.

In Figure 3 it is seen that the fringe pattern has multiple interference
pattern groupings on either side of a central principal pattern. The
diffraction pattern of $\rho(x,t)$ has infinitely many regularly spaced
intervals

$\dots,[-30,-20],\,[-20,-10],[-10,10],[10,20],[20,30],\dots,$

with approximately regularly spaced interior fringes repeated in each of the
intervals. This illustration is consistent with the experimental data in
Figure 2. The spacing is determined by $s/b=10$.

##  3\. The nonlocal advection diffusion equation model

An alternative to the Schrödinger equation formulation of the two slit
experiment is the nonlocal advection diffusion equation formulation NLAD. This
formulation incorporates the movement at a given spatial $x$ coordinate as
dependent on nearby spatial $x$ coordinates. Such models have been developed
in [1] and [10]. The NLAD model supports the ensemble interpretation of
quantum mechanics, which maintains that mathematical description of particle
behavior should correspond to communal particle behavior, rather than to
individual particle behavior [3].

In [1] the authors examine an interpretation of the two-slit experiment based
on the nonlocal interaction of a single particle with both slit openings. In
their interpretation a single particle only passes through one slit, but with
its corpuscular motion affected by the other slit. The authors provide a
deterministic nonlocal dynamic equation of motion for the density of particles
and relate the behavior of the solutions to the interference patterns observed
in the two-slit experiment. This interpretation provides an alternative to the
classical Schrödinger equation interpretation of this experiment as an
ensemble wave of particles passing through both slits.

In the NLAD model in [10], the nonlocal advection term represents directed
particle movement due to the influence of nearby particles, and the diffusion
term represents variability of particle movement due to stochastic variation.
The nonlocal advection term involves an integral corresponding to the slit
separation widths $s$ in the $\pm x$-directions. In [10], the NLAD equation
analyzed was the following:

$\displaystyle\frac{\partial}{\partial
t}\omega(x,t)\,=\,\alpha\,\frac{\partial^{2}}{\partial
x^{2}}\omega(x,t)+\frac{\partial}{\partial
x}\int_{-s}^{s}\,\beta_{0}\,\omega(x+\hat{x},t)\frac{\hat{x}}{|\hat{x}|}d\hat{x}\hskip
267.39877pt$ (3.1) $\displaystyle=\alpha\,\frac{\partial^{2}}{\partial
x^{2}}\omega(x,t)+\beta_{0}\bigg{(}\omega(x+s,t)-2\omega(x,t)+\omega(x-s,t)\bigg{)},\,t>0,\,-\infty\,<x<\infty,\hskip
180.67499pt$ (3.2)
$\omega(x,0)\,=\,\omega_{0}(x),-\infty\,<x<\infty,\,\omega_{0}\in
L_{+}^{1}(-\infty,\infty),\,\int_{-\infty}^{\infty}\omega_{0}(x)dx=1.$

In equation (3.1) $x$ is the spatial coordinate of particles, $t$ is the
downstream distance perpendicular to the slits openings, $\alpha$ is the
diffusion parameter, $\beta_{0}$ is the advection parameter, $2s$ is the slit
width, $\omega_{0}(x)$ is the initial data, and $\omega(x,t)$ is the
probability density function for the distribution of particle positions. The
values of $\alpha$ and $\beta_{0}$ are chosen so that the solutions of (2.1)
and (3.1) are similar for $x\in[-20,20]$ at $t=1/\pi$.

In this work we analyze an extension of equation (3.2) to include higher order
fringe pattern levels as seen in Figure (2). The extension of (3.2) to include
higher order fringe pattern levels was illustrated in ([11]) with numerical
examples. Equation (3.2) is modified as follows:

$\displaystyle\frac{\partial}{\partial
t}\omega(x,t)\,=\,\alpha\,\frac{\partial^{2}}{\partial
x^{2}}\omega(x,t)+\frac{\partial}{\partial
x}\int_{-s}^{s}\,\beta_{0}\,\omega(x+\hat{x},t)\frac{\hat{x}}{|\hat{x}|}d\hat{x}\hskip
317.98755pt$ (3.3) $\displaystyle+\frac{\partial}{\partial
x}\int_{-\frac{3}{2b}}^{\frac{3}{2b}}\beta_{1}\,\omega(x+\hat{x},t)\frac{\hat{x}}{|\hat{x}|}d\hat{x}\,+\frac{\partial}{\partial
x}\int_{-\frac{5}{2b}}^{\frac{5}{2b}}\beta_{2}\,\omega(x+\hat{x},t)\frac{\hat{x}}{|\hat{x}|}d\hat{x}\hskip
267.39877pt$ $\displaystyle=\alpha\,\frac{\partial^{2}}{\partial
x^{2}}\omega(x,t)+\beta_{0}\bigg{(}\omega(x+s,t)-2\omega(x,t)+\omega(x-s,t)\bigg{)}\hskip
267.39877pt$
$\displaystyle+\beta_{1}\bigg{(}\omega(x+\frac{3s}{2b},t)-2\omega(x,t)+\omega(x-\frac{3s}{2b},t)\bigg{)}\hskip
361.34999pt$
$\displaystyle+\beta_{2}\bigg{(}\omega(x+\frac{5s}{2b},t)-2\omega(x,t)+\omega(x-\frac{5s}{2b},t)\bigg{)},\,t>0,\,-\infty\,<x<\infty,\hskip
245.71754pt$ (3.4)
$\displaystyle\omega(x,0)\,=\,\omega_{0}(x),-\infty\,<x<\infty,\,\omega_{0}\in
L_{+}^{1}(-\infty,\infty),\,\int_{-\infty}^{\infty}\omega_{0}(x)dx=1.\hskip
252.94499pt$

Equation (3.4) has two additional fringe pattern levels on either sided of the
origin: $[-\frac{4s}{b},-\frac{3s}{b}]$, $[-\frac{3s}{b},-\frac{2s}{b}]$,
$[\frac{2s}{b},\frac{3s}{b}]$, $[\frac{3s}{b},\frac{4s}{b}]$. Additional
levels can be added.

###  3.1. Analysis of Equation (3.4)

Let $X\,=\,L^{1}(-\infty,\infty)$, the space of integrable functions on
$(-\infty,\infty)$ with norm $\|f\|\,=\,\int_{-\infty}^{\infty}|f(x)|dx$. For
$\sigma>0$ let

$\displaystyle(T_{\sigma}(t)f)(x)=$
$\displaystyle\frac{1}{2\,\sqrt{\sigma\,t}}\,\int_{-\infty}^{\infty}\exp\bigg{(}-\frac{(x-y)^{2}}{4\,\sigma\,t}\bigg{)}\,f(y)\,dy,\,\,f\in
X,\,\,t\,>\,0,\,\,-\infty\,<x\,<\,\infty.$ (3.5)

$T_{\sigma}(t),t\geq 0$ is a strongly continuous holomorphic semigroup of
positive linear operators in $X$ with infinitesimal generator
$(A_{\sigma}f)(x)=\sigma\,d^{2}f(x)/dx^{2}$ satisfying $|T_{\sigma}(t)|\leq
1,t\geq 0$ ([5]). Further, $(T_{\sigma}(t)f)(x)$ is the strong solution in $X$
to the diffusion equation

$\frac{\partial}{\partial x}u(x,t)\,=\,\sigma\,\frac{\partial^{2}}{\partial
x^{2}}u(x,t),\,\,u(x,0)\,=\,f(x),\,\,t\,>\,0,\,\,-\infty\,<x\,<\,\infty,\,f\in
X.$ (3.6)

and ([12])

$\int_{-\infty}^{\infty}(T_{\sigma}(t)f)(x)dx\,=\,\int_{-\infty}^{\infty}f(x)dxf\in
X.$ (3.7)

For a bounded linear operator $B$ in $X$ define the exponential of $B$ as

$\exp(tB)f\,=\,\sum_{n=0}^{\infty}\frac{(tB)^{n}}{n!}f,\,f\in X,\,t\geq 0.$

Define the bounded linear operators $B_{0,\pm},B_{1,\pm},B_{2,\pm}$ in $X$ as
follows: for $f\in X$, $-\infty<x<\infty$,

$(B_{0,\pm}f)(x)=\beta_{0}f(x\pm s),$
$(B_{1,\pm}f)(x)=\beta_{1}f(x\pm\frac{3s}{2b}),$
$(B_{2,\pm}f)(x)=\beta_{2}f(x\pm\frac{5s}{2b}).$

Define the bounded linear operators $B_{0},\,B_{1},\,B_{2}$ in $X$ as follows:

$(B_{0}f)(x)=\beta_{0}\bigg{(}f(x+s)-2f(x)+f(x-s)\bigg{)},$
$(B_{1}f)(x)=\beta_{1}\bigg{(}f(x+\frac{3s}{2b})-2f(x)+f(x-\frac{3s}{2b}))\bigg{)},$
$(B_{2}f)(x)=\beta_{2}\bigg{(}f(x+s)-2f(x)+f(x-s)\bigg{)}.$

Then, since $B_{0,+},\,B_{0,-}$, $B_{1,+},\,B_{1,-}$, $B_{2,+},\,B_{2,-}$
commute,

$exp(tB_{0})=e^{-2\beta_{0}t}exp(tB_{0})=e^{-2\beta_{0}t}exp(tB_{0,+})exp(tB_{0,-}),$
$exp(tB_{1})=e^{-2\beta_{1}t}exp(tB_{1})=e^{-2\beta_{1}t}exp(tB_{1,+})exp(tB_{1,-}),$
$exp(tB_{2})=e^{-2\beta_{0}t}exp(tB_{2})=e^{-2\beta_{2}t}exp(tB_{2,+})exp(tB_{2,-}).$

######  Theorem 3.1.

Let $X\,=\,L^{1}(-\infty,\infty)$, let
$\alpha,\,s,\,b,\,\beta_{0},\beta_{1},\beta_{2}>0$, Let $T_{\alpha}(t),t\geq
0$ be the semigroup of linear operators as in equation (3.1). The unique
generalized solution of equation (3.4) is given by the strongly continuous
semigroup of linear operators $T(t),t\geq 0$ in $X$ with

$T(t)\omega_{0}\,=\,T_{\alpha}(t)\,exp(tB_{0})\,exp(tB_{1})\,exp(tB_{2})\,\omega_{0},\,t\geq
0,\,\omega_{0}\in X.$ (3.8)

Further, if

$\omega_{0}(x)\geq 0\,\text{ a.e.
on}\,(-\infty,\infty)\,\text{and}\,\int_{-\infty}^{\infty}\omega_{0}(x)dx\,=\,1,$
(3.9)

then

$(T(t)\omega_{0})(x)\geq 0\,\text{ a.e.
on}\,(-\infty,\infty)\,\text{and}\,\int_{-\infty}^{\infty}(T(t)\omega_{0})(x)dx\,=\,1.$
(3.10)

###### Proof.

For $\omega_{0}\in X$, $\omega_{0}(x)\geq 0\,\text{ a.e.
on}\,(-\infty,\infty)$,

$(B_{0,+}\omega_{0})(x)=\beta_{0}\,\omega_{0}(x+s)\,\geq 0\,\text{ a.e.
on}\,(-\infty,\infty),$
$(B^{2}_{0,+}\omega_{0})(x)=B_{0,+}\,(\beta_{0}(\omega_{0}(x+s))=\beta_{0}^{2}\,\omega_{0}(x+2s)\,\geq
0\,\text{ a.e. on}\,(-\infty,\infty),$
$(B^{3}_{0,+}\omega_{0})(x)=B^{2}_{0,+}\,(\beta_{0}(\omega_{0}(x+s))=\beta_{0}^{3}\,\omega_{0}(x+3s)\,\geq
0\,\text{ a.e. on}\,(-\infty,\infty)\dots.$

Thus,

$\exp(tB_{0,+})\omega_{0}\,=\,\sum_{n=0}^{\infty}\frac{t^{n}\,B_{0,+}^{n}}{n!}\omega_{0}\geq
0\,\text{ a.e. on}\,(-\infty,\infty).$

Similarly, $\exp(tB_{0,-})\omega_{0}$, $\exp(tB_{1,+})\omega_{0}$,
$\exp(tB_{1,-})\omega_{0}$, $\exp(tB_{2,+})\omega_{0}$,
$\exp(tB_{2,-})\omega_{0}$ $\geq 0\,\text{ a.e. on}\,(-\infty,\infty)$. From
([5]) $T_{\alpha}(t)\omega_{0}(x)\geq 0\,\text{ a.e. on}\,(-\infty,\infty)$.
Thus, for $t\geq 0$,

$T(t)\omega_{0}\,=\,T_{\alpha}(t)\,exp(tB_{0})\,exp(tB_{1})\,exp(tB_{2})\geq
0$

.

Let $\omega_{0}\in X$, $\omega_{0}(x)\geq 0\,\text{ a.e.
on}\,(-\infty,\infty)$, and $\int_{-\infty}^{\infty}\omega_{0}(x)dx\,=\,1$.
Then,

$\int_{-\infty}^{\infty}(B_{0}\omega_{0})(x)dx\,=\,\beta_{0}\,\Bigg{(}\int_{-\infty}^{\infty}\Bigg{(}\omega_{0}(x+s)-2\omega_{0}(x)+\omega_{0}(x-s)\bigg{)}dx\,=\,0,$
$\int_{-\infty}^{\infty}(B^{2}_{0}\omega_{0})(x)dx\,=\,\beta_{0}\,\Bigg{(}\int_{-\infty}^{\infty}\bigg{(}\omega_{0}(x+2s)-2\omega_{0}(x+s)+\omega_{0}(x)\bigg{)}\hskip
108.405pt$
$-2\bigg{(}\omega_{0}(x+s)-2\omega_{0}(x)\,+\omega_{0}(x-s)\bigg{)}+\bigg{(}\omega_{0}(x-2s)-2\omega_{0}(x-s)+\omega_{0}(x)\bigg{)}\Bigg{)}dx\,=\,0,$
$\int_{-\infty}^{\infty}(B^{3}_{0}\omega_{0})(x)dx=0,\,\dots.\hskip
252.94499pt$

Similarly,

$\int_{-\infty}^{\infty}(B^{n}_{1}\omega_{0})(x)dx=0,\,\int_{-\infty}^{\infty}(B^{n}_{2}\omega_{0})(x)dx=0,\,n=1,2,3\dots.$

Then,

$\int_{-\infty}^{\infty}(exp(tB_{0})\omega_{0})(x)dx\,=\,\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\int_{-\infty}^{\infty}(B_{0}^{n}\omega_{0})(x)dx\,=\,\int_{-\infty}^{\infty}\omega_{0}(x)dx\,=\,1.$

Similarly,

$\int_{-\infty}^{\infty}(exp(tB_{1})\omega_{0})(x)dx\,=\,1,\,\text{ and
}\,\int_{-\infty}^{\infty}(exp(tB_{2})\omega_{0})(x)dx\,=\,1.$

From (3.7)

$\int_{-\infty}^{\infty}(T_{\alpha}(t)\omega_{0})(x)dx\,=\,1,\,t\geq 0.$

Thus, for $t\geq 0$,

$\int_{-\infty}^{\infty}T(t)\omega_{0}dx\,=\,\int_{-\infty}^{\infty}T_{\alpha}(t)\,exp(tB_{0})\,exp(tB_{1})\,exp(tB_{2})dx\,=\,1$

. ∎

##  4\. Numerical simulations of the models

In this section we provide numerical simulations of the solutions $\rho(x,t)$
of the SE model (2.3) and the solutions $\omega(x,t)$ of the NLAD model (3.4).
The SE solutions and the NLAD solutions are similar, depending on the values
of parameters and the values of $t$. There are two significant differences in
the model outputs, which we will demonstrate in the simulations.

One difference is in the spacing of the local minima and maxima in the fringe
patterns in the first phase. This spacing is very regular in the solutions of
the NLAD model, but irregular in the solutions of the SE model. Another
difference is in the second phase, where the solutions of the SE model
demonstrate the space-dilation property, but the solutions of the NLAD model
elevate above the $x$-axis and demonstrate a dispersion property of the fringe
pattern peaks.

In Figures (4), (5), (6), (7), simulations are given for the first phase of
the SE and NLAD models at four different time values. The spacing of peaks for
the NLAD model is very regular, whereas the spacing of peaks for the SE model
is very irregular. In Figure (7) the spacing of the local minima of
$\omega(x,1/\pi)$ occurs regularly at $x\approx.5,1.5,2.5,3.5,\dots$. The
spacing of local minima of $\rho(x,1/\pi)$ occurs irregularly at
$x\approx.5,1.5,2.5,3.5,\dots,7.5,8.5,9.25,10.0,10.75,11.5,12.5,\dots$,
$17.5,18.5,19.25,20.0$, $20.75,21.5,\dots$.

Figure 4. The interference diffraction pattern of the solution $\rho(x,t)$ of
SE (blue) in equation (2.3) and the solution $\omega(x,t)$ of NLAD (red) in
equation (3.4). $\alpha=1/\pi^{3}$, $s=1$, $b=.1$, $\beta_{0}=1/(8\,b^{2})$,
$\beta_{1}=\pi/(2\,b\,15^{2})$, $\beta_{2}=\pi/(2\,b\,25^{2})$, $t=.1/\pi$.
The bottom graphs are $log(\rho(x,t))$ (blue) and $log(\omega(x,t))$ (red).
Figure 5. The interference diffraction pattern of the solution $\rho(x,t)$ of
SE (blue) in equation (2.3) and the solution $\omega(x,t)$ of NLAD (red) in
equation (3.4). $\alpha=1/\pi^{3}$, $s=1$, $b=.1$, $\beta_{0}=1/(8\,b^{2})$,
$\beta_{1}=\pi/(2\,b\,15^{2})$, $\beta_{2}=\pi/(2\,b\,25^{2})$, $t=.25/\pi$.
The bottom graphs are $log(\rho(x,t))$ (blue) and $log(\omega(x,t))$ (red).
Figure 6. The interference diffraction pattern of the solution $\rho(x,t)$ of
SE (blue) in equation (2.3) and the solution $\omega(x,t)$ of NLAD (red) in
equation (3.4). $\alpha=1/\pi^{3}$, $s=1$, $b=.1$, $\beta_{0}=1/(8\,b^{2})$,
$\beta_{1}=\pi/(2\,b\,15^{2})$, $\beta_{2}=\pi/(2\,b\,25^{2})$, $t=.5/\pi$.
The bottom graphs are $log(\rho(x,t))$ (blue) and $log(\omega(x,t))$ (red).
Figure 7. The interference diffraction pattern of the solution $\rho(x,t)$ of
SE (blue) in equation (2.3) and the solution $\omega(x,t)$ of NLAD (red) in
equation (3.4). $\alpha=1/\pi^{3}$, $s=1$, $b=.1$, $\beta_{0}=1/(8\,b^{2})$,
$\beta_{1}=\pi/(2\,b\,15^{2})$, $\beta_{2}=\pi/(2\,b\,25^{2})$, $t=1.0/\pi$.
The bottom graphs are $log(\rho(x,t))$ (blue) and $log(\omega(x,t))$ (red).

In Figures (8), (9), (10), (11), simulations are given for the second phase of
the SE and NLAD models at four different time values
$t_{2}=2/\pi,\,t_{3}=3/\pi,\,t_{4}=4/\pi,\,t_{6}=6/\pi$. In these simulations
for the second phase, the solutions $\omega(x,t)$ of the nonlocal advection-
diffusion equation (3.4) are space-time dilated according to the formulas
$\hat{\omega}(x,t_{2})=\frac{1}{2}\,\omega(\frac{x}{2},t_{2}),\,\,\,\,\hat{\omega}(x,t_{3})=\frac{1}{3}\,\omega(\frac{x}{3},t_{3}),\,\,\,\,\hat{\omega}(x,t_{4})=\frac{1}{4}\,\omega(\frac{x}{4},t_{4}),\,\,\,\,\hat{\omega}(x,t_{6})=\frac{1}{6}\,\omega(\frac{x}{6},t_{6}).$
This dilation of $\omega(x,t)$ to $\hat{\omega}(x,t)$ in the second phase
corresponds to an extension of the fringe pattern established in the first
phase, with increasing distance of the detection plate. The simulations
$\rho(x,t)$ of the SE model exhibit the space-time dilation property, with
local minima remaining on the $x$-axis. The simulations $\omega(x,t)$ of the
NLAD model exhibit an elevation of local minima above the $x$-axis and a
dissipation of the fringe pattern peaks.

Figure 8. The interference diffraction pattern of the solution $\rho(x,t_{2})$
of SE (blue) in equation (2.3) and the dilated solution
$\hat{\omega}(x,t_{2})$ (red) in the NLAD equation (3.4). $\alpha=1/\pi^{3}$,
$s=1$, $b=.1$, $\beta_{0}=1/(8\,b^{2})$, $\beta_{1}=\pi/(2\,b\,15^{2})$,
$\beta_{2}=\pi/(2\,b\,25^{2})$, $t_{2}=2.0/\pi$. Figure 9. The interference
diffraction pattern of the solution $\rho(x,t_{3})$ of SE (blue) in equation
(2.3) and the dilated solution $\hat{\omega}(x,t_{3})$ (red) in the NLAD
equation (3.4). $\alpha=1/\pi^{3}$, $s=1$, $b=.1$, $\beta_{0}=1/(8\,b^{2})$,
$\beta_{1}=\pi/(2\,b\,15^{2})$, $\beta_{2}=\pi/(2\,b\,25^{2})$,
$t_{3}=3.0/\pi$. Figure 10. The interference diffraction pattern of the
solution $\rho(x,t_{4})$ of SE (blue) in equation (2.3) and the dilated
solution $\hat{\omega}(x,t_{4})$ (red) in the NLAD equation (3.4).
$\alpha=1/\pi^{3}$, $s=1$, $b=.1$, $\beta_{0}=1/(8\,b^{2})$,
$\beta_{1}=\pi/(2\,b\,15^{2})$, $\beta_{2}=\pi/(2\,b\,25^{2})$,
$t_{4}=4.0/\pi$. Figure 11. The interference diffraction pattern of the
solution $\rho(x,t_{6})$ of SE (blue) in equation (2.3) and the dilated
solution $\hat{\omega}(x,t_{6})$ (red) in the NLAD equation (3.4).
$\alpha=1/\pi^{3}$, $s=1$, $b=.1$, $\beta_{0}=1/(8\,b^{2})$,
$\beta_{1}=\pi/(2\,b\,15^{2})$, $\beta_{2}=\pi/(2\,b\,25^{2})$,
$t_{6}=6.0/\pi$.

##  5\. Summary

We have developed a nonlocal advection-diffusion model NLAD for the two-slit
experiment of quantum mechanics, in which quantum particles are projected
forward, one at a time, through two stits, and detected downstream on a
detection surface. Our work here is an extension of the models in [10] and
[11], which allowed higher order fringe pattern levels observed in experiments
(Figure 2). We compare the NLAD equation model to the Schrödinger equation
model SE with initial data consisting of two rectangular steps.

In both formulations NLAD and SE, there is a two-phase development of the
multi-level fringe pattern. In the first phase the initial data transitions to
an established multi-level fringe pattern with local minima located
approximately on the $x$-axis. This transition is very simple for the NLAD
model, but very complex for the SE model.

In the second phase the multi-level fringe pattern established in the first
phase evolves in a space-time dilation in both models. In the second phase of
the SE model, the pattern established in the first phase is preserved almost
perfectly in the space-time dilation with constant speed. In the second phase
of the NLAD model, the pattern established in the first phase dissipates, with
local minima rising above the $x$-axis and with magnitude of the fringe
pattern oscillations decreasing.

The SE formulation and the NLAD formulation of the 2-slit experiment have very
different interpretations. The interpretation of the SE model is that an
individual quantum particle exists as a wave moving forward in space. The
interpretation of the NLAD model is that an individual quantum particle exists
as a component of an ensemble, with its forward movement influenced by
nonlocal reaction to its environment within a sensing radius of its spatial
position.

Scientific understanding of the two-slit experiment, which is one of the most
fundamental experiments in science, requires mathematical formulations, which
provide descriptive connection to experiments, and interpretation of physical
processes. The two formulations SE and NLAD can be compared for these
purposes.

## References

  * [1] Y. Aharonov, E. Cohen, F. Columbo, and J. Tollaksen, _Finally making sense of the double-slit experiment_ , Proc. Nat. Acad. Sci., 114.25 (2017), 6480–6485.
  * [2] J. Goldstein, _Semigroups of Linear Operators and Applications_ , Oxford University Press, Oxford, 1985.
  * [3] A. Einstein, B. Podolsky, and N. Rosen, _Can quantum-mechanical description of physical reality be considered complete?_ , Phys. Rev., 41 (1935), 777–780.
  * [4] A. Friedman, _Partial Differential Equations_ , Holt, Rinehart and Winston, New York and London, 1969.
  * [5] T. Kato, _Perturbation Theory for Linear Operators_ , Springer-Verlag, New York, 1966.
  * [6] John W. Neuberger, _Analyticity and quasi-analyticity for one-parameter semigroups_ , Proc. Amer. Math. Soc., 25 (1970), 488–494.
  * [7] John W. Neuberger, _Quasi-analyticity and semigroups_ , Bull. Amer. Math. Soc., 78(6) (1972), 909-922.
  * [8] A. Tonomura, _The Quantum World Unveiled by Electron Waves_ , World Scientific, Singapore, 1998.
  * [9] G. F. Webb, _Event based interpretation of Schrödinger’s equation for the two-slit experiment_ , Int. J. Theor. Phys., 50(11) (2011), 3571–3601.
  * [10] G. F. Webb, _The ensemble interpretation of quantum mechanics and the two-slit experiment_ , Comp. Meth. Appl. Sci., 47 (2018), 433–452.
  * [11] G. F. Webb, _The Schrödinger equation and the two-slit experiment of quantum mechanics_ , to appear.
  * [12] K. Yosida, _Functional Analysis_ , Springer-Verlag, New York, 1968.